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JEE Main · Mathematics · 2026

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1
[JEE Main 2026, 8 Apr Shift 2, Q5] Let α=3+4+8+9+13+14+…\alpha=3+4+8+9+13+14+\dots upto 40 terms. If (tan⁡β)α1020(\tan\beta)^{\frac{\alpha}{1020}} is a root of the equation x2+x−2=0x^2+x-2=0, β∈(0,π2)\beta\in\left(0,\frac\pi2\right), then sin⁡2β+3cos⁡2β\sin^2\beta+3\cos^2\beta is equal to:
Not attempted
A
22
B
74\frac74
C
52\frac52
D
32\frac32
2
[JEE Main 2026, 8 Apr Shift 2, Q12] Let α=3sin⁡−1(611)\alpha=3\sin^{-1}\left(\frac6{11}\right) and β=3cos⁡−1(49)\beta=3\cos^{-1}\left(\frac49\right), where inverse trigonometric functions take only the principal values. Given below are two statements: Statement I: cos⁡(α+β)>0\cos(\alpha+\beta)>0. Statement II: cos⁡(α)<0\cos(\alpha)<0. In the light of the above statements, choose the correct answer from the options given below:
Not attempted
A
Both Statement I and Statement II are true
B
Both Statement I and Statement II are false
C
Statement I is true but Statement II is false
D
Statement I is false but Statement II is true
3
[JEE Main 2026, 8 Apr Shift 2, Q6] A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are 25,15\frac25,\frac15 and 25\frac25. The probabilities that the candidate reaches late at the examination centre are 15,13\frac15,\frac13 and 14\frac14 if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is:
Not attempted
A
1137\frac{11}{37}
B
1237\frac{12}{37}
C
1337\frac{13}{37}
D
1437\frac{14}{37}
4
[JEE Main 2026, 8 Apr Shift 2, Q7] A set of four observations has mean 1 and variance 13. Another set of six observations has mean 2 and variance 1. Then, the variance of all these 10 observations is equal to:
Not attempted
A
5.965.96
B
6.146.14
C
6.046.04
D
6.246.24
5
[JEE Main 2026, 8 Apr Shift 2, Q14] Let a⃗=4i^−j^+3k^\vec a=4\hat i-\hat j+3\hat k, b⃗=10i^+2j^−k^\vec b=10\hat i+2\hat j-\hat k and a vector c⃗\vec c be such that 2(a⃗×b⃗)+3(b⃗×c⃗)=0⃗2(\vec a\times\vec b)+3(\vec b\times\vec c)=\vec0. If a⃗⋅c⃗=15\vec a\cdot\vec c=15, then c⃗⋅(i^+j^−3k^)\vec c\cdot(\hat i+\hat j-3\hat k) is equal to:
Not attempted
A
−6-6
B
−5-5
C
−4-4
D
−3-3
6
[JEE Main 2026, 8 Apr Shift 2, Q15] Let the foot of perpendicular from the point (λ,2,3)(\lambda,2,3) on the line x−41=y−92=z−51\frac{x-4}{1}=\frac{y-9}{2}=\frac{z-5}{1} be the point (1,μ,2)(1,\mu,2). Then the distance between the lines x−12=y−23=z+46\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6} and x−λ2=y−μ3=z+56\frac{x-\lambda}{2}=\frac{y-\mu}{3}=\frac{z+5}{6} is equal to:
Not attempted
A
127\frac{12}{7}
B
1457\frac{\sqrt{145}}{7}
C
1467\frac{\sqrt{146}}{7}
D
1437\frac{\sqrt{143}}{7}
7
[JEE Main 2026, 8 Apr Shift 2, Q23] Let a line L1L_1 pass through the origin and be perpendicular to the lines L2:r⃗=(3+t)i^+(2t−1)j^+(2t+4)k^L_2:\vec r=(3+t)\hat i+(2t-1)\hat j+(2t+4)\hat k and L3:r⃗=(3+2s)i^+(3+2s)j^+(2+s)k^L_3:\vec r=(3+2s)\hat i+(3+2s)\hat j+(2+s)\hat k, t,s∈Rt,s\in\mathbb{R}. If (a,b,c)(a,b,c), a∈Za\in\mathbb{Z}, is the point on L3L_3 at a distance of 17\sqrt{17} from the point of intersection of L1L_1 and L2L_2, then (a+b+c)2(a+b+c)^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 4
8
[JEE Main 2026, 8 Apr Shift 2, Q10] If a straight line drawn through the point of intersection of the lines 4x+3y−1=04x+3y-1=0 and 3x+4y−1=03x+4y-1=0, meets the co-ordinate axes at the points P and Q, then the locus of the mid point of PQ is:
Not attempted
A
x+y−7=0x+y-7=0
B
x+y−14xy=0x+y-14xy=0
C
2x+y+14xy=02x+y+14xy=0
D
x+2y−14xy=0x+2y-14xy=0
9
[JEE Main 2026, 8 Apr Shift 2, Q11] Let O be the vertex of the parabola y2=4xy^2=4x and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is:
Not attempted
A
11
B
22
C
44
D
88
10
[JEE Main 2026, 8 Apr Shift 2, Q20] Let x2f(a2+7a+3)+y2f(3a+15)=1\frac{x^2}{f(a^2+7a+3)}+\frac{y^2}{f(3a+15)}=1 represent an ellipse with major axis along y-axis, where f is a strictly decreasing positive function on R\mathbb{R}. If the set of all possible values of a is R−[α,β]\mathbb{R}-[\alpha,\beta], then α2+β2\alpha^2+\beta^2 is equal to:
Not attempted
A
2828
B
4040
C
6161
D
2424
11
[JEE Main 2026, 8 Apr Shift 2, Q24] Consider the circle C:x2+y2−6x−8y−11=0C:x^2+y^2-6x-8y-11=0. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle x2+y2−αx−βy−γ=0x^2+y^2-\alpha x-\beta y-\gamma=0, then α+β+2γ\alpha+\beta+2\gamma is equal to ______.
Not attempted
Your answer: —
Correct answer: 18
12
[JEE Main 2026, 8 Apr Shift 2, Q17] Let y=y(x)y=y(x) be the solution of the differential equation x1−x2dy+(y1−x2−xcos⁡−1x)dx=0x\sqrt{1-x^2}dy+(y\sqrt{1-x^2}-x\cos^{-1}x)dx=0, x∈(0,1)x\in(0,1), lim⁡x→1−y(x)=1\lim_{x\to1^-}y(x)=1. Then y(12)y\left(\frac12\right) equals:
Not attempted
A
3−π33-\frac{\pi}{\sqrt3}
B
4−3π4-\sqrt3\pi
C
4−2π34-\frac{2\pi}{\sqrt3}
D
3−π233-\frac{\pi}{2\sqrt3}
13
[JEE Main 2026, 8 Apr Shift 2, Q16] The value of the integral ∫02xx2+x+1x+1 x4+x2+1dx\int_0^2\frac{\sqrt x\sqrt{x^2+x+1}}{\sqrt{x+1}\,\sqrt{x^4+x^2+1}}dx is equal to:
Not attempted
A
13log⁡e(3−22)\frac13\log_e(3-2\sqrt2)
B
23log⁡e(4+2)\frac23\log_e(4+\sqrt2)
C
23log⁡e(3+22)\frac23\log_e(3+2\sqrt2)
D
13log⁡e(1+62)\frac13\log_e(1+6\sqrt2)
14
[JEE Main 2026, 8 Apr Shift 2, Q18] Let f:(1,∞)→Rf:(1,\infty)\to\mathbb{R} be a function defined as f(x)=x−1x+1f(x)=\frac{x-1}{x+1}. Let fi+1(x)=f(fi(x))f^{i+1}(x)=f(f^i(x)), i=1,2,…,25i=1,2,\dots,25, where f1(x)=f(x)f^1(x)=f(x). If g(x)+f26(x)=0g(x)+f^{26}(x)=0, x∈(1,∞)x\in(1,\infty), then the area of the region bounded by the curves y=g(x)y=g(x), 2y=2x−32y=2x-3, y=0y=0 and x=4x=4 is:
Not attempted
A
18+log⁡e2\frac18+\log_e2
B
14+log⁡e2\frac14+\log_e2
C
56+3log⁡e2\frac56+3\log_e2
D
56+log⁡e2\frac56+\log_e2
15
[JEE Main 2026, 8 Apr Shift 2, Q22] If ∫π/6π/4(cot⁡(x−π3)cot⁡(x+π3)+1)dx=αlog⁡e(3−1)\int_{\pi/6}^{\pi/4}\left(\cot\left(x-\frac\pi3\right)\cot\left(x+\frac\pi3\right)+1\right)dx=\alpha\log_e(\sqrt3-1), then 9α29\alpha^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 12
16
[JEE Main 2026, 8 Apr Shift 2, Q13] For the function f(x)=esin⁡∣x∣−∣x∣f(x)=e^{\sin|x|}-|x|, x∈Rx\in\mathbb{R}, consider the following statements: Statement I: f is differentiable for all x∈Rx\in\mathbb{R}. Statement II: f is increasing in (−π,−π2)\left(-\pi,-\frac\pi2\right). In the light of the above statements, choose the correct answer from the options given below:
Not attempted
A
Both Statement I and Statement II are true
B
Both Statement I and Statement II are false
C
Statement I is true but Statement II is false
D
Statement I is false but Statement II is true
17
[JEE Main 2026, 8 Apr Shift 2, Q19] Let f(x)=13f(x)=\frac13 for x≤π2x\le\frac\pi2 and f(x)=b(1−sin⁡x)(π−2x)2f(x)=\frac{b(1-\sin x)}{(\pi-2x)^2} for x>π2x>\frac\pi2. If f is continuous at x=π2x=\frac\pi2, then the value of ∫03b−6∣x2+2x−3∣dx\int_0^{3b-6}|x^2+2x-3|dx is:
Not attempted
A
55
B
22
C
33
D
44
18
[JEE Main 2026, 8 Apr Shift 2, Q8] If 26(233(122)+255(124)+277(126)+⋯+21313(1212))=313−α26\left(\frac{2^3}{3}\binom{12}{2}+\frac{2^5}{5}\binom{12}{4}+\frac{2^7}{7}\binom{12}{6}+\dots+\frac{2^{13}}{13}\binom{12}{12}\right)=3^{13}-\alpha, then α\alpha is equal to:
Not attempted
A
4545
B
4848
C
5151
D
5454
19
[JEE Main 2026, 8 Apr Shift 2, Q9] A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is:
Not attempted
A
1818
B
3636
C
3939
D
7272
20
[JEE Main 2026, 8 Apr Shift 2, Q3] If the system of linear equations: x+y+z=6x+y+z=6 x+2y+5z=10x+2y+5z=10 2x+3y+λz=μ2x+3y+\lambda z=\mu has infinitely many solutions, then the value of λ+μ\lambda+\mu equals:
Not attempted
A
1212
B
1616
C
2222
D
2828
21
[JEE Main 2026, 8 Apr Shift 2, Q4] Let A=[α12230045]A=\begin{bmatrix}\alpha&1&2\\2&3&0\\0&4&5\end{bmatrix} and B=[1000−5α004α−2α]+adj(A)B=\begin{bmatrix}1&0&0\\0&-5\alpha&0\\0&4\alpha&-2\alpha\end{bmatrix}+\mathrm{adj}(A). If det⁡(B)=66\det(B)=66, then det⁡(adj(A))\det(\mathrm{adj}(A)) equals:
Not attempted
A
289289
B
361361
C
441441
D
529529
22
[JEE Main 2026, 8 Apr Shift 2, Q2] The number of values of z∈Cz\in\mathbb{C}, satisfying the equations ∣z−(4+8i)∣=10|z-(4+8i)|=\sqrt{10} and ∣z−(3+5i)∣+∣z−(5+11i)∣=45|z-(3+5i)|+|z-(5+11i)|=4\sqrt5, is:
Not attempted
A
00
B
22
C
11
D
44
23
[JEE Main 2026, 8 Apr Shift 2, Q1] Consider the relation R on the set {−2,−1,0,1,2}\{-2,-1,0,1,2\} defined by (a,b)∈R(a,b)\in R if and only if 1+ab>01+ab>0. Then, among the statements: I. The number of elements in R is 17; II. R is an equivalence relation,
Not attempted
A
Only I is true
B
Only II is true
C
Both I and II are true
D
Neither I nor II is true
24
[JEE Main 2026, 8 Apr Shift 2, Q21] The sum of squares of all the real solutions of the equation log⁡(x+1)(2x2+5x+3)=4−log⁡(2x+3)(x2+2x+1)\log_{(x+1)}(2x^2+5x+3)=4-\log_{(2x+3)}(x^2+2x+1) is equal to ______.
Not attempted
Your answer: —
Correct answer: 2
25
[JEE Main 2026, 8 Apr Shift 2, Q25] Let f be a polynomial function such that log⁡2(f(x))=(log⁡2(2+23+29+…∞))⋅log⁡3(1+f(x)f(1/x))\log_2(f(x))=\left(\log_2\left(2+\frac23+\frac29+\dots\infty\right)\right)\cdot\log_3\left(1+\frac{f(x)}{f(1/x)}\right), x>0x>0 and f(6)=37f(6)=37. Then ∑n=110f(n)\sum_{n=1}^{10}f(n) is equal to ______.
Not attempted
Your answer: —
Correct answer: 395
26
[JEE Main 2026, 6 Apr Shift 2, Q13] If sin⁡(tan⁡−1(x2))=cot⁡(sin⁡−11−x2)\sin(\tan^{-1}(x\sqrt2))=\cot\left(\sin^{-1}\sqrt{1-x^2}\right), x∈(0,1)x\in(0,1), then the value of x is:
Not attempted
A
12\frac12
B
13\frac13
C
23\frac23
D
58\frac58
27
[JEE Main 2026, 6 Apr Shift 2, Q8] Let the mean and the variance of seven observations 2,4,α,8,β,12,142,4,\alpha,8,\beta,12,14, α<β\alpha<\beta, be 8 and 16 respectively. Then the quadratic equation whose roots are 3α+23\alpha+2 and 2β+12\beta+1 is:
Not attempted
A
x2−35x+306=0x^2-35x+306=0
B
x2−41x+420=0x^2-41x+420=0
C
x2−45x+506=0x^2-45x+506=0
D
x2−37x+342=0x^2-37x+342=0
28
[JEE Main 2026, 6 Apr Shift 2, Q9] A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is:
Not attempted
A
63925\frac{63}{925}
B
17231\frac{17}{231}
C
16231\frac{16}{231}
D
64925\frac{64}{925}
29
[JEE Main 2026, 6 Apr Shift 2, Q15] Let a⃗=2i^+3j^+3k^\vec a=2\hat i+3\hat j+3\hat k and b⃗=6i^+3j^+3k^\vec b=6\hat i+3\hat j+3\hat k. Then the square of the area of the triangle with adjacent sides determined by the vectors (2a⃗+3b⃗)(2\vec a+3\vec b) and (a⃗−b⃗)(\vec a-\vec b) is:
Not attempted
A
450450
B
900900
C
18001800
D
24002400
30
[JEE Main 2026, 6 Apr Shift 2, Q14] The shortest distance between the lines x−41=y−32=z−2−3\frac{x-4}{1}=\frac{y-3}{2}=\frac{z-2}{-3} and x+22=y−64=z−5−5\frac{x+2}{2}=\frac{y-6}{4}=\frac{z-5}{-5} is:
Not attempted
A
566\frac{5\sqrt6}{6}
B
252\sqrt5
C
353\sqrt5
D
454\sqrt5
31
[JEE Main 2026, 6 Apr Shift 2, Q24] Let the image of the point P(0,−5,0)P(0,-5,0) in the line x−12=y1=z+1−2\frac{x-1}{2}=\frac y1=\frac{z+1}{-2} be the point R and the image of the point Q(0,−12,0)Q\left(0,-\frac12,0\right) in the line x−1−1=y+94=z+11\frac{x-1}{-1}=\frac{y+9}{4}=\frac{z+1}{1} be the point S. Then the square of the area of the parallelogram PQRS is ______.
Not attempted
Your answer: —
Correct answer: 162
32
[JEE Main 2026, 6 Apr Shift 2, Q10] Let C be a circle having centre in the first quadrant and touching the x-axis at a distance of 3 units from the origin. If the circle C has an intercept of length 636\sqrt3 on y-axis, then the length of the chord of the circle C on the line x−y=3x-y=3 is:
Not attempted
A
88
B
66
C
626\sqrt2
D
828\sqrt2
33
[JEE Main 2026, 6 Apr Shift 2, Q11] The eccentricity of an ellipse E with centre at the origin O is 32\frac{\sqrt3}{2} and its directrices are x=±463x=\pm\frac{4\sqrt6}{3}. Let H:x2a2−y2b2=1H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 be a hyperbola whose eccentricity is equal to the length of semi-major axis of E, and whose length of latus rectum is equal to the length of minor axis of E. Then the distance between the foci of H is:
Not attempted
A
427\frac{4\sqrt2}{\sqrt7}
B
427\frac{4\sqrt2}{7}
C
47\frac4{\sqrt7}
D
87\frac87
34
[JEE Main 2026, 6 Apr Shift 2, Q12] Let x=9x=9 be a directrix of an ellipse E, whose centre is at the origin and eccentricity is 13\frac13. Let P(α,0)P(\alpha,0), α>0\alpha>0, be a focus of E and AB be a chord passing through P. Then the locus of the mid point of AB is:
Not attempted
A
9y2=8x(1−x)9y^2=8x(1-x)
B
3y2=4x(1−x)3y^2=4x(1-x)
C
9y2=8x(x−1)9y^2=8x(x-1)
D
3y2=4x(x−1)3y^2=4x(x-1)
35
[JEE Main 2026, 6 Apr Shift 2, Q23] Let the line x−y=4x-y=4 intersect the circle C:(x−4)2+(y+3)2=9C:(x-4)^2+(y+3)^2=9 at the points Q and R. If P(α,β)P(\alpha,\beta) is a point on C such that PQ=PRPQ=PR, then (6α+8β)2(6\alpha+8\beta)^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 18
36
[JEE Main 2026, 6 Apr Shift 2, Q18] Let f:R→Rf:\mathbb{R}\to\mathbb{R} be such that f(xy)=f(x)f(y)f(xy)=f(x)f(y), for all x,y∈Rx,y\in\mathbb{R} and f(0)≠0f(0)\ne0. Let g:[1,∞)→Rg:[1,\infty)\to\mathbb{R} be a differentiable function such that x2g(x)=∫1x(t2f(t)−tg(t))dtx^2g(x)=\int_1^x(t^2f(t)-tg(t))dt. Then g(2)g(2) is equal to:
Not attempted
A
138\frac{13}{8}
B
1116\frac{11}{16}
C
1532\frac{15}{32}
D
1764\frac{17}{64}
37
[JEE Main 2026, 6 Apr Shift 2, Q17] Let A=[13−121α01−1]A=\begin{bmatrix}1&3&-1\\2&1&\alpha\\0&1&-1\end{bmatrix} be a singular matrix. Let f(x)=∫0x(t2+2t+3)dtf(x)=\int_0^x(t^2+2t+3)dt, x∈[1,α]x\in[1,\alpha]. If M and m are respectively the maximum and the minimum values of f in [1,α][1,\alpha], then 3(M−m)3(M-m) is equal to:
Not attempted
A
6464
B
6868
C
7272
D
7676
38
[JEE Main 2026, 6 Apr Shift 2, Q19] The area of the region {(x,y):x2−8x≤y≤−x}\{(x,y):x^2-8x\le y\le-x\} is:
Not attempted
A
3436\frac{343}{6}
B
6376\frac{637}{6}
C
4376\frac{437}{6}
D
5236\frac{523}{6}
39
[JEE Main 2026, 6 Apr Shift 2, Q20] The value of the integral ∫−11(x3+∣x∣+1x2+2∣x∣+1)dx\int_{-1}^{1}\left(\frac{x^3+|x|+1}{x^2+2|x|+1}\right)dx is equal to:
Not attempted
A
3log⁡e23\log_e2
B
2log⁡e22\log_e2
C
5log⁡e35\log_e3
D
3log⁡e33\log_e3
40
[JEE Main 2026, 6 Apr Shift 2, Q25] Let f(x)=x3+8f(x)=x^3+8 for x<0x<0, f(x)=x2−4f(x)=x^2-4 for x≥0x\ge0, and g(x)=(x−8)1/3g(x)=(x-8)^{1/3} for x<0x<0, g(x)=(x+4)1/2g(x)=(x+4)^{1/2} for x≥0x\ge0. Then the number of points, where the function g∘fg\circ f is discontinuous, is ______.
Not attempted
Your answer: —
Correct answer: 3
41
[JEE Main 2026, 6 Apr Shift 2, Q2] Consider the quadratic equation (n2−2n+2)x2−3x+(n2−2n+2)2=0(n^2-2n+2)x^2-3x+(n^2-2n+2)^2=0, n∈Rn\in\mathbb{R}. Let α\alpha be the minimum value of the product of its roots and β\beta be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is α\alpha and the common ratio is αβ\frac\alpha\beta, is:
Not attempted
A
6137\frac{61}{37}
B
12181\frac{121}{81}
C
364243\frac{364}{243}
D
1093729\frac{1093}{729}
42
[JEE Main 2026, 6 Apr Shift 2, Q6] The sum 1+12(12+22)+13(12+22+32)+…1+\frac12(1^2+2^2)+\frac13(1^2+2^2+3^2)+\dots upto 10 terms is equal to:
Not attempted
A
130130
B
155155
C
3152\frac{315}{2}
D
3252\frac{325}{2}
43
[JEE Main 2026, 6 Apr Shift 2, Q22] If (1−x3)10=∑r=010arxr(1−x)30−2r(1-x^3)^{10}=\sum_{r=0}^{10}a_rx^r(1-x)^{30-2r}, then 9a9a10\frac{9a_9}{a_{10}} is equal to ______.
Not attempted
Your answer: —
Correct answer: 30
44
[JEE Main 2026, 6 Apr Shift 2, Q7] A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the 10th10^{th} floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to:
Not attempted
A
21842184
B
30643064
C
70567056
D
1134011340
45
[JEE Main 2026, 6 Apr Shift 2, Q4] The sum of all possible values of θ∈[0,2π]\theta\in[0,2\pi], for which the system of equations: xcos⁡3θ−8y−12z=0x\cos3\theta-8y-12z=0 xcos⁡2θ+3y+3z=0x\cos2\theta+3y+3z=0 x+y+3z=0x+y+3z=0 has a non-trivial solution, is equal to:
Not attempted
A
π\pi
B
2π2\pi
C
3π3\pi
D
4π4\pi
46
[JEE Main 2026, 6 Apr Shift 2, Q5] Let A=[100310931]A=\begin{bmatrix}1&0&0\\3&1&0\\9&3&1\end{bmatrix} and B=[bij]B=[b_{ij}], 1≤i,j≤31\le i,j\le3. If B=A99−IB=A^{99}-I, then the value of b31−b21b32\frac{b_{31}-b_{21}}{b_{32}} is:
Not attempted
A
9999
B
199199
C
149149
D
159159
47
[JEE Main 2026, 6 Apr Shift 2, Q3] Let S={z∈C:z2+6iz−3=0}S=\{z\in\mathbb{C}:z^2+\sqrt6iz-3=0\}. Then ∑z∈Sz8\sum_{z\in S}z^8 is equal to:
Not attempted
A
162162
B
184184
C
262262
D
324324
48
[JEE Main 2026, 6 Apr Shift 2, Q16] Let lim⁡x→2(tan⁡(x−2))(rx2+(p−2)x−2p)(x−2)2=5\lim_{x\to2}\frac{(\tan(x-2))(rx^2+(p-2)x-2p)}{(x-2)^2}=5 for some r,p∈Rr,p\in\mathbb{R}. If the set of all possible values of q, such that the roots of the equation rx2−px+q=0rx^2-px+q=0 lie in (0,2)(0,2), be the interval (α,β](\alpha,\beta], then 4(α+β)4(\alpha+\beta) equals:
Not attempted
A
1111
B
1313
C
1717
D
2121
49
[JEE Main 2026, 6 Apr Shift 2, Q1] Let f:R→Rf:\mathbb{R}\to\mathbb{R} be defined as f(x)=2x2−3x+23x2+x+3f(x)=\frac{2x^2-3x+2}{3x^2+x+3}. Then f is:
Not attempted
A
both one-one and onto
B
one-one but not onto
C
onto but not one-one
D
neither one-one nor onto
50
[JEE Main 2026, 6 Apr Shift 2, Q21] Let R={(x,y)∈N×N:log⁡e(x+y)≤2}R=\{(x,y)\in\mathbb{N}\times\mathbb{N}:\log_e(x+y)\le2\}. Then the minimum number of elements, required to be added in R to make it a transitive relation, is ______.
Not attempted
Your answer: —
Correct answer: 15
51
[JEE Main 2026, 6 Apr Shift 1, Q13] Let 0<α<10<\alpha<1, β=13α\beta=\frac1{3\alpha} and tan⁡−1(1−α)+tan⁡−1(1−β)=π4\tan^{-1}(1-\alpha)+\tan^{-1}(1-\beta)=\frac\pi4. Then 6(α+β)6(\alpha+\beta) is equal to:
Not attempted
A
66
B
77
C
88
D
99
52
[JEE Main 2026, 6 Apr Shift 1, Q14] Let S={θ∈(−2π,2π):cos⁡θ+1=3sin⁡θ}S=\{\theta\in(-2\pi,2\pi):\cos\theta+1=\sqrt3\sin\theta\}. Then ∑θ∈Sθ\sum_{\theta\in S}\theta is equal to:
Not attempted
A
−2π3-\frac{2\pi}3
B
−4π3-\frac{4\pi}3
C
2π3\frac{2\pi}3
D
4π3\frac{4\pi}3
53
[JEE Main 2026, 6 Apr Shift 1, Q9] A data consists of 20 observations x1,x2,…,x20x_1,x_2,\dots,x_{20}. If ∑i=120(xi+5)2=2500\sum_{i=1}^{20}(x_i+5)^2=2500 and ∑i=120(xi−5)2=100\sum_{i=1}^{20}(x_i-5)^2=100, then the ratio of mean to standard deviation of this data is:
Not attempted
A
2 : 1
B
3 : 1
C
3 : 2
D
4 : 1
54
[JEE Main 2026, 6 Apr Shift 1, Q10] A bag contains (N+1) coins — N fair coins, and one coin with 'Head' on both sides. A coin is selected at random and tossed. If the probability of getting 'Head' is 916\frac9{16}, then N is equal to:
Not attempted
A
55
B
77
C
88
D
99
55
[JEE Main 2026, 6 Apr Shift 1, Q23] If a⃗=i^+j^+k^\vec a=\hat i+\hat j+\hat k, b⃗=j^−k^\vec b=\hat j-\hat k and c⃗\vec c be three vectors such that a⃗×c⃗=b⃗\vec a\times\vec c=\vec b and a⃗⋅c⃗=3\vec a\cdot\vec c=3, then c⃗⋅(a⃗−2b⃗)\vec c\cdot(\vec a-2\vec b) is equal to ______.
Not attempted
Your answer: —
Correct answer: 3
56
[JEE Main 2026, 6 Apr Shift 1, Q15] Let the image of the point P(1,6,a)P(1,6,a) in the line L:x1=y−12=z−a+1bL:\frac x1=\frac{y-1}{2}=\frac{z-a+1}{b}, b>0b>0, be (a3,0,a+c)\left(\frac a3,0,a+c\right). If S(α,β,γ)S(\alpha,\beta,\gamma), α>0\alpha>0, is the point on L such that the distance of S from the foot of perpendicular from the point P on L is 2142\sqrt{14}, then α+β+γ\alpha+\beta+\gamma is equal to:
Not attempted
A
1919
B
2020
C
2121
D
2222
57
[JEE Main 2026, 6 Apr Shift 1, Q16] Let a line L be perpendicular to both the lines L1:x+13=y+35=z+57L_1:\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7} and L2:x−21=y−44=z−67L_2:\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-6}{7}. If θ\theta is the acute angle between the lines L and L3:x−872=y−471=z2L_3:\frac{x-\frac87}{2}=\frac{y-\frac47}{1}=\frac z2, then tan⁡θ\tan\theta is equal to:
Not attempted
A
322\frac32\sqrt2
B
522\frac52\sqrt2
C
532\frac53\sqrt2
D
432\frac43\sqrt2
58
[JEE Main 2026, 6 Apr Shift 1, Q2] Let one root of the quadratic equation in x: (k2−15k+27)x2+9(k−1)x+18=0(k^2-15k+27)x^2+9(k-1)x+18=0 be twice the other. Then the length of the latus rectum of the parabola y2=6kxy^2=6kx is equal to:
Not attempted
A
44
B
66
C
88
D
1212
59
[JEE Main 2026, 6 Apr Shift 1, Q3] Let e1e_1 and e2e_2 be two distinct roots of the equation x2−ax+2=0x^2-ax+2=0. Let the sets {a∈R:e1 and e2 are the eccentricities of hyperbolas}=(α,β)\{a\in\mathbb{R}:e_1\text{ and }e_2\text{ are the eccentricities of hyperbolas}\}=(\alpha,\beta), and {a∈R:e1 and e2 are the eccentricities of an ellipse and a hyperbola, respectively}=(γ,∞)\{a\in\mathbb{R}:e_1\text{ and }e_2\text{ are the eccentricities of an ellipse and a hyperbola, respectively}\}=(\gamma,\infty). Then α2+β2+γ2\alpha^2+\beta^2+\gamma^2 is equal to:
Not attempted
A
1818
B
2222
C
2626
D
3434
60
[JEE Main 2026, 6 Apr Shift 1, Q11] If the eccentricity e of the hyperbola x2a2−y2b2=1\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, passing through (6,43)(6,4\sqrt3), satisfies 15(e2+1)=34e15(e^2+1)=34e, then the length of the latus rectum of the hyperbola x2b2−y22(a2+1)=1\frac{x^2}{b^2}-\frac{y^2}{2(a^2+1)}=1 is:
Not attempted
A
1010
B
2020
C
2525
D
3030
61
[JEE Main 2026, 6 Apr Shift 1, Q12] Let chord PQ of length 3133\sqrt{13} of the parabola y2=12xy^2=12x be such that the ordinates of points P and Q are in the ratio 1 : 2. If the chord PQ subtends an angle α\alpha at the focus of the parabola, then sin⁡α\sin\alpha is equal to:
Not attempted
A
35\frac35
B
45\frac45
C
513\frac5{13}
D
1213\frac{12}{13}
62
[JEE Main 2026, 6 Apr Shift 1, Q22] Let the centre of the circle x2+y2+2gx+2fy+25=0x^2+y^2+2gx+2fy+25=0 be in the first quadrant and lie on the line 2x−y=42x-y=4. Let the area of an equilateral triangle inscribed in the circle be 27327\sqrt3. Then the square of the length of the chord of the circle on the line x=1x=1 is ______.
Not attempted
Your answer: —
Correct answer: 80
63
[JEE Main 2026, 6 Apr Shift 1, Q25] Let y=y(x)y=y(x) be the solution of the differential equation (x2−xx2−1)dy+(y(x−x2−1)−x)dx=0(x^2-x\sqrt{x^2-1})dy+(y(x-\sqrt{x^2-1})-x)dx=0, x≥1x\ge1. If y(1)=1y(1)=1, then the greatest integer less than y(5)y(\sqrt5) is ______.
Not attempted
Your answer: —
Correct answer: 3
64
[JEE Main 2026, 6 Apr Shift 1, Q18] The value of the integral ∫−π/4π/4(32cos⁡4x1+esin⁡x)dx\int_{-\pi/4}^{\pi/4}\left(\frac{32\cos^4x}{1+e^{\sin x}}\right)dx is:
Not attempted
A
4π+24\pi+2
B
3π+83\pi+8
C
3π+43\pi+4
D
4π+34\pi+3
65
[JEE Main 2026, 6 Apr Shift 1, Q19] The area of the region {(x,y):0≤y≤6−x, y2≥4x−3, x≥0}\{(x,y):0\le y\le6-x,\ y^2\ge4x-3,\ x\ge0\} is:
Not attempted
A
88
B
99
C
1212
D
1515
66
[JEE Main 2026, 6 Apr Shift 1, Q20] Let e be the base of natural logarithm and let f:{1,2,3,4}→{1,e,e2,e3}f:\{1,2,3,4\}\to\{1,e,e^2,e^3\} and g:{1,e,e2,e3}→{1,12,13,14}g:\{1,e,e^2,e^3\}\to\left\{1,\frac12,\frac13,\frac14\right\} be two bijective functions such that f is strictly decreasing and g is strictly increasing. If ϕ(x)=[f−1{g−1(12)}]x\phi(x)=\left[f^{-1}\left\{g^{-1}\left(\frac12\right)\right\}\right]^x, then the area of the region R={(x,y):x2≤y≤ϕ(x),0≤x≤1}R=\{(x,y):x^2\le y\le\phi(x),0\le x\le1\} is:
Not attempted
A
3−log⁡e(2)3log⁡e(2)\frac{3-\log_e(2)}{3\log_e(2)}
B
13log⁡e(2)\frac1{3\log_e(2)}
C
3+log⁡e(2)3+\log_e(2)
D
3+log⁡e(2)2+log⁡e(3)\frac{3+\log_e(2)}{2+\log_e(3)}
67
[JEE Main 2026, 6 Apr Shift 1, Q17] The value of lim⁡x→0(x2sin⁡2xx2−sin⁡2x)\lim_{x\to0}\left(\frac{x^2\sin^2x}{x^2-\sin^2x}\right) is:
Not attempted
A
22
B
33
C
44
D
66
68
[JEE Main 2026, 6 Apr Shift 1, Q5] The value of 13−23+33−⋯+1531^3-2^3+3^3-\dots+15^3 is:
Not attempted
A
17061706
B
18561856
C
19821982
D
24032403
69
[JEE Main 2026, 6 Apr Shift 1, Q6] The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:
Not attempted
A
349\frac{34}{9}
B
3413\frac{34}{13}
C
329\frac{32}{9}
D
3213\frac{32}{13}
70
[JEE Main 2026, 6 Apr Shift 1, Q24] For the functions f(θ)=αtan⁡2θ+βcot⁡2θf(\theta)=\alpha\tan^2\theta+\beta\cot^2\theta and g(θ)=αsin⁡2θ+βcos⁡2θg(\theta)=\alpha\sin^2\theta+\beta\cos^2\theta, α>β>0\alpha>\beta>0, let min⁡0<θ<π/2f(θ)=max⁡0<θ<πg(θ)\min_{0<\theta<\pi/2}f(\theta)=\max_{0<\theta<\pi}g(\theta). If the first term of a G.P. is (α2β)\left(\frac{\alpha}{2\beta}\right), its common ratio is (2βα)\left(\frac{2\beta}{\alpha}\right) and the sum of its first 10 terms is mn\frac mn, gcd⁡(m,n)=1\gcd(m,n)=1, then m+nm+n is equal to ______.
Not attempted
Your answer: —
Correct answer: 1279
71
[JEE Main 2026, 6 Apr Shift 1, Q8] If the coefficients of the middle terms in the binomial expansions of (1+αx)26(1+\alpha x)^{26} and (1−αx)28(1-\alpha x)^{28}, α≠0\alpha\ne0, are equal, then the value of α\alpha is:
Not attempted
A
11
B
1413\frac{14}{13}
C
277\frac{27}{7}
D
727\frac7{27}
72
[JEE Main 2026, 6 Apr Shift 1, Q7] The number of 4-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is:
Not attempted
A
26702670
B
28402840
C
29202920
D
36003600
73
[JEE Main 2026, 6 Apr Shift 1, Q21] Let A=[−11−1101001]A=\begin{bmatrix}-1&1&-1\\1&0&1\\0&0&1\end{bmatrix} satisfy A2+α(adj(adj(A)))+β(adj(A)(adj(adj(A))))=[2−22−20−100−1]A^2+\alpha(\mathrm{adj}(\mathrm{adj}(A)))+\beta(\mathrm{adj}(A)(\mathrm{adj}(\mathrm{adj}(A))))=\begin{bmatrix}2&-2&2\\-2&0&-1\\0&0&-1\end{bmatrix} for some α,β∈R\alpha,\beta\in\mathbb{R}. Then (α−β)2(\alpha-\beta)^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 4
74
[JEE Main 2026, 6 Apr Shift 1, Q4] Let the set of all values of k∈Rk\in\mathbb{R} such that the equation z(zˉ+2+i)+k(2+3i)=0z(\bar z+2+i)+k(2+3i)=0, z∈Cz\in\mathbb{C}, has at least one solution, be the interval [α,β][\alpha,\beta]. Then 9(α+β)9(\alpha+\beta) is equal to:
Not attempted
A
−10-10
B
−8-8
C
101310\sqrt{13}
D
8138\sqrt{13}
75
[JEE Main 2026, 6 Apr Shift 1, Q1] Let [⋅][\cdot] denote the greatest integer function. If the domain of the function f(x)=sin⁡−1(x+[x]3)f(x)=\sin^{-1}\left(\frac{x+[x]}{3}\right) is [α,β)[\alpha,\beta), then α2+β2\alpha^2+\beta^2 is equal to:
Not attempted
A
22
B
55
C
1010
D
1313
76
[JEE Main 2026, 5 Apr Shift 2, Q23] If S={θ∈[−π,π]:cos⁡θcos⁡5θ2=cos⁡7θcos⁡7θ2}S=\left\{\theta\in[-\pi,\pi]:\cos\theta\cos\frac{5\theta}{2}=\cos7\theta\cos\frac{7\theta}{2}\right\}, then n(S)n(S) is equal to ______.
Not attempted
Your answer: —
Correct answer: 19
77
[JEE Main 2026, 5 Apr Shift 2, Q8] The probabilities that players A and B of a team are selected for the captaincy for a tournament are 0.6 and 0.4, respectively. If A is selected the captain, the probability that the team wins the tournament is 0.8 and if B is selected the captain, the probability that the team wins the tournament is 0.7. Then the probability, that the team wins the tournament, is:
Not attempted
A
0.740.74
B
0.760.76
C
0.720.72
D
0.780.78
78
[JEE Main 2026, 5 Apr Shift 2, Q10] A variable X takes values 0,0,2,6,12,20,…,n(n−1)0,0,2,6,12,20,\dots,n(n-1) with frequencies nC0,nC1,nC2,nC3,nC4,nC5,…,nCn{}^nC_0,{}^nC_1,{}^nC_2,{}^nC_3,{}^nC_4,{}^nC_5,\dots,{}^nC_n, respectively. If the mean of this data is 60, then its median is:
Not attempted
A
5656
B
4242
C
7272
D
9090
79
[JEE Main 2026, 5 Apr Shift 2, Q16] Let O be the origin, OP⃗=a⃗\vec{OP}=\vec a and OQ⃗=b⃗\vec{OQ}=\vec b. If R is the point on OP⃗\vec{OP} such that OP⃗=5OR⃗\vec{OP}=5\vec{OR}, and M is the point such that OQ⃗=5RM⃗\vec{OQ}=5\vec{RM}, then PM⃗\vec{PM} is equal to:
Not attempted
A
15(a⃗−4b⃗)\frac15(\vec a-4\vec b)
B
15(b⃗−4a⃗)\frac15(\vec b-4\vec a)
C
15(−a⃗+4b⃗)\frac15(-\vec a+4\vec b)
D
15(−b⃗+4a⃗)\frac15(-\vec b+4\vec a)
80
[JEE Main 2026, 5 Apr Shift 2, Q14] Let a triangle PQR be such that P and Q lie on the line x+38=y−42=z+12\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2} and are at a distance of 6 units from R(1,2,3)R(1,2,3). If (α,β,γ)(\alpha,\beta,\gamma) is the centroid of △PQR\triangle PQR, then α+β+γ\alpha+\beta+\gamma is equal to:
Not attempted
A
44
B
55
C
66
D
88
81
[JEE Main 2026, 5 Apr Shift 2, Q15] If the distance of the point (a,2,5)(a,2,5) from the image of the point (1,2,7)(1,2,7) in the line x1=y−11=z−22\frac x1=\frac{y-1}{1}=\frac{z-2}{2} is 4, then the sum of all possible values of a is equal to:
Not attempted
A
1111
B
99
C
66
D
44
82
[JEE Main 2026, 5 Apr Shift 2, Q11] Let the point P be the vertex of the parabola y=x2−6x+12y=x^2-6x+12. If a line passing through the point P intersects the circle x2+y2−2x−4y+3=0x^2+y^2-2x-4y+3=0 at the points R and S, then the maximum value of (PR+PS)2(PR+PS)^2 is:
Not attempted
A
1010
B
2020
C
2525
D
55
83
[JEE Main 2026, 5 Apr Shift 2, Q12] Let the directrix of the parabola P:y2=8xP:y^2=8x, cut x-axis at the point A. Let B(α,β)B(\alpha,\beta), α>1\alpha>1, be a point on P such that the slope of AB is 3/5. If BC is a focal chord of P, then six times the area of △ABC\triangle ABC is:
Not attempted
A
8080
B
160160
C
174174
D
192192
84
[JEE Main 2026, 5 Apr Shift 2, Q13] Let the eccentricity e of a hyperbola satisfy the equation 6e2−11e+3=06e^2-11e+3=0. If the foci of the hyperbola are (3,5)(3,5) and (3,−4)(3,-4), then the length of its latus rectum is:
Not attempted
A
11/311/3
B
17/317/3
C
15/215/2
D
17/217/2
85
[JEE Main 2026, 5 Apr Shift 2, Q22] From the point (−1,−1)(-1,-1), two rays are sent making angles of 45∘45^\circ with the line x+y=0x+y=0. These rays get reflected from the mirror x+2y=1x+2y=1. If the equations of the reflected rays are ax+by=9ax+by=9 and cx+dy=7cx+dy=7, a,b,c,d∈Za,b,c,d\in\mathbb{Z}, then the value of ad+bcad+bc is ______.
Not attempted
Your answer: —
Correct answer: 7
86
[JEE Main 2026, 5 Apr Shift 2, Q19] Let f:[1,∞)→Rf:[1,\infty)\to\mathbb{R} be a differentiable function defined as f(x)=∫1xf(t) dt+(1−x)(log⁡ex−1)+ef(x)=\int_1^xf(t)\,dt+(1-x)(\log_ex-1)+e. Then the value of f(f(1))f(f(1)) is:
Not attempted
A
(1+ee)(1+e^e)
B
(1+e)(1+e)
C
(1+e+ee)(1+e+e^e)
D
1+2e1+2e
87
[JEE Main 2026, 5 Apr Shift 2, Q20] Let f(x)f(x) and g(x)g(x) be twice differentiable functions satisfying f′′(x)=g′′(x)f''(x)=g''(x) for all x∈Rx\in\mathbb{R}, f′(1)=2g′(1)=4f'(1)=2g'(1)=4 and g(2)=3f(2)=9g(2)=3f(2)=9. Then f(25)−g(25)f(25)-g(25) is equal to:
Not attempted
A
2020
B
4040
C
−20-20
D
−40-40
88
[JEE Main 2026, 5 Apr Shift 2, Q25] Let y=y(x)y=y(x) be the solution of the differential equation (tan⁡x)1/2dy=(sec⁡3x−(tan⁡x)3/2y)dx(\tan x)^{1/2}dy=(\sec^3x-(\tan x)^{3/2}y)dx, 0<x<π20<x<\frac\pi2, y(π4)=625y\left(\frac\pi4\right)=\frac{6\sqrt2}{5}. If y(π3)=45αy\left(\frac\pi3\right)=\frac45\alpha, then α4\alpha^4 equals ______.
Not attempted
Your answer: —
Correct answer: 48
89
[JEE Main 2026, 5 Apr Shift 2, Q18] Let (21−a+21+a)(2^{1-a}+2^{1+a}), f(a)f(a), (3a+3−a)(3^a+3^{-a}) be in A.P. and α\alpha be the minimum value of f(a)f(a). Then the value of the integral ∫log⁡e(α−1)log⁡e(α)dx(e2x−e−2x)\int_{\log_e(\alpha-1)}^{\log_e(\alpha)}\frac{dx}{(e^{2x}-e^{-2x})} is:
Not attempted
A
12log⁡e(43)\frac12\log_e\left(\frac43\right)
B
14log⁡e(43)\frac14\log_e\left(\frac43\right)
C
12log⁡e(85)\frac12\log_e\left(\frac85\right)
D
14log⁡e(85)\frac14\log_e\left(\frac85\right)
90
[JEE Main 2026, 5 Apr Shift 2, Q24] Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a function such that f(x)+3f(π2−x)=sin⁡xf(x)+3f\left(\frac\pi2-x\right)=\sin x, x∈Rx\in\mathbb{R}. Let the maximum value of f on R\mathbb{R} be α\alpha. If the area of the region bounded by the curves g(x)=x2g(x)=x^2 and h(x)=βx3h(x)=\beta x^3, β>0\beta>0, is α2\alpha^2, then 30β330\beta^3 is equal to ______.
Not attempted
Your answer: —
Correct answer: 16
91
[JEE Main 2026, 5 Apr Shift 2, Q17] Let f(x)=lim⁡y→0(1−cos⁡(xy))tan⁡(xy)y3f(x)=\lim_{y\to0}\frac{(1-\cos(xy))\tan(xy)}{y^3}. Then the number of solutions of the equation f(x)=sin⁡xf(x)=\sin x, x∈Rx\in\mathbb{R} is:
Not attempted
A
00
B
22
C
33
D
11
92
[JEE Main 2026, 5 Apr Shift 2, Q5] If the sum of the first 10 terms of the series 11+14×4+21+24×4+31+34×4+41+44×4+…\frac{1}{1+1^4\times4}+\frac{2}{1+2^4\times4}+\frac{3}{1+3^4\times4}+\frac{4}{1+4^4\times4}+\dots is mn\frac mn, gcd⁡(m,n)=1\gcd(m,n)=1, then m+nm+n is equal to:
Not attempted
A
256256
B
264264
C
276276
D
284284
93
[JEE Main 2026, 5 Apr Shift 2, Q6] Let A1,A2,A3,…,A39A_1,A_2,A_3,\dots,A_{39} be 39 arithmetic means between the numbers 59 and 159. Then the mean of A25A_{25}, A28A_{28}, A31A_{31} and A36A_{36} is equal to:
Not attempted
A
129129
B
136136
C
131.50131.50
D
134134
94
[JEE Main 2026, 5 Apr Shift 2, Q7] The coefficient of x2x^2 in the expansion of (2x2+1x)10\left(2x^2+\frac1x\right)^{10}, x≠0x\ne0, is:
Not attempted
A
32403240
B
33603360
C
34803480
D
36003600
95
[JEE Main 2026, 5 Apr Shift 2, Q9] A box contains 5 blue, 6 yellow and 4 red balls. The number of ways, of drawing 8 balls containing at least two balls of each colour, is:
Not attempted
A
41004100
B
41404140
C
42304230
D
42904290
96
[JEE Main 2026, 5 Apr Shift 2, Q3] If f:N→Zf:\mathbb{N}\to\mathbb{Z} is defined by f(n)=∣n−1−5−2n23(2k+1)2k+1−3n33k(2k+1)3k(k+2)+1∣f(n)=\begin{vmatrix}n&-1&-5\\-2n^2&3(2k+1)&2k+1\\-3n^3&3k(2k+1)&3k(k+2)+1\end{vmatrix}, k∈Nk\in\mathbb{N}, and ∑n=1kf(n)=98\sum_{n=1}^{k}f(n)=98, then k is equal to:
Not attempted
A
33
B
44
C
55
D
66
97
[JEE Main 2026, 5 Apr Shift 2, Q4] Let M be a 3×33\times3 matrix such that M(100)=(123)M\begin{pmatrix}1\\0\\0\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix}, M(010)=(012)M\begin{pmatrix}0\\1\\0\end{pmatrix}=\begin{pmatrix}0\\1\\2\end{pmatrix} and M(001)=(−111)M\begin{pmatrix}0\\0\\1\end{pmatrix}=\begin{pmatrix}-1\\1\\1\end{pmatrix}. If M(xyz)=(1711)M\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\7\\11\end{pmatrix}, then x+y+zx+y+z equals:
Not attempted
A
44
B
55
C
77
D
1111
98
[JEE Main 2026, 5 Apr Shift 2, Q1] Let α,β\alpha,\beta be the roots of the equation x2−x+p=0x^2-x+p=0 and γ,δ\gamma,\delta be the roots of the equation x2−4x+q=0x^2-4x+q=0; p,q∈Zp,q\in\mathbb{Z}. If α,β,γ,δ\alpha,\beta,\gamma,\delta are in G.P., then ∣p+q∣|p+q| equals:
Not attempted
A
1616
B
3232
C
3434
D
3838
99
[JEE Main 2026, 5 Apr Shift 2, Q2] Let z1,z2∈Cz_1,z_2\in\mathbb{C} be the distinct solutions of the equation z2+4z−(1+12i)=0z^2+4z-(1+12i)=0. Then ∣z1∣2+∣z2∣2|z_1|^2+|z_2|^2 is equal to:
Not attempted
A
1818
B
2222
C
2929
D
3434
100
[JEE Main 2026, 5 Apr Shift 2, Q21] Let A={1,4,7}A=\{1,4,7\} and B={2,3,8}B=\{2,3,8\}. Then the number of elements, in the relation R={((a1,b1),(a2,b2))∈((A×B)×(A×B)):a1+b2 divides a2+b1}R=\{((a_1,b_1),(a_2,b_2))\in((A\times B)\times(A\times B)):a_1+b_2\text{ divides }a_2+b_1\} is ______.
Not attempted
Your answer: —
Correct answer: 18
101
[JEE Main 2026, 5 Apr Shift 1, Q6] Let tan⁡A\tan A, tan⁡B\tan B, where A,B∈(−π2,π2)A,B\in\left(-\frac\pi2,\frac\pi2\right), be the roots of the quadratic equation x2−2x−5=0x^2-2x-5=0. Then 20sin⁡2(A+B2)20\sin^2\left(\frac{A+B}{2}\right) is equal to:
Not attempted
A
10+1010+\sqrt{10}
B
10−21010-2\sqrt{10}
C
10−31010-3\sqrt{10}
D
10−1010-\sqrt{10}
102
[JEE Main 2026, 5 Apr Shift 1, Q12] The sum of all the integral values of p such that the equation 3sin⁡2x+12cos⁡x−3=p3\sin^2x+12\cos x-3=p, x∈Rx\in\mathbb{R}, has at least one solution, is:
Not attempted
A
−54-54
B
−60-60
C
−75-75
D
−84-84
103
[JEE Main 2026, 5 Apr Shift 1, Q24] If π4+∑p=111tan⁡−1(2p−11+22p−1)=α\frac\pi4+\sum_{p=1}^{11}\tan^{-1}\left(\frac{2^{p-1}}{1+2^{2p-1}}\right)=\alpha, then tan⁡α\tan\alpha is equal to ______.
Not attempted
Your answer: —
Correct answer: 2048
104
[JEE Main 2026, 5 Apr Shift 1, Q7] A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is:
Not attempted
A
710\frac7{10}
B
1017\frac{10}{17}
C
1219\frac{12}{19}
D
719\frac7{19}
105
[JEE Main 2026, 5 Apr Shift 1, Q8] The mean deviation about the mean for the data xi579101215fi862226\displaystyle\begin{array}{|c|c|c|c|c|c|c|}\hline x_i & 5 & 7 & 9 & 10 & 12 & 15 \\ \hline f_i & 8 & 6 & 2 & 2 & 2 & 6 \\ \hline \end{array} is equal to:
Not attempted
A
4013\frac{40}{13}
B
4213\frac{42}{13}
C
4413\frac{44}{13}
D
4613\frac{46}{13}
106
[JEE Main 2026, 5 Apr Shift 1, Q14] Let a⃗=7i^+j^−k^\vec a=\sqrt7\hat i+\hat j-\hat k and b⃗=j^+2k^\vec b=\hat j+2\hat k. If r⃗\vec r is a vector such that r⃗×a⃗+a⃗×b⃗=0⃗\vec r\times\vec a+\vec a\times\vec b=\vec0 and r⃗⋅a⃗=0\vec r\cdot\vec a=0, then ∣3r⃗∣2|3\vec r|^2 is equal to:
Not attempted
A
4444
B
5454
C
8686
D
132132
107
[JEE Main 2026, 5 Apr Shift 1, Q13] The square of the distance of the point P(5,6,7)P(5,6,7) from the line x−22=y−53=z−24\frac{x-2}{2}=\frac{y-5}{3}=\frac{z-2}{4} is equal to:
Not attempted
A
33
B
55
C
66
D
88
108
[JEE Main 2026, 5 Apr Shift 1, Q15] The square of the distance of the point of intersection of the lines r⃗=(i^+j^−k^)+λ(ai^−j^)\vec r=(\hat i+\hat j-\hat k)+\lambda(a\hat i-\hat j), a≠0a\ne0 and r⃗=(4i^−k^)+μ(2i^+ak^)\vec r=(4\hat i-\hat k)+\mu(2\hat i+a\hat k) from the origin is:
Not attempted
A
55
B
1010
C
1717
D
2626
109
[JEE Main 2026, 5 Apr Shift 1, Q9] Let a focus of the ellipse E:x2a2+y2b2=1E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 be S(4,0)S(4,0) and its eccentricity be 45\frac45. If the point P(3,α)P(3,\alpha) lies on E and O is the origin, then the area of △POS\triangle POS is equal to:
Not attempted
A
12/5
B
14/5
C
24/5
D
48/5
110
[JEE Main 2026, 5 Apr Shift 1, Q10] Let P be a moving point on the circle x2+y2−6x−8y+21=0x^2+y^2-6x-8y+21=0. Then, the maximum distance of P from the vertex of the parabola x2+6x+y+13=0x^2+6x+y+13=0 is equal to:
Not attempted
A
88
B
1010
C
1212
D
99
111
[JEE Main 2026, 5 Apr Shift 1, Q11] In an equilateral triangle PQR, let the vertex P be at (3,5)(3,5) and the side QR be along the line x+y=4x+y=4. If the orthocentre of the triangle PQR is (α,β)(\alpha,\beta), then 9(α+β)9(\alpha+\beta) is equal to:
Not attempted
A
1616
B
2727
C
3636
D
4848
112
[JEE Main 2026, 5 Apr Shift 1, Q25] Let y=y(x)y=y(x) be the solution of the differential equation xsin⁡(yx)dy=(ysin⁡(yx)−x)dxx\sin\left(\frac yx\right)dy=\left(y\sin\left(\frac yx\right)-x\right)dx, y(1)=π2y(1)=\frac\pi2 and let α=cos⁡(y(e12)e12)\alpha=\cos\left(\frac{y(e^{12})}{e^{12}}\right). Then the number of integral value of p, for which the equation x2+y2−2px+2py+α+2=0x^2+y^2-2px+2py+\alpha+2=0 represents a circle of radius r≤6r\le6, is ______.
Not attempted
Your answer: —
Correct answer: 11
113
[JEE Main 2026, 5 Apr Shift 1, Q16] The area of the region R={(x,y):xy≤27, 1≤y≤x2}R=\{(x,y):xy\le27,\ 1\le y\le x^2\} is equal to:
Not attempted
A
78log⁡e3−52378\log_e3-\frac{52}{3}
B
54log⁡e3−52354\log_e3-\frac{52}{3}
C
54log⁡e3−26354\log_e3-\frac{26}{3}
D
54log⁡e3+26354\log_e3+\frac{26}{3}
114
[JEE Main 2026, 5 Apr Shift 1, Q18] The value of the integral ∫0∞log⁡e(x)x2+4dx\int_0^\infty\frac{\log_e(x)}{x^2+4}dx is:
Not attempted
A
πlog⁡e(2)2\frac{\pi\log_e(2)}{2}
B
πlog⁡e(2)4\frac{\pi\log_e(2)}{4}
C
1+πlog⁡e(2)1+\pi\log_e(2)
D
2+πlog⁡e(2)2+\pi\log_e(2)
115
[JEE Main 2026, 5 Apr Shift 1, Q20] The value of the integral ∫π/6π/3(4−cosec2xcos⁡4x)dx\int_{\pi/6}^{\pi/3}\left(\frac{4-\mathrm{cosec}^2x}{\cos^4x}\right)dx is:
Not attempted
A
113\frac{11}{\sqrt3}
B
163\frac{16}{\sqrt3}
C
3233\frac{32}{3\sqrt3}
D
6433\frac{64}{3\sqrt3}
116
[JEE Main 2026, 5 Apr Shift 1, Q17] The product of all possible values of α\alpha, for which lim⁡x→0(1−cos⁡(αx)cos⁡((α+1)x)cos⁡((α+2)x)sin⁡2((α+1)x))=2\lim_{x\to0}\left(\frac{1-\cos(\alpha x)\cos((\alpha+1)x)\cos((\alpha+2)x)}{\sin^2((\alpha+1)x)}\right)=2, is:
Not attempted
A
−2-2
B
11
C
−1-1
D
54\frac54
117
[JEE Main 2026, 5 Apr Shift 1, Q19] Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a differentiable function such that f(x+y3)=f(x)+f(y)3f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)}{3} for all x,y∈Rx,y\in\mathbb{R}, and f′(0)=3f'(0)=3. Then the minimum value of the function g(x)=3+exf(x)g(x)=3+e^xf(x), is:
Not attempted
A
3(e+1e)3\left(\frac{e+1}{e}\right)
B
3(e−1e)3\left(\frac{e-1}{e}\right)
C
3−ee\frac{3-e}{e}
D
3e3e
118
[JEE Main 2026, 5 Apr Shift 1, Q2] Let the sum of the first n terms of an A.P. be 3n2+5n3n^2+5n. Then the sum of squares of the first 10 terms of the A.P. is:
Not attempted
A
1022010220
B
1286012860
C
1522015220
D
1978019780
119
[JEE Main 2026, 5 Apr Shift 1, Q5] ∑n=110(528n(n+1)(n+2))\sum_{n=1}^{10}\left(\frac{528}{n(n+1)(n+2)}\right) is equal to:
Not attempted
A
6565
B
130130
C
220220
D
440440
120
[JEE Main 2026, 5 Apr Shift 1, Q23] If the sum of the coefficients of x7x^7 and x14x^{14} in the expansion of (1x3−x4)n\left(\frac1{x^3}-x^4\right)^n, x≠0x\ne0, is zero, then the value of n is ______.
Not attempted
Your answer: —
Correct answer: 21
121
[JEE Main 2026, 5 Apr Shift 1, Q22] Two players A and B play a series of games of badminton. The player, who wins 5 games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player A wins the series is ______.
Not attempted
Your answer: —
Correct answer: 126
122
[JEE Main 2026, 5 Apr Shift 1, Q3] Let A be a 3×33\times3 matrix such that AT[101]=[522]A^T\begin{bmatrix}1\\0\\1\end{bmatrix}=\begin{bmatrix}5\\2\\2\end{bmatrix}, AT[001]=[311]A^T\begin{bmatrix}0\\0\\1\end{bmatrix}=\begin{bmatrix}3\\1\\1\end{bmatrix}, A[101]=[344]A\begin{bmatrix}1\\0\\1\end{bmatrix}=\begin{bmatrix}3\\4\\4\end{bmatrix} and A[001]=[131]A\begin{bmatrix}0\\0\\1\end{bmatrix}=\begin{bmatrix}1\\3\\1\end{bmatrix}. If det⁡(A)=1\det(A)=1, then det⁡(adj(A2+A))\det(\mathrm{adj}(A^2+A)) is equal to:
Not attempted
A
1616
B
2525
C
4949
D
6464
123
[JEE Main 2026, 5 Apr Shift 1, Q4] Consider the system of linear equations in x, y, z: x+2y+tz=0x+2y+tz=0 6x+y+5tz=06x+y+5tz=0 3x+t2y+f(t)z=03x+t^2y+f(t)z=0 where f:R→Rf:\mathbb{R}\to\mathbb{R} is a differentiable function. If this system has infinitely many solutions for all t∈Rt\in\mathbb{R}, then f
Not attempted
A
is a constant function
B
is strictly increasing on R\mathbb{R}
C
is strictly decreasing on R\mathbb{R}
D
has two critical points
124
[JEE Main 2026, 5 Apr Shift 1, Q1] Let a,b∈Ca,b\in\mathbb{C}. Let α,β\alpha,\beta be the roots of the equation x2+ax+b=0x^2+ax+b=0. If β−α=11\beta-\alpha=\sqrt{11} and β2−α2=3i11\beta^2-\alpha^2=3i\sqrt{11}, then (β3−α3)2(\beta^3-\alpha^3)^2 is equal to:
Not attempted
A
160160
B
176176
C
194194
D
187187
125
[JEE Main 2026, 5 Apr Shift 1, Q21] Let A={1,2,3,4,5,6}A=\{1,2,3,4,5,6\}. The number of one-one functions f:A→Af:A\to A such that f(1)≥3f(1)\ge3, f(3)≤4f(3)\le4 and f(2)+f(3)=5f(2)+f(3)=5, is ______.
Not attempted
Your answer: —
Correct answer: 72
126
[JEE Main 2026, 4 Apr Shift 2, Q8] For 10 observations x1,x2,…,x10x_1,x_2,\dots,x_{10}, if ∑i=110(xi+2)2=180\sum_{i=1}^{10}(x_i+2)^2=180 and ∑i=110(xi−1)2=90\sum_{i=1}^{10}(x_i-1)^2=90, then their standard deviation is:
Not attempted
A
22
B
3\sqrt3
C
222\sqrt2
D
33
127
[JEE Main 2026, 4 Apr Shift 2, Q21] From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to ab\frac ab, where a,b∈Na,b\in\mathbb{N} and gcd⁡(a,b)=1\gcd(a,b)=1, then a+ba+b is equal to ______.
Not attempted
Your answer: —
Correct answer: 944
128
[JEE Main 2026, 4 Apr Shift 2, Q15] Let u^\hat u and v^\hat v be unit vectors inclined at an acute angle such that ∣u^×v^∣=32|\hat u\times\hat v|=\frac{\sqrt3}{2}. If A⃗=λu^+v^+(u^×v^)\vec A=\lambda\hat u+\hat v+(\hat u\times\hat v), then λ\lambda is equal to:
Not attempted
A
43(A⃗⋅u^)−23(A⃗⋅v^)\frac43(\vec A\cdot\hat u)-\frac23(\vec A\cdot\hat v)
B
23(A⃗⋅u^)−13(A⃗⋅v^)\frac23(\vec A\cdot\hat u)-\frac13(\vec A\cdot\hat v)
C
43(A⃗⋅u^)+23(A⃗⋅v^)\frac43(\vec A\cdot\hat u)+\frac23(\vec A\cdot\hat v)
D
(A⃗⋅u^)−12(A⃗⋅v^)(\vec A\cdot\hat u)-\frac12(\vec A\cdot\hat v)
129
[JEE Main 2026, 4 Apr Shift 2, Q13] The shortest distance between the lines r⃗=(13i^+2j^+83k^)+λ(2i^−5j^+6k^)\vec r=\left(\frac13\hat i+2\hat j+\frac83\hat k\right)+\lambda(2\hat i-5\hat j+6\hat k) and r⃗=(−23i^−13k^)+μ(j^−k^)\vec r=\left(-\frac23\hat i-\frac13\hat k\right)+\mu(\hat j-\hat k), λ,μ∈R\lambda,\mu\in\mathbb{R}, is:
Not attempted
A
5\sqrt5
B
33
C
232\sqrt3
D
15\sqrt{15}
130
[JEE Main 2026, 4 Apr Shift 2, Q14] If (2α+1,α2−3α,α−12)\left(2\alpha+1,\alpha^2-3\alpha,\frac{\alpha-1}{2}\right) is the image of (α,2α,1)(\alpha,2\alpha,1) in the line x−23=y−12=z1\frac{x-2}{3}=\frac{y-1}{2}=\frac z1, then the possible value(s) of α\alpha is (are)
Not attempted
A
Only 3
B
Only 3 and −1-1
C
Only 3, 14\frac14 and −1-1
D
Only 3 and 14\frac14
131
[JEE Main 2026, 4 Apr Shift 2, Q10] Let P(3cos⁡α,2sin⁡α)P(3\cos\alpha,2\sin\alpha), α≠0\alpha\ne0, be a point on the ellipse x29+y24=1\frac{x^2}{9}+\frac{y^2}{4}=1, Q be a point on the circle x2+y2−14x−14y+82=0x^2+y^2-14x-14y+82=0 and R be a point on the line x+y=5x+y=5 such that the centroid of the triangle PQR is (2+cos⁡α,3+23sin⁡α)\left(2+\cos\alpha,3+\frac23\sin\alpha\right). Then the sum of the ordinates of all possible points R is:
Not attempted
A
66
B
22
C
44
D
88
132
[JEE Main 2026, 4 Apr Shift 2, Q11] Let H:x2a2−y2b2=1H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is 83\frac83. If the line x=αx=\alpha intersects the hyperbola H at the points A and B such that the area of the triangle AOB is 4154\sqrt{15}, where O is the origin, then α2\alpha^2 equals
Not attempted
A
1212
B
1616
C
2424
D
2525
133
[JEE Main 2026, 4 Apr Shift 2, Q23] Let A, B be points on the two half-lines x−3∣y∣=αx-\sqrt3|y|=\alpha, α>0\alpha>0 at a distance of α\alpha from their point of intersection P. The line segment AB meets the angle bisector of the given half-lines at the point Q. If PQ=92PQ=\frac92 and R is the radius of the circumcircle of △PAB\triangle PAB, then α2R\frac{\alpha^2}{R} is equal to ______.
Not attempted
Your answer: —
Correct answer: 9
134
[JEE Main 2026, 4 Apr Shift 2, Q24] Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola y2=16xy^2=16x. Let the vertex B containing the right angle be (4,8)(4,8) and the locus of the centroid of △ABC\triangle ABC be a conic C0C_0. Then three times the length of latus rectum of C0C_0 is ______.
Not attempted
Your answer: —
Correct answer: 16
135
[JEE Main 2026, 4 Apr Shift 2, Q19] Let y=y(x)y=y(x) be the solution of the differential equation: dydx+(6x2+(3x2+2x3+4)e−2x(x3+2)(2+e−2x))y=2+e−2x\frac{dy}{dx}+\left(\frac{6x^2+(3x^2+2x^3+4)e^{-2x}}{(x^3+2)(2+e^{-2x})}\right)y=2+e^{-2x}, x∈(−1,2)x\in(-1,2), satisfying y(0)=32y(0)=\frac32. If y(1)=α(2+e−2)y(1)=\alpha(2+e^{-2}), then α\alpha is equal to:
Not attempted
A
138\frac{13}{8}
B
613\frac{6}{13}
C
1213\frac{12}{13}
D
1312\frac{13}{12}
136
[JEE Main 2026, 4 Apr Shift 2, Q25] Let f be a twice differentiable function such that f(x)=∫0xtan⁡(t−x) dt−∫0xf(t)tan⁡t dtf(x)=\int_0^x\tan(t-x)\,dt-\int_0^xf(t)\tan t\,dt, x∈(−π2,π2)x\in\left(-\frac\pi2,\frac\pi2\right). Then f′′(π6)+12f′(−π6)+f(π6)f''\left(\frac\pi6\right)+12f'\left(-\frac\pi6\right)+f\left(\frac\pi6\right) is equal to ______.
Not attempted
Your answer: —
Correct answer: 5
137
[JEE Main 2026, 4 Apr Shift 2, Q4] If α=1\alpha=1 and β=1+i2\beta=1+i\sqrt2, where i=−1i=\sqrt{-1} are two roots of the equation x3+ax2+bx+c=0x^3+ax^2+bx+c=0, a,b,c∈Ra,b,c\in\mathbb{R}, then ∫−11(x3+ax2+bx+c) dx\int_{-1}^{1}(x^3+ax^2+bx+c)\,dx is equal to:
Not attempted
A
−2-2
B
−4-4
C
−8-8
D
−10-10
138
[JEE Main 2026, 4 Apr Shift 2, Q18] The area of the region bounded by the curves x+3y2=0x+3y^2=0 and x+4y2=1x+4y^2=1 is equal to:
Not attempted
A
13\frac13
B
23\frac23
C
43\frac43
D
53\frac53
139
[JEE Main 2026, 4 Apr Shift 2, Q20] The integral ∫01cot⁡−1(1+x+x2) dx\int_0^1\cot^{-1}(1+x+x^2)\,dx is equal to:
Not attempted
A
2tan⁡−12+12log⁡e(54)+π22\tan^{-1}2+\frac12\log_e\left(\frac54\right)+\frac\pi2
B
2tan⁡−12+12log⁡e(54)−π22\tan^{-1}2+\frac12\log_e\left(\frac54\right)-\frac\pi2
C
2tan⁡−12−12log⁡e(54)+π22\tan^{-1}2-\frac12\log_e\left(\frac54\right)+\frac\pi2
D
2tan⁡−12−12log⁡e(54)−π22\tan^{-1}2-\frac12\log_e\left(\frac54\right)-\frac\pi2
140
[JEE Main 2026, 4 Apr Shift 2, Q12] max⁡0≤x≤π(16sin⁡(x2)cos⁡3(x2))\max_{0\le x\le\pi}\left(16\sin\left(\frac x2\right)\cos^3\left(\frac x2\right)\right) is equal to:
Not attempted
A
332\frac{3\sqrt3}{2}
B
333\sqrt3
C
434\sqrt3
D
636\sqrt3
141
[JEE Main 2026, 4 Apr Shift 2, Q22] Let f(x)=ex−1f(x)=e^{x-1} for x<0x<0 and f(x)=x2−5x+6f(x)=x^2-5x+6 for x≥0x\ge0, and g(x)=f(∣x∣)+∣f(x)∣g(x)=f(|x|)+|f(x)|. If the number of points where g is not continuous and is not differentiable are α\alpha and β\beta respectively, then α+β\alpha+\beta is equal to ______.
Not attempted
Your answer: —
Correct answer: 4
142
[JEE Main 2026, 4 Apr Shift 2, Q7] Let α=14+18+116+…∞\alpha=\frac14+\frac18+\frac1{16}+\dots\infty and β=13+19+127+…∞\beta=\frac13+\frac19+\frac1{27}+\dots\infty. Then the value of (0.2)log⁡5(α)+(0.04)log⁡5(β)(0.2)^{\log_{\sqrt5}(\alpha)}+(0.04)^{\log_5(\beta)} is equal to:
Not attempted
A
44
B
55
C
88
D
2525
143
[JEE Main 2026, 4 Apr Shift 2, Q9] In the expansion of (9x−13x)18\left(9x-\frac{1}{3\sqrt x}\right)^{18}, x>0x>0, if the term independent of x is (221)k(221)k, then k is equal to:
Not attempted
A
8484
B
7878
C
168168
D
198198
144
[JEE Main 2026, 4 Apr Shift 2, Q17] Let A={(a,b,c):a,b,c are non-negative integers and a+b+2c=22}A=\{(a,b,c):a,b,c\text{ are non-negative integers and }a+b+2c=22\}. Then n(A)n(A) is equal to:
Not attempted
A
121121
B
124124
C
144144
D
169169
145
[JEE Main 2026, 4 Apr Shift 2, Q3] If the system of equations: x+y+z=5x+y+z=5 x+2y+3z=9x+2y+3z=9 x+3y+λz=μx+3y+\lambda z=\mu has infinitely many solutions, then the value of λ+μ\lambda+\mu is:
Not attempted
A
1616
B
1818
C
1919
D
2121
146
[JEE Main 2026, 4 Apr Shift 2, Q6] Let A=[1274−2838−7]A=\begin{bmatrix}1&2&7\\4&-2&8\\3&8&-7\end{bmatrix} and det⁡(A−αI)=0\det(A-\alpha I)=0, where α\alpha is a real number. If the largest possible value of α\alpha is p, then the circle (x−p)2+(y−2p)2=320(x-p)^2+(y-2p)^2=320, intersects the co-ordinate axes at
Not attempted
A
1 point
B
2 points
C
3 points
D
4 points
147
[JEE Main 2026, 4 Apr Shift 2, Q2] Let S={z∈C:z2+4z+16=0}S=\{z\in\mathbb{C}:z^2+4z+16=0\}. Then ∑z∈S∣z+3i∣2\sum_{z\in S}|z+\sqrt3i|^2 is equal to:
Not attempted
A
4242
B
2323
C
2727
D
3838
148
[JEE Main 2026, 4 Apr Shift 2, Q5] If the quadratic equation (λ+2)x2−3λx+4λ=0(\lambda+2)x^2-3\lambda x+4\lambda=0, λ≠−2\lambda\ne-2, has two positive roots, then the number of possible integral values of λ\lambda is:
Not attempted
A
11
B
22
C
33
D
44
149
[JEE Main 2026, 4 Apr Shift 2, Q1] For the function f:[1,∞)→[1,∞)f:[1,\infty)\to[1,\infty) defined by f(x)=(x−1)4+1f(x)=(x-1)^4+1, among the two statements: (I) The set S={x∈[1,∞):f(x)=f−1(x)}S=\{x\in[1,\infty):f(x)=f^{-1}(x)\} contains exactly two elements, and (II) The set S={x∈[1,∞):f(x)=f−1(x+1)}S=\{x\in[1,\infty):f(x)=f^{-1}(x+1)\} is an empty set,
Not attempted
A
only (I) is TRUE
B
only (II) is TRUE
C
both (I) and (II) are TRUE
D
neither (I) nor (II) is TRUE
150
[JEE Main 2026, 4 Apr Shift 2, Q16] Let for some α∈R\alpha\in\mathbb{R}, f:R→Rf:\mathbb{R}\to\mathbb{R} be a function satisfying f(x+y)=f(x)+2y2+y+αxyf(x+y)=f(x)+2y^2+y+\alpha xy for all x,y∈Rx,y\in\mathbb{R}. If f(0)=−1f(0)=-1 and f(1)=2f(1)=2, then the value of ∑n=15(α+f(n))\sum_{n=1}^{5}(\alpha+f(n)) is:
Not attempted
A
110110
B
140140
C
150150
D
170170
151
[JEE Main 2026, 4 Apr Shift 1, Q23] If A=sin⁡3∘cos⁡9∘+sin⁡9∘cos⁡27∘+sin⁡27∘cos⁡81∘A=\frac{\sin3^\circ}{\cos9^\circ}+\frac{\sin9^\circ}{\cos27^\circ}+\frac{\sin27^\circ}{\cos81^\circ} and B=tan⁡81∘−tan⁡3∘B=\tan81^\circ-\tan3^\circ, then BA\frac BA is equal to ______.
Not attempted
Your answer: —
Correct answer: 2
152
[JEE Main 2026, 4 Apr Shift 1, Q10] Suppose that the mean and median of the non-negative numbers 21, 8, 17, a, 51, 103, b, 13, 67, (a>b)(a>b), are 40 and 21, respectively. If the mean deviation about the median is 26, then 2a is equal to:
Not attempted
A
109109
B
117117
C
161161
D
131131
153
[JEE Main 2026, 4 Apr Shift 1, Q21] A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is p, then 96p is equal to ______.
Not attempted
Your answer: —
Correct answer: 9
154
[JEE Main 2026, 4 Apr Shift 1, Q24] Let ak⃗=(tan⁡θk)i^+j^\vec{a_k}=(\tan\theta_k)\hat i+\hat j and bk⃗=i^−(cot⁡θk)j^\vec{b_k}=\hat i-(\cot\theta_k)\hat j, where θk=2k−1π2n+1\theta_k=\frac{2^{k-1}\pi}{2^n+1}, for some n∈Nn\in\mathbb{N}, n>5n>5. Then the value of ∑k=1n∣ak⃗∣2∑k=1n∣bk⃗∣2\frac{\sum_{k=1}^n|\vec{a_k}|^2}{\sum_{k=1}^n|\vec{b_k}|^2} is ______.
Not attempted
Your answer: —
Correct answer: 3
155
[JEE Main 2026, 4 Apr Shift 1, Q14] A line with direction ratios 1,−1,21,-1,2 intersects the lines x2=y3=z+13\frac x2=\frac y3=\frac{z+1}{3} and x+1−1=y−21=z4\frac{x+1}{-1}=\frac{y-2}{1}=\frac z4 at the points P and Q, respectively. If the length of the line segment PQ is α\alpha, then 225α2225\alpha^2 is equal to:
Not attempted
A
10241024
B
10141014
C
11041104
D
12041204
156
[JEE Main 2026, 4 Apr Shift 1, Q15] The square of the distance of the point (−2,−8,6)(-2,-8,6) from the line x−11=y−12=z−1\frac{x-1}{1}=\frac{y-1}{2}=\frac{z}{-1} along the line x+51=y+5−1=z2\frac{x+5}{1}=\frac{y+5}{-1}=\frac z2 is equal to:
Not attempted
A
33
B
66
C
88
D
1212
157
[JEE Main 2026, 4 Apr Shift 1, Q11] Let the line L1:x+3=0L_1:x+3=0 intersect the lines L2:x−y=0L_2:x-y=0 and L3:3x+y=0L_3:3x+y=0 at the points A and B, respectively. Let the bisector of the obtuse angle between the lines L2L_2 and L3L_3 intersect the line L1L_1 at the point C. Then BC2:AC2BC^2:AC^2 is equal to:
Not attempted
A
5:15:1
B
1:51:5
C
2:32:3
D
3:23:2
158
[JEE Main 2026, 4 Apr Shift 1, Q12] Let the vertex A of a triangle ABC be (1,2)(1,2), and the mid-point of the side AB be (5,−1)(5,-1). If the centroid of this triangle is (3,4)(3,4) and its circumcenter is (α,β)(\alpha,\beta), then 21(α+β)21(\alpha+\beta) is equal to:
Not attempted
A
309309
B
403403
C
497497
D
524524
159
[JEE Main 2026, 4 Apr Shift 1, Q13] Suppose that two chords, drawn from the point (1,2)(1,2) on the circle x2+y2+x−3y=0x^2+y^2+x-3y=0 are bisected by the y-axis. If the other ends of these chords are R and S, and the mid point of the line segment RS is (α,β)(\alpha,\beta), then 6(α+β)6(\alpha+\beta) is equal to:
Not attempted
A
11
B
33
C
44
D
66
160
[JEE Main 2026, 4 Apr Shift 1, Q22] Consider the parabola P:y2=4kxP:y^2=4kx and the ellipse E:x2a2+y2b2=1E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1. Let the line segment joining the points of intersection of P and E, be their latus rectums. If the eccentricity of E is e, then e2+22e^2+2\sqrt2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 3
161
[JEE Main 2026, 4 Apr Shift 1, Q20] Let y=y(x)y=y(x) be the solution of the differential equation dydx=(1+x+x2)(1−y+y2)\frac{dy}{dx}=(1+x+x^2)(1-y+y^2), y(0)=12y(0)=\frac12. Then (2y(1)−1)(2y(1)-1) is equal to
Not attempted
A
3tan⁡(1136)\sqrt3\tan\left(\frac{11\sqrt3}{6}\right)
B
32tan⁡(11312)\frac{\sqrt3}{2}\tan\left(\frac{11\sqrt3}{12}\right)
C
3tan⁡(11312)\sqrt3\tan\left(\frac{11\sqrt3}{12}\right)
D
32tan⁡(1136)\frac{\sqrt3}{2}\tan\left(\frac{11\sqrt3}{6}\right)
162
[JEE Main 2026, 4 Apr Shift 1, Q18] The area of the region {(x,y):y≤π−∣x∣, y≤∣xsin⁡x∣, y≥0}\{(x,y):y\le\pi-|x|,\ y\le|x\sin x|,\ y\ge0\} is:
Not attempted
A
1+π281+\frac{\pi^2}{8}
B
2+π242+\frac{\pi^2}{4}
C
π28−1\frac{\pi^2}{8}-1
D
4+π224+\frac{\pi^2}{2}
163
[JEE Main 2026, 4 Apr Shift 1, Q19] Let ∫−22(∣sin⁡x∣+[xsin⁡x])dx=2(3−cos⁡2)+β\int_{-2}^{2}(|\sin x|+[x\sin x])dx=2(3-\cos2)+\beta, where [⋅][\cdot] is the greatest integer function. Then βsin⁡(β2)\beta\sin\left(\frac\beta2\right) equals:
Not attempted
A
11
B
22
C
44
D
88
164
[JEE Main 2026, 4 Apr Shift 1, Q16] If y=tan⁡−1(3cos⁡x−4sin⁡x4cos⁡x+3sin⁡x)+2tan⁡−1(x1+1−x2)y=\tan^{-1}\left(\frac{3\cos x-4\sin x}{4\cos x+3\sin x}\right)+2\tan^{-1}\left(\frac{x}{1+\sqrt{1-x^2}}\right), then dydx\frac{dy}{dx} at x=32x=\frac{\sqrt3}{2} is equal to:
Not attempted
A
33
B
−1-1
C
11
D
22
165
[JEE Main 2026, 4 Apr Shift 1, Q17] Let f be a real polynomial of degree n such that f(x)=f′(x)f′′(x)f(x)=f'(x)f''(x), for all x∈Rx\in\mathbb{R}. If f(0)=0f(0)=0, then 36(f′(2)+f′′(2)+∫02f(x)dx)36\left(f'(2)+f''(2)+\int_0^2f(x)dx\right) is equal to:
Not attempted
A
4242
B
4646
C
5656
D
6666
166
[JEE Main 2026, 4 Apr Shift 1, Q25] The number of points, at which the function f(x)=max⁡{6x,2+3x2}+∣x−1∣cos⁡∣x2−14∣f(x)=\max\{6x,2+3x^2\}+|x-1|\cos\left|x^2-\frac14\right|, x∈(−π,π)x\in(-\pi,\pi), is not differentiable, is ______.
Not attempted
Your answer: —
Correct answer: 3
167
[JEE Main 2026, 4 Apr Shift 1, Q7] The first term of an A.P. of 30 non-negative terms is 103\frac{10}{3}. If the sum of this A.P. is the cube of its last term, then its common difference is:
Not attempted
A
587\frac{5}{87}
B
2583\frac{25}{83}
C
1529\frac{15}{29}
D
529\frac{5}{29}
168
[JEE Main 2026, 4 Apr Shift 1, Q9] Let the smallest value of k∈Nk\in\mathbb{N}, for which the coefficient of x3x^3 in (1+x)3+(1+x)4+(1+x)5+⋯+(1+x)99+(1+kx)100(1+x)^3+(1+x)^4+(1+x)^5+\dots+(1+x)^{99}+(1+kx)^{100}, x≠0x\ne0, is (43n+1014)(100C3)\left(43n+\frac{101}{4}\right)({}^{100}C_3) for some n∈Nn\in\mathbb{N}, be p. Then the value of p+np+n is:
Not attempted
A
1010
B
1111
C
1212
D
1313
169
[JEE Main 2026, 4 Apr Shift 1, Q8] The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is:
Not attempted
A
50405040
B
30503050
C
34103410
D
43204320
170
[JEE Main 2026, 4 Apr Shift 1, Q5] Let S={A=[abcd]:a,b,c,d∈{0,1,2,3,4} and A2−4A+3I=O}S=\left\{A=\begin{bmatrix}a&b\\c&d\end{bmatrix}:a,b,c,d\in\{0,1,2,3,4\}\text{ and }A^2-4A+3I=O\right\}. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is:
Not attempted
A
2020
B
1717
C
2121
D
1919
171
[JEE Main 2026, 4 Apr Shift 1, Q6] Let A=[112−201135]A=\begin{bmatrix}1&1&2\\-2&0&1\\1&3&5\end{bmatrix}. Then the sum of all elements of the matrix adj(adj(2(adjA)−1))\mathrm{adj}(\mathrm{adj}(2(\mathrm{adj}A)^{-1})) is equal to:
Not attempted
A
33
B
44
C
−4-4
D
−3-3
172
[JEE Main 2026, 4 Apr Shift 1, Q2] If the set of all solutions of ∣x2+x−9∣=∣x∣+∣x2−9∣|x^2+x-9|=|x|+|x^2-9| is [α,β]∪[γ,∞)[\alpha,\beta]\cup[\gamma,\infty), then (α2+β2+γ2)(\alpha^2+\beta^2+\gamma^2) is equal to:
Not attempted
A
99
B
1818
C
3636
D
7272
173
[JEE Main 2026, 4 Apr Shift 1, Q3] Let z be a complex number such that ∣z+2∣=∣z−2∣|z+2|=|z-2| and arg⁡(z+3z−i)=π4\arg\left(\frac{z+3}{z-i}\right)=\frac\pi4. Then ∣z∣2|z|^2 is equal to:
Not attempted
A
99
B
44
C
55
D
11
174
[JEE Main 2026, 4 Apr Shift 1, Q1] Let [⋅][\cdot] denote the greatest integer function. If the domain of the function f(x)=cos⁡−1(4x+2[x]3)f(x)=\cos^{-1}\left(\frac{4x+2[x]}{3}\right) is [α,β][\alpha,\beta], then 12(α+β)12(\alpha+\beta) is equal to:
Not attempted
A
66
B
88
C
99
D
44
175
[JEE Main 2026, 4 Apr Shift 1, Q4] The number of functions f:{1,2,3,4}→{a,b,c}f:\{1,2,3,4\}\to\{a,b,c\}, which are not onto, is:
Not attempted
A
4848
B
4545
C
5151
D
3535
176
[JEE Main 2026, 2 Apr Shift 2, Q13] Let P={θ∈[0,4π]:tan⁡2θ≠1}P=\{\theta\in[0,4\pi]:\tan^2\theta\ne1\} and S={a∈Z:2(cos⁡8θ−sin⁡8θ)sec⁡2θ=a2,θ∈P}S=\{a\in\mathbb{Z}:2(\cos^8\theta-\sin^8\theta)\sec2\theta=a^2,\theta\in P\}. Then n(S)n(S) is:
Not attempted
A
00
B
11
C
22
D
33
177
[JEE Main 2026, 2 Apr Shift 2, Q8] A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is mn\frac mn, gcd⁡(m,n)=1\gcd(m,n)=1, then m+nm+n is equal to:
Not attempted
A
5353
B
5555
C
107107
D
105105
178
[JEE Main 2026, 2 Apr Shift 2, Q9] The mean and variance of n observations are 8 and 16, respectively. If the sum of the first (n−1)(n-1) observations is 48 and the sum of squares of the first (n−1)(n-1) observations is 496, then the value of n is:
Not attempted
A
2121
B
1616
C
1313
D
77
179
[JEE Main 2026, 2 Apr Shift 2, Q14] Let the vectors a⃗=−i^+j^+3k^\vec a=-\hat i+\hat j+3\hat k and b⃗=i^+3j^+k^\vec b=\hat i+3\hat j+\hat k. For some λ,μ∈R\lambda,\mu\in\mathbb{R}, let c⃗=λa⃗+μb⃗\vec c=\lambda\vec a+\mu\vec b. If c⃗⋅(3i^−6j^+2k^)=10\vec c\cdot(3\hat i-6\hat j+2\hat k)=10 and c⃗⋅(i^+j^+k^)=−2\vec c\cdot(\hat i+\hat j+\hat k)=-2, then ∣c⃗∣2|\vec c|^2 is equal to:
Not attempted
A
88
B
1212
C
1414
D
1515
180
[JEE Main 2026, 2 Apr Shift 2, Q16] Two adjacent sides of a parallelogram PQRS are given by PQ⃗=j^+k^\vec{PQ}=\hat j+\hat k and PS⃗=i^−j^\vec{PS}=\hat i-\hat j. If the side PS is rotated about the point P by an acute angle α\alpha in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then sin⁡2(5α2)−sin⁡2(α2)\sin^2\left(\frac{5\alpha}{2}\right)-\sin^2\left(\frac\alpha2\right) is equal to:
Not attempted
A
12\frac12
B
32\frac{\sqrt3}{2}
C
34\frac{\sqrt3}{4}
D
235\frac{2\sqrt3}{5}
181
[JEE Main 2026, 2 Apr Shift 2, Q15] Let the point A be the foot of perpendicular drawn from the point P(a,b,0)P(a,b,0) on the line x−12=y−21=z−α3\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-\alpha}{3}. If the midpoint of the line segment PA is (0,34,−14)\left(0,\frac34,-\frac14\right), then the value of a2+b2+α2a^2+b^2+\alpha^2 is equal to:
Not attempted
A
11
B
22
C
66
D
99
182
[JEE Main 2026, 2 Apr Shift 2, Q10] Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines x+(k−1)y+3=0x+(k-1)y+3=0 and 2x+k2y−4=02x+k^2y-4=0. If the line x−y+2=0x-y+2=0 intersects the circle at the points A and B, then (AB)2(AB)^2 is equal to:
Not attempted
A
1010
B
2727
C
1818
D
3434
183
[JEE Main 2026, 2 Apr Shift 2, Q11] Let O be the origin, and P and Q be two points on the rectangular hyperbola xy=12xy=12 such that the mid point of the line segment PQ is (12,−12)\left(\frac12,-\frac12\right). Then the area of the triangle OPQ equals:
Not attempted
A
32\frac32
B
52\frac52
C
72\frac72
D
92\frac92
184
[JEE Main 2026, 2 Apr Shift 2, Q12] Let the parabola y=x2+px+qy=x^2+px+q passing through the point (1,−1)(1,-1) be such that the distance between its vertex and the x-axis is minimum. Then the value of p2+q2p^2+q^2 is:
Not attempted
A
22
B
44
C
55
D
88
185
[JEE Main 2026, 2 Apr Shift 2, Q23] Let A be the point (3,0)(3,0) and circles with variable diameter AB touch the circle x2+y2=36x^2+y^2=36 internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then 72e272e^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 18
186
[JEE Main 2026, 2 Apr Shift 2, Q20] Let x=x(y)x=x(y) be the solution of the differential equation 2y2dxdy−2xy+x2=02y^2\frac{dx}{dy}-2xy+x^2=0, y>1y>1, x(e)=ex(e)=e. Then x(e2)x(e^2) is equal to:
Not attempted
A
32e2\frac32e^2
B
23e2\frac23e^2
C
e2e^2
D
2e22e^2
187
[JEE Main 2026, 2 Apr Shift 2, Q17] The value of ∫020π(sin⁡4x+cos⁡4x) dx\int_0^{20\pi}(\sin^4x+\cos^4x)\,dx is equal to:
Not attempted
A
15π2\frac{15\pi}{2}
B
25π25\pi
C
15π15\pi
D
25π2\frac{25\pi}{2}
188
[JEE Main 2026, 2 Apr Shift 2, Q19] Let f(x)=∫(16x+24x2+2x−15)dxf(x)=\int\left(\frac{16x+24}{x^2+2x-15}\right)dx. If f(4)=14log⁡e3f(4)=14\log_e3 and f(7)=log⁡e(2α⋅3β)f(7)=\log_e(2^\alpha\cdot3^\beta), α,β∈N\alpha,\beta\in\mathbb{N}, then α+β\alpha+\beta is equal to:
Not attempted
A
3131
B
3737
C
3939
D
4141
189
[JEE Main 2026, 2 Apr Shift 2, Q24] If the area of the region bounded by 16x2−9y2=14416x^2-9y^2=144 and 8x−3y=248x-3y=24 is A, then 3(A+6log⁡e3)3(A+6\log_e3) is equal to ______.
Not attempted
Your answer: —
Correct answer: 24
190
[JEE Main 2026, 2 Apr Shift 2, Q18] Let f(x)f(x) be a polynomial of degree 5, and have extrema at x=1x=1 and x=−1x=-1. If lim⁡x→0(f(x)x3)=−5\lim_{x\to0}\left(\frac{f(x)}{x^3}\right)=-5, then f(2)−f(−2)f(2)-f(-2) is equal to:
Not attempted
A
00
B
5050
C
9292
D
112112
191
[JEE Main 2026, 2 Apr Shift 2, Q25] The number of points in the interval [2,4][2,4], at which the function f(x)=[x2−x−12]f(x)=\left[x^2-x-\frac12\right], where [⋅][\cdot] denotes the greatest integer function, is discontinuous, is ______.
Not attempted
Your answer: —
Correct answer: 10
192
[JEE Main 2026, 2 Apr Shift 2, Q4] Let a1,a2,a3,…a_1,a_2,a_3,\dots be an A.P. and g1=a1,g2,g3,…g_1=a_1,g_2,g_3,\dots be an increasing G.P. If a1=a2+g2=1a_1=a_2+g_2=1 and a3+g3=4a_3+g_3=4, then a10+g5a_{10}+g_5 is equal to:
Not attempted
A
8181
B
7676
C
6262
D
5555
193
[JEE Main 2026, 2 Apr Shift 2, Q5] The sum 131+13+231+3+13+23+331+3+5+⋯\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\cdots up to 8 terms, is:
Not attempted
A
7070
B
7171
C
7272
D
7373
194
[JEE Main 2026, 2 Apr Shift 2, Q6] If for 3≤r≤303\le r\le30, (30C30−r)+3(30C31−r)+3(30C32−r)+(30C33−r)=mCr({}^{30}C_{30-r})+3({}^{30}C_{31-r})+3({}^{30}C_{32-r})+({}^{30}C_{33-r})={}^mC_r, then m equals:
Not attempted
A
3131
B
3232
C
3333
D
3434
195
[JEE Main 2026, 2 Apr Shift 2, Q7] Let pnp_n denote the total number of triangles formed by joining the vertices of an n-side regular polygon. If pn+1−pn=66p_{n+1}-p_n=66, then the sum of all distinct prime divisors of n is:
Not attempted
A
77
B
88
C
55
D
66
196
[JEE Main 2026, 2 Apr Shift 2, Q3] If the system of equations x+5y+6z=4x+5y+6z=4 2x+3y+4z=72x+3y+4z=7 x+6y+az=bx+6y+az=b has infinitely many solutions, then the point (a,b)(a,b) lies on the line
Not attempted
A
y−x=3y-x=3
B
x−y=3x-y=3
C
x+y=11x+y=11
D
x+y=12x+y=12
197
[JEE Main 2026, 2 Apr Shift 2, Q22] Consider the matrices A=[2−24−2]A=\begin{bmatrix}2&-2\\4&-2\end{bmatrix} and B=[3913]B=\begin{bmatrix}3&9\\1&3\end{bmatrix}. If matrices P and Q are such that PA=BPA=B and AQ=BAQ=B, then the absolute value of the sum of the diagonal elements of 2(P+Q)2(P+Q) is ______.
Not attempted
Your answer: —
Correct answer: 34
198
[JEE Main 2026, 2 Apr Shift 2, Q1] Let α,β\alpha,\beta be the roots of the equation x2−3x+r=0x^2-3x+r=0, and α2,2β\frac\alpha2,2\beta be the roots of the equation x2+3x+r=0x^2+3x+r=0. If the roots of the equation x2+6x=mx^2+6x=m are 2α+β+2r2\alpha+\beta+2r and α−2β−r2\alpha-2\beta-\frac r2, then m is equal to:
Not attempted
A
−135-135
B
−567-567
C
135135
D
567567
199
[JEE Main 2026, 2 Apr Shift 2, Q2] Let the circles C1:∣z∣=rC_1:|z|=r and C2:∣z−3−4i∣=5C_2:|z-3-4i|=5, z∈Cz\in\mathbb{C}, be such that C2C_2 lies within C1C_1. If z1z_1 moves on C1C_1, z2z_2 moves on C2C_2 and min⁡∣z1−z2∣=2\min|z_1-z_2|=2, then max⁡∣z1−z2∣\max|z_1-z_2| is equal to:
Not attempted
A
1212
B
1717
C
2222
D
2424
200
[JEE Main 2026, 2 Apr Shift 2, Q21] Let A={2,3,4,5,6}A=\{2,3,4,5,6\}. Let R be a relation on the set A×AA\times A given by (x,y)R(z,w)(x,y)R(z,w) if and only if x divides z and y≤wy\le w. Then the number of elements in R is ______.
Not attempted
Your answer: —
Correct answer: 120
201
[JEE Main 2026, 2 Apr Shift 1, Q11] If sin⁡(π18)sin⁡(5π18)sin⁡(7π18)=K\sin\left(\frac{\pi}{18}\right)\sin\left(\frac{5\pi}{18}\right)\sin\left(\frac{7\pi}{18}\right)=K, then the value of sin⁡(10Kπ3)\sin\left(\frac{10K\pi}{3}\right) is:
Not attempted
A
3+122\frac{\sqrt3+1}{2\sqrt2}
B
3−12\frac{\sqrt3-1}{\sqrt2}
C
32\frac{\sqrt3}{2}
D
12\frac12
202
[JEE Main 2026, 2 Apr Shift 1, Q12] Let S={x∈[−π,π]:sin⁡x(sin⁡x+cos⁡x)=a, a∈Z}S=\{x\in[-\pi,\pi]:\sin x(\sin x+\cos x)=a,\ a\in\mathbb{Z}\}. Then n(S)n(S) is equal to:
Not attempted
A
33
B
66
C
77
D
99
203
[JEE Main 2026, 2 Apr Shift 1, Q8] If the mean of the data Class5−1010−1515−2020−2525−3030−35Frequency2k2854k+15\displaystyle\begin{array}{|c|c|c|c|c|c|c|}\hline \text{Class} & 5-10 & 10-15 & 15-20 & 20-25 & 25-30 & 30-35 \\ \hline \text{Frequency} & 2 & k & 28 & 54 & k+1 & 5 \\ \hline \end{array} is 21, then k is one of the roots of the equation:
Not attempted
A
2x2−23x−10=02x^2-23x-10=0
B
4x2−35x+24=04x^2-35x+24=0
C
2x2−19x−10=02x^2-19x-10=0
D
2x2−35x+98=02x^2-35x+98=0
204
[JEE Main 2026, 2 Apr Shift 1, Q23] Let a,b,c∈{1,2,3,4}a,b,c\in\{1,2,3,4\}. If the probability, that ax2+22bx+c>0ax^2+2\sqrt2bx+c>0 for all x∈Rx\in\mathbb{R}, is mn\frac mn, gcd⁡(m,n)=1\gcd(m,n)=1, then m+nm+n is equal to ______.
Not attempted
Your answer: —
Correct answer: 81
205
[JEE Main 2026, 2 Apr Shift 1, Q14] If a⃗\vec a and b⃗\vec b are two vectors such that ∣a⃗∣=2|\vec a|=2 and ∣b⃗∣=3|\vec b|=3, then the maximum value of 3∣3a⃗+2b⃗∣+4∣3a⃗−2b⃗∣3|3\vec a+2\vec b|+4|3\vec a-2\vec b| is:
Not attempted
A
3030
B
3636
C
6060
D
7272
206
[JEE Main 2026, 2 Apr Shift 1, Q13] If the point of intersection of the lines x+13=y+a5=z+b+17\frac{x+1}{3}=\frac{y+a}{5}=\frac{z+b+1}{7} and x−21=y−b4=z−2a7\frac{x-2}{1}=\frac{y-b}{4}=\frac{z-2a}{7} lies on xy-plane, then the value of a+ba+b is:
Not attempted
A
22
B
55
C
77
D
99
207
[JEE Main 2026, 2 Apr Shift 1, Q15] Let a line L passing through the point (1,1,1)(1,1,1) be perpendicular to both the vectors 2i^+2j^+k^2\hat i+2\hat j+\hat k and i^+2j^+2k^\hat i+2\hat j+2\hat k. If P(a,b,c)P(a,b,c) is the foot of perpendicular from the origin on the line L, then the value of 34(a+b+c)34(a+b+c) is:
Not attempted
A
5050
B
8080
C
100100
D
120120
208
[JEE Main 2026, 2 Apr Shift 1, Q9] Let the mid points of the sides of a triangle ABC be (52,7)\left(\frac52,7\right), (52,3)\left(\frac52,3\right) and (4,5)(4,5). If its incentre is (h,k)(h,k), then 3h+k3h+k is equal to:
Not attempted
A
1111
B
1212
C
1313
D
1414
209
[JEE Main 2026, 2 Apr Shift 1, Q10] Let an ellipse x2a2+y2b2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a<ba<b, pass through the point (4,3)(4,3) and have eccentricity 53\frac{\sqrt5}{3}. Then the length of its latus rectum is:
Not attempted
A
453\frac{4\sqrt5}{3}
B
252\sqrt5
C
753\frac{7\sqrt5}{3}
D
853\frac{8\sqrt5}{3}
210
[JEE Main 2026, 2 Apr Shift 1, Q24] Let a circle C have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of C on the line x+y=1x+y=1 is 14\sqrt{14}, then the square of the radius of C is ______.
Not attempted
Your answer: —
Correct answer: 8
211
[JEE Main 2026, 2 Apr Shift 1, Q17] If the curve y=f(x)y=f(x) passes through the point (1,e)(1,e) and satisfies the differential equation dy=y(2+log⁡ex)dxdy=y(2+\log_e x)dx, x>0x>0, then f(e)f(e) is equal to:
Not attempted
A
eee^e
B
ee2e^{e^2}
C
e2ee^{2e}
D
e2ee^{2^e}
212
[JEE Main 2026, 2 Apr Shift 1, Q20] Let y=y(x)y=y(x) be the solution curve of the differential equation (1+sin⁡x)dydx+(y+1)cos⁡x=0(1+\sin x)\frac{dy}{dx}+(y+1)\cos x=0, y(0)=0y(0)=0. If the curve y=y(x)y=y(x) passes through the point (α,−12)\left(\alpha,-\frac12\right), then a value of α\alpha is:
Not attempted
A
π6\frac\pi6
B
π4\frac\pi4
C
π3\frac\pi3
D
π2\frac\pi2
213
[JEE Main 2026, 2 Apr Shift 1, Q19] Let [⋅][\cdot] denote the greatest integer function. Then the value of ∫03(ex+e−x[x]!)dx\int_0^3\left(\frac{e^x+e^{-x}}{[x]!}\right)dx is:
Not attempted
A
e2+e3−1e2−1e3e^2+e^3-\frac1{e^2}-\frac1{e^3}
B
12(e2+e3−1e2−1e3)\frac12\left(e^2+e^3-\frac1{e^2}-\frac1{e^3}\right)
C
e2+e3−12e2−12e3e^2+e^3-\frac1{2e^2}-\frac1{2e^3}
D
12(e2+e3)−1e2−1e3\frac12(e^2+e^3)-\frac1{e^2}-\frac1{e^3}
214
[JEE Main 2026, 2 Apr Shift 1, Q25] If α=∫023log⁡2(x2+4) dx+∫242x−4 dx\alpha=\int_0^{2\sqrt3}\log_2(x^2+4)\,dx+\int_2^4\sqrt{2^x-4}\,dx, then α2\alpha^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 192
215
[JEE Main 2026, 2 Apr Shift 1, Q16] If lim⁡x→2sin⁡(x3−5x2+ax+b)(x−1−1)log⁡e(x−1)=m\lim_{x\to2}\frac{\sin(x^3-5x^2+ax+b)}{(\sqrt{x-1}-1)\log_e(x-1)}=m, then a+b+ma+b+m is equal to:
Not attempted
A
55
B
66
C
88
D
1010
216
[JEE Main 2026, 2 Apr Shift 1, Q18] The number of critical points of the function f(x)=∣sin⁡xx∣f(x)=\left|\frac{\sin x}{x}\right| for x≠0x\ne0, f(0)=1f(0)=1, in the interval (−2π,2π)(-2\pi,2\pi) is equal to:
Not attempted
A
11
B
33
C
55
D
77
217
[JEE Main 2026, 2 Apr Shift 1, Q5] Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in A∩BA\cap B, which are divisible by 3, is:
Not attempted
A
44
B
55
C
66
D
77
218
[JEE Main 2026, 2 Apr Shift 1, Q22] If ∑k=1nak=6n3\sum_{k=1}^{n}a_k=6n^3, then ∑k=16(ak+1−ak36)2\sum_{k=1}^{6}\left(\frac{a_{k+1}-a_k}{36}\right)^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 91
219
[JEE Main 2026, 2 Apr Shift 1, Q7] The number of elements in the set S={(r,k):k∈Z and 36Cr+1=6 (35Cr)k2−3}S=\left\{(r,k):k\in\mathbb{Z}\text{ and }{}^{36}C_{r+1}=\frac{6\,({}^{35}C_r)}{k^2-3}\right\}, is:
Not attempted
A
22
B
44
C
88
D
1616
220
[JEE Main 2026, 2 Apr Shift 1, Q6] The number of seven-digit numbers, that can be formed by using the digits 1, 2, 3, 5 and 7 such that each digit is used at least once, is:
Not attempted
A
1540015400
B
1780017800
C
1680016800
D
2940029400
221
[JEE Main 2026, 2 Apr Shift 1, Q3] Let α,β∈R\alpha,\beta\in\mathbb{R} be such that the system of linear equations x+2y+z=5x+2y+z=5 2x+y+αz=52x+y+\alpha z=5 8x+4y+βz=188x+4y+\beta z=18 has no solution. Then βα\frac\beta\alpha is equal to:
Not attempted
A
−4-4
B
44
C
88
D
−8-8
222
[JEE Main 2026, 2 Apr Shift 1, Q4] Let A=[121α]A=\begin{bmatrix}1&2\\1&\alpha\end{bmatrix} and B=[33β2]B=\begin{bmatrix}3&3\\\beta&2\end{bmatrix}. If A2−4A+I=OA^2-4A+I=O and B2−5B−6I=OB^2-5B-6I=O, then among the two statements: (S1): [(B−A)(B+A)]T=[1315710][(B-A)(B+A)]^T=\begin{bmatrix}13&15\\7&10\end{bmatrix} and (S2): det⁡(adj(A+B))=−5\det(\mathrm{adj}(A+B))=-5,
Not attempted
A
only (S1) is correct
B
only (S2) is correct
C
both (S1) and (S2) are correct
D
both (S1) and (S2) are wrong
223
[JEE Main 2026, 2 Apr Shift 1, Q1] Let α,α+2\alpha,\alpha+2, α∈Z\alpha\in\mathbb{Z}, be the roots of the quadratic equation x(x+2)+(x+1)(x+3)+(x+2)(x+4)+⋯+(x+n−1)(x+n+1)=4nx(x+2)+(x+1)(x+3)+(x+2)(x+4)+\dots+(x+n-1)(x+n+1)=4n for some n∈Nn\in\mathbb{N}. Then n+αn+\alpha is equal to:
Not attempted
A
00
B
11
C
22
D
33
224
[JEE Main 2026, 2 Apr Shift 1, Q2] Let x and y be real numbers such that 50(2x1+3i−y1−2i)=31+17i50\left(\frac{2x}{1+3i}-\frac{y}{1-2i}\right)=31+17i, i=−1i=\sqrt{-1}. Then the value of 10(x−3y)10(x-3y) is:
Not attempted
A
2020
B
3131
C
3535
D
7575
225
[JEE Main 2026, 2 Apr Shift 1, Q21] If the domain of the function f(x)=log⁡(0.6)(∣2x−5x2−4∣)f(x)=\sqrt{\log_{(0.6)}\left(\left|\frac{2x-5}{x^2-4}\right|\right)} is (−∞,a]∪{b}∪[c,d)∪(e,∞)(-\infty,a]\cup\{b\}\cup[c,d)\cup(e,\infty), then the value of a+b+c+d+ea+b+c+d+e is ______.
Not attempted
Your answer: —
Correct answer: 4
226
[JEE Main 2026, 28 Jan Shift 2, Q17] The sum of all the elements in the range of f(x)=Sgn(sin⁡x)+Sgn(cos⁡x)+Sgn(tan⁡x)+Sgn(cot⁡x)f(x)=\mathrm{Sgn}(\sin x)+\mathrm{Sgn}(\cos x)+\mathrm{Sgn}(\tan x)+\mathrm{Sgn}(\cot x), x≠nπ2x\ne\frac{n\pi}{2}, n∈Zn\in\mathbb{Z}, where Sgn(t)=1\mathrm{Sgn}(t)=1 if t>0t>0 and Sgn(t)=−1\mathrm{Sgn}(t)=-1 if t<0t<0, is:
Not attempted
A
22
B
44
C
−2-2
D
00
227
[JEE Main 2026, 28 Jan Shift 2, Q18] Considering the principal values of inverse trigonometric functions, the value of the expression tan⁡(2sin⁡−1213−2cos⁡−1310)\tan\left(2\sin^{-1}\frac{2}{\sqrt{13}}-2\cos^{-1}\frac{3}{\sqrt{10}}\right) is equal to:
Not attempted
A
−3356-\frac{33}{56}
B
1663\frac{16}{63}
C
3356\frac{33}{56}
D
−1663-\frac{16}{63}
228
[JEE Main 2026, 28 Jan Shift 2, Q4] The probability distribution of a random variable X is given below: X4k307k327k347k367k387k407k6kP(X)2151152151511521515115\displaystyle\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline X & 4k & \frac{30}{7}k & \frac{32}{7}k & \frac{34}{7}k & \frac{36}{7}k & \frac{38}{7}k & \frac{40}{7}k & 6k \\ \hline P(X) & \frac{2}{15} & \frac{1}{15} & \frac{2}{15} & \frac15 & \frac{1}{15} & \frac{2}{15} & \frac15 & \frac{1}{15} \\ \hline \end{array} If E(X)=26315E(X)=\frac{263}{15}, then P(X<20)P(X<20) is equal to:
Not attempted
A
1415\frac{14}{15}
B
815\frac{8}{15}
C
1115\frac{11}{15}
D
35\frac35
229
[JEE Main 2026, 28 Jan Shift 2, Q2] Let P be a point in the plane of the vectors AB⃗=3i^+j^−k^\vec{AB}=3\hat i+\hat j-\hat k and AC⃗=i^−j^+3k^\vec{AC}=\hat i-\hat j+3\hat k such that P is equidistant from the lines AB and AC. If ∣AP⃗∣=52|\vec{AP}|=\frac{\sqrt5}{2}, then the area of the triangle ABP is:
Not attempted
A
32\frac32
B
264\frac{\sqrt{26}}{4}
C
22
D
304\frac{\sqrt{30}}{4}
230
[JEE Main 2026, 28 Jan Shift 2, Q7] Let Q(a,b,c)Q(a,b,c) be the image of the point P(3,2,1)P(3,2,1) in the line x−11=y2=z−11\frac{x-1}{1}=\frac y2=\frac{z-1}{1}. Then the distance of Q from the line x−93=y−92=z−5−2\frac{x-9}{3}=\frac{y-9}{2}=\frac{z-5}{-2} is
Not attempted
A
88
B
77
C
66
D
55
231
[JEE Main 2026, 28 Jan Shift 2, Q25] If the distance of the point P(43,α,β)P(43,\alpha,\beta), β<0\beta<0, from the line r⃗=4i^−k^+μ(2i^+3k^)\vec r=4\hat i-\hat k+\mu(2\hat i+3\hat k), μ∈R\mu\in\mathbb{R} along a line with direction ratios 3,−1,03,-1,0 is 131013\sqrt{10}, then α2+β2\alpha^2+\beta^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 170
232
[JEE Main 2026, 28 Jan Shift 2, Q8] Let the ellipse E:x2144+y2169=1E:\frac{x^2}{144}+\frac{y^2}{169}=1 and the hyperbola H:x216−y2λ2=−1H:\frac{x^2}{16}-\frac{y^2}{\lambda^2}=-1 have the same foci. If e and L respectively denote the eccentricity and the length of the latus rectum of H, then the value of 24(e+L)24(e+L) is:
Not attempted
A
148148
B
126126
C
296296
D
6767
233
[JEE Main 2026, 28 Jan Shift 2, Q10] An ellipse has its center at (1,−2)(1,-2), one focus at (3,−2)(3,-2) and one vertex at (5,−2)(5,-2). Then the length of its latus rectum is:
Not attempted
A
163\frac{16}{\sqrt3}
B
434\sqrt3
C
66
D
636\sqrt3
234
[JEE Main 2026, 28 Jan Shift 2, Q15] Let A be the focus of the parabola y2=8xy^2=8x. Let the line y=mx+cy=mx+c intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is (73,43)\left(\frac73,\frac43\right), then (BC)2(BC)^2 is equal to:
Not attempted
A
4141
B
8989
C
8080
D
3232
235
[JEE Main 2026, 28 Jan Shift 2, Q16] Let the circle x2+y2=4x^2+y^2=4 intersect x-axis at the points A(a,0)A(a,0), a>0a>0 and B(b,0)B(b,0). Let P(2cos⁡α,2sin⁡α)P(2\cos\alpha,2\sin\alpha), 0<α<π20<\alpha<\frac\pi2 and Q(2cos⁡β,2sin⁡β)Q(2\cos\beta,2\sin\beta) be two points such that (α−β)=π2(\alpha-\beta)=\frac\pi2. Then the point of intersection of AQ and BP lies on:
Not attempted
A
x2+y2−4y−4=0x^2+y^2-4y-4=0
B
x2+y2−4x−4y=0x^2+y^2-4x-4y=0
C
x2+y2−4x−4y−4=0x^2+y^2-4x-4y-4=0
D
x2+y2−4x−4=0x^2+y^2-4x-4=0
236
[JEE Main 2026, 28 Jan Shift 2, Q6] Let y=y(x)y=y(x) be the solution of the differential equation xdydx−y=x2cot⁡xx\frac{dy}{dx}-y=x^2\cot x, x∈(0,π)x\in(0,\pi). If y(π2)=π2y\left(\frac\pi2\right)=\frac\pi2, then 6y(π6)−8y(π4)6y\left(\frac\pi6\right)-8y\left(\frac\pi4\right) is equal to:
Not attempted
A
π\pi
B
−3π-3\pi
C
3π3\pi
D
−π-\pi
237
[JEE Main 2026, 28 Jan Shift 2, Q1] Let f(x)=∫dxx2/3+2x1/2f(x)=\int\frac{dx}{x^{2/3}+2x^{1/2}} be such that f(0)=−26+24log⁡e2f(0)=-26+24\log_e 2. If f(1)=a+blog⁡e3f(1)=a+b\log_e 3, where a,b∈Za,b\in\mathbb{Z}, then a+ba+b is equal to:
Not attempted
A
−26-26
B
−18-18
C
−5-5
D
−11-11
238
[JEE Main 2026, 28 Jan Shift 2, Q3] Let [⋅][\cdot] denote the greatest integer function. Then ∫−π/2π/212(3+[x])3+[sin⁡x]+[cos⁡x] dx\int_{-\pi/2}^{\pi/2}\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\,dx is equal to:
Not attempted
A
13π+113\pi+1
B
12π+512\pi+5
C
11π+211\pi+2
D
15π+415\pi+4
239
[JEE Main 2026, 28 Jan Shift 2, Q5] Let P1:y=4x2P_1:y=4x^2 and P2:y=x2+27P_2:y=x^2+27 be two parabolas. If the area of the bounded region enclosed between P1P_1 and P2P_2 is six times the area of the bounded region enclosed between the line y=αxy=\alpha x, α>0\alpha>0 and P1P_1, then α\alpha is equal to:
Not attempted
A
1515
B
1212
C
88
D
66
240
[JEE Main 2026, 28 Jan Shift 2, Q11] Let f(x)=lim⁡θ→0(cos⁡πx−x(2/θ)sin⁡(x−1)1+x(2/θ)(x−1))f(x)=\lim_{\theta\to0}\left(\frac{\cos\pi x-x^{(2/\theta)}\sin(x-1)}{1+x^{(2/\theta)}(x-1)}\right), x∈Rx\in\mathbb{R}. Consider the following two statements: (I) f(x)f(x) is discontinuous at x=1x=1. (II) f(x)f(x) is continuous at x=−1x=-1. Then,
Not attempted
A
Only (I) is True
B
Neither (I) nor (II) is True
C
Only (II) is True
D
Both (I) and (II) are True
241
[JEE Main 2026, 28 Jan Shift 2, Q21] Let f be a differentiable function satisfying f(x)=1−2x+∫0xe(x−t)f(t) dtf(x)=1-2x+\int_0^x e^{(x-t)}f(t)\,dt, x∈Rx\in\mathbb{R} and let g(x)=∫0x(f(t)+2)15(t−4)6(t+12)17dtg(x)=\int_0^x(f(t)+2)^{15}(t-4)^6(t+12)^{17}dt, x∈Rx\in\mathbb{R}. If p and q are respectively the points of local minima and local maxima of g, then the value of ∣p+q∣|p+q| is equal to ______.
Not attempted
Your answer: —
Correct answer: 9
242
[JEE Main 2026, 28 Jan Shift 2, Q12] 6326+10⋅1325+10⋅2324+10⋅22323+⋯+10⋅2243\frac{6}{3^{26}}+\frac{10\cdot1}{3^{25}}+\frac{10\cdot2}{3^{24}}+\frac{10\cdot2^2}{3^{23}}+\dots+\frac{10\cdot2^{24}}{3} is equal to:
Not attempted
A
2252^{25}
B
3253^{25}
C
3263^{26}
D
2262^{26}
243
[JEE Main 2026, 28 Jan Shift 2, Q24] If ∑r=125(rr4+r2+1)=pq\sum_{r=1}^{25}\left(\frac{r}{r^4+r^2+1}\right)=\frac pq, where p and q are positive integers such that gcd⁡(p,q)=1\gcd(p,q)=1, then p+qp+q is equal to ______.
Not attempted
Your answer: —
Correct answer: 976
244
[JEE Main 2026, 28 Jan Shift 2, Q13] The sum of the coefficients of x499x^{499} and x500x^{500} in (1+x)1000+x(1+x)999+x2(1+x)998+⋯+x1000(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\dots+x^{1000} is:
Not attempted
A
1000C501{}^{1000}C_{501}
B
1002C501{}^{1002}C_{501}
C
1001C501{}^{1001}C_{501}
D
1002C500{}^{1002}C_{500}
245
[JEE Main 2026, 28 Jan Shift 2, Q19] Given below are two statements: Statement I: 2513+2013+813+31325^{13}+20^{13}+8^{13}+3^{13} is divisible by 7. Statement II: The integral part of (7+43)25(7+4\sqrt3)^{25} is an odd number. In the light of the above statements, choose the correct answer from the options given below:
Not attempted
A
Statement I is false but Statement II is true
B
Both Statement I and Statement II are false
C
Statement I is true but Statement II is false
D
Both Statement I and Statement II are true
246
[JEE Main 2026, 28 Jan Shift 2, Q22] Three persons enter in a lift at the ground floor. The lift will go upto 10th10^{th} floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to ______.
Not attempted
Your answer: —
Correct answer: 210
247
[JEE Main 2026, 28 Jan Shift 2, Q23] Let A=[3−41−1]A=\begin{bmatrix}3&-4\\1&-1\end{bmatrix} and B be two matrices such that A100=100B+IA^{100}=100B+I. Then the sum of all the elements of B100B^{100} is ______.
Not attempted
Your answer: —
Correct answer: 0
248
[JEE Main 2026, 28 Jan Shift 2, Q14] Let A={z∈C:∣z−2∣≤4}A=\{z\in\mathbb{C}:|z-2|\le4\} and B={z∈C:∣z−2∣+∣z+2∣=5}B=\{z\in\mathbb{C}:|z-2|+|z+2|=5\}. Then max⁡{∣z1−z2∣:z1∈A and z2∈B}\max\{|z_1-z_2|:z_1\in A\text{ and }z_2\in B\} is:
Not attempted
A
172\frac{17}{2}
B
88
C
152\frac{15}{2}
D
99
249
[JEE Main 2026, 28 Jan Shift 2, Q20] Let the arithmetic mean of 1a\frac1a and 1b\frac1b be 516\frac{5}{16}, a>2a>2. If α\alpha is such that a,4,α,ba,4,\alpha,b are in A.P., then the equation αx2−ax+2(α−2b)=0\alpha x^2-ax+2(\alpha-2b)=0 has:
Not attempted
A
one root in (0,2)(0,2) and another in (−4,−2)(-4,-2)
B
one root in (1,4)(1,4) and another in (−2,0)(-2,0)
C
complex roots of magnitude less than 2
D
both roots in the interval (−2,0)(-2,0)
250
[JEE Main 2026, 28 Jan Shift 2, Q9] Given below are two statements: Statement I: The function f:R→Rf:\mathbb{R}\to\mathbb{R} defined by f(x)=x1+∣x∣f(x)=\frac{x}{1+|x|} is one-one. Statement II: The function f:R→Rf:\mathbb{R}\to\mathbb{R} defined by f(x)=x2+4x−30x2−8x+18f(x)=\frac{x^2+4x-30}{x^2-8x+18} is many-one. In the light of the above statements, choose the correct answer from the options given below:
Not attempted
A
Statement I is true but Statement II is false
B
Statement I is false but Statement II is true
C
Both Statement I and Statement II are false
D
Both Statement I and Statement II are true
251
[JEE Main 2026, 28 Jan Shift 1, Q12] If tan⁡(A−B)tan⁡A+sin⁡2Csin⁡2A=1\frac{\tan(A-B)}{\tan A}+\frac{\sin^2C}{\sin^2A}=1, where A,B,C∈(0,π2)A,B,C\in\left(0,\frac\pi2\right), then
Not attempted
A
tan⁡A,tan⁡C,tan⁡B\tan A,\tan C,\tan B are in G.P.
B
tan⁡A,tan⁡C,tan⁡B\tan A,\tan C,\tan B are in A.P.
C
tan⁡A,tan⁡B,tan⁡C\tan A,\tan B,\tan C are in A.P.
D
tan⁡A,tan⁡B,tan⁡C\tan A,\tan B,\tan C are in G.P.
252
[JEE Main 2026, 28 Jan Shift 1, Q23] If k=tan⁡(π4+12cos⁡−123)+tan⁡(12sin⁡−123)k=\tan\left(\frac\pi4+\frac12\cos^{-1}\frac23\right)+\tan\left(\frac12\sin^{-1}\frac23\right), then the number of solutions of the equation sin⁡−1(kx−1)=sin⁡−1x−cos⁡−1x\sin^{-1}(kx-1)=\sin^{-1}x-\cos^{-1}x is ____
Not attempted
Your answer: —
Correct answer: 1
253
[JEE Main 2026, 28 Jan Shift 1, Q1] The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are 2, 3, 5, 10, 11, 13, 15, 21, then the mean deviation about the median of all the 10 observations is
Not attempted
A
55
B
66
C
44
D
77
254
[JEE Main 2026, 28 Jan Shift 1, Q6] A bag contains 10 balls out of which k are red and (10−k)(10-k) are black, where 0≤k≤100\le k\le10. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:
Not attempted
A
7110\frac{7}{110}
B
711\frac{7}{11}
C
1455\frac{14}{55}
D
755\frac{7}{55}
255
[JEE Main 2026, 28 Jan Shift 1, Q8] For three unit vectors a⃗,b⃗,c⃗\vec a,\vec b,\vec c satisfying ∣a⃗−b⃗∣2+∣b⃗−c⃗∣2+∣c⃗−a⃗∣2=9|\vec a-\vec b|^2+|\vec b-\vec c|^2+|\vec c-\vec a|^2=9 and ∣2a⃗+kb⃗+kc⃗∣=3|2\vec a+k\vec b+k\vec c|=3, the positive value of k is
Not attempted
A
33
B
44
C
66
D
55
256
[JEE Main 2026, 28 Jan Shift 1, Q25] Let PQR be a triangle such that PQ→=−2i^−j^+2k^\overrightarrow{PQ}=-2\hat i-\hat j+2\hat k and PR→=ai^+bj^−4k^\overrightarrow{PR}=a\hat i+b\hat j-4\hat k, a,b∈Za,b\in\mathbb{Z}. Let S be the point on QR which is equidistant from the lines PQ and PR. If ∣PR→∣=9|\overrightarrow{PR}|=9 and PS→\overrightarrow{PS} is along i^−7j^+2k^\hat i-7\hat j+2\hat k, then the value of 3a−4b3a-4b is ____
Not attempted
Your answer: —
Correct answer: 37
257
[JEE Main 2026, 28 Jan Shift 1, Q20] If the distances of the point (1,2,a)(1,2,a) from the line x−11=y2=z−11\frac{x-1}{1}=\frac y2=\frac{z-1}{1} along the lines L1:x−13=y−24=z−abL_1:\frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b} and L2:x−11=y−24=z−acL_2:\frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c} are equal, then a+b+ca+b+c is equal to
Not attempted
A
77
B
66
C
44
D
55
258
[JEE Main 2026, 28 Jan Shift 1, Q16] Let y=xy=x be the equation of a chord of the circle C1C_1 (in the closed half-plane x≥0x\ge0) of diameter 10 passing through the origin. Let C2C_2 be another circle described on the given chord as its diameter. If the equation of the chord of the circle C2C_2, which passes through the point (2,3)(2,3) and is farthest from the center of C2C_2, is x+ay+b=0x+ay+b=0, then a−ba-b is equal to
Not attempted
A
−6-6
B
−2-2
C
66
D
1010
259
[JEE Main 2026, 28 Jan Shift 1, Q17] Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line x+22y=4x+2\sqrt2y=4. If the co-ordinates of the vertex A are (α,β)(\alpha,\beta), then the greatest integer less than or equal to ∣α+2β∣|\alpha+\sqrt2\beta| is
Not attempted
A
44
B
55
C
33
D
22
260
[JEE Main 2026, 28 Jan Shift 1, Q22] For some θ∈(0,π2)\theta\in\left(0,\frac\pi2\right), let the eccentricity and the length of the latus rectum of the hyperbola x2−y2sec⁡2θ=8x^2-y^2\sec^2\theta=8 be e1e_1 and l1l_1, respectively, and let the eccentricity and the length of the latus rectum of the ellipse x2sec⁡2θ+y2=6x^2\sec^2\theta+y^2=6 be e2e_2 and l2l_2, respectively. If e12=e22(sec⁡2θ+1)e_1^2=e_2^2(\sec^2\theta+1), then (l1l2e1e2)tan⁡2θ\left(\frac{l_1l_2}{e_1e_2}\right)\tan^2\theta is equal to ____
Not attempted
Your answer: —
Correct answer: 8
261
[JEE Main 2026, 28 Jan Shift 1, Q19] Let y=y(x)y=y(x) be the solution of the differential equation xdydx−sin⁡2y=x3(2−x3)cos⁡2yx\frac{dy}{dx}-\sin2y=x^3(2-x^3)\cos^2y, x≠0x\ne0. If y(2)=0y(2)=0, then tan⁡(y(1))\tan(y(1)) is equal to
Not attempted
A
34\frac34
B
74\frac74
C
−34-\frac34
D
−74-\frac74
262
[JEE Main 2026, 28 Jan Shift 1, Q9] If ∫(1−5cos⁡2xsin⁡5xcos⁡2x)dx=f(x)+C\int\left(\frac{1-5\cos^2x}{\sin^5x\cos^2x}\right)dx=f(x)+C, where C is the constant of integration, then f(π6)−f(π4)f\left(\frac\pi6\right)-f\left(\frac\pi4\right) is equal to
Not attempted
A
23(4+6)\frac{2}{\sqrt3}(4+\sqrt6)
B
13(26−3)\frac{1}{\sqrt3}(26-\sqrt3)
C
13(26+3)\frac{1}{\sqrt3}(26+\sqrt3)
D
43(8−6)\frac{4}{\sqrt3}(8-\sqrt6)
263
[JEE Main 2026, 28 Jan Shift 1, Q14] Find the area of the region R={(x,y):xy≤8, 1≤y≤x2, x≥0}R=\{(x,y):xy\le8,\ 1\le y\le x^2,\ x\ge0\}
Not attempted
A
23(20log⁡e2+9)\frac23(20\log_e2+9)
B
13(40log⁡e2+27)\frac13(40\log_e2+27)
C
23(24log⁡e2−7)\frac23(24\log_e2-7)
D
13(49log⁡e2−15)\frac13(49\log_e2-15)
264
[JEE Main 2026, 28 Jan Shift 1, Q18] Let f be a polynomial function such that f(x2+1)=x4+5x2+2f(x^2+1)=x^4+5x^2+2, for all x∈Rx\in\mathbb{R}. Then ∫03f(x) dx\int_0^3f(x)\,dx is equal to
Not attempted
A
413\frac{41}{3}
B
272\frac{27}{2}
C
332\frac{33}{2}
D
53\frac53
265
[JEE Main 2026, 28 Jan Shift 1, Q24] The value of ∑r=120∣π(∫0rx∣sin⁡πx∣dx)∣\sum_{r=1}^{20}\left|\sqrt{\pi\left(\int_0^rx|\sin\pi x|dx\right)}\right| is ____
Not attempted
Your answer: —
Correct answer: 210
266
[JEE Main 2026, 28 Jan Shift 1, Q7] The value of lim⁡x→0log⁡e(sec⁡(ex)⋅sec⁡(e2x)⋯sec⁡(e10x))e2−e2cos⁡x\lim_{x\to0}\frac{\log_e(\sec(ex)\cdot\sec(e^2x)\cdots\sec(e^{10}x))}{e^2-e^{2\cos x}} is equal to
Not attempted
A
e10−12e2(e2−1)\frac{e^{10}-1}{2e^2(e^2-1)}
B
e20−12(e2−1)\frac{e^{20}-1}{2(e^2-1)}
C
e20−12e2(e2−1)\frac{e^{20}-1}{2e^2(e^2-1)}
D
e10−12(e2−1)\frac{e^{10}-1}{2(e^2-1)}
267
[JEE Main 2026, 28 Jan Shift 1, Q11] The value of ∑k=1∞(−1)k+1(k(k+1)k!)\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{k(k+1)}{k!}\right) is
Not attempted
A
2e\frac2e
B
e\sqrt e
C
e2\frac e2
D
1e\frac1e
268
[JEE Main 2026, 28 Jan Shift 1, Q13] The common difference of the A.P.: a1,a2,…,ama_1,a_2,\ldots,a_m is 13 more than the common difference of the A.P. b1,b2,…,bnb_1,b_2,\ldots,b_n. If b31=−277b_{31}=-277, b43=−385b_{43}=-385 and a78=327a_{78}=327, then a1a_1 is equal to
Not attempted
A
1616
B
1919
C
2424
D
2121
269
[JEE Main 2026, 28 Jan Shift 1, Q21] In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is R−(a,b)\mathbb{R}-(a,b), then a2+b2a^2+b^2 is equal to ____ .
Not attempted
Your answer: —
Correct answer: 90
270
[JEE Main 2026, 28 Jan Shift 1, Q15] Let S={1,2,…,9}S=\{1,2,\ldots,9\}. Let x be the number of 9-digit numbers formed using the digits of S such that only one digit is repeated and it is repeated exactly twice. Let y be the number of 9-digit numbers formed using the digits of S such that only two digits are repeated and each of these is repeated exactly twice. Then,
Not attempted
A
45x=7y45x=7y
B
21x=4y21x=4y
C
29x=5y29x=5y
D
56x=9y56x=9y
271
[JEE Main 2026, 28 Jan Shift 1, Q10] Let A, B and C be three 2×22\times2 matrices with real entries such that B=(I+A)−1B=(I+A)^{-1} and A+C=IA+C=I. If BC=[1−5−12]BC=\begin{bmatrix}1&-5\\-1&2\end{bmatrix} and CB[x1x2]=[12−6]CB\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}12\\-6\end{bmatrix}, then x1+x2x_1+x_2 is
Not attempted
A
−2-2
B
00
C
44
D
22
272
[JEE Main 2026, 28 Jan Shift 1, Q2] Let S={x3+ax2+bx+c:a,b,c∈N and a,b,c≤20}S=\{x^3+ax^2+bx+c:a,b,c\in\mathbb{N}\text{ and }a,b,c\le20\} be a set of polynomials. Then the number of polynomials in S which are divisible by x2+2x^2+2 is
Not attempted
A
1010
B
120120
C
66
D
2020
273
[JEE Main 2026, 28 Jan Shift 1, Q4] Let z be a complex number such that ∣z−6∣=5|z-6|=5 and ∣z+2−6i∣=5|z+2-6i|=5. Then the value of z3+3z2−15z+141z^3+3z^2-15z+141 is equal to
Not attempted
A
3737
B
4242
C
5050
D
6161
274
[JEE Main 2026, 28 Jan Shift 1, Q5] If α,β\alpha,\beta, where α<β\alpha<\beta, are the roots of the equation λx2−(λ+3)x+3=0\lambda x^2-(\lambda+3)x+3=0 such that 1α−1β=13\frac1\alpha-\frac1\beta=\frac13, then the sum of all possible values of λ\lambda is
Not attempted
A
44
B
22
C
66
D
88
275
[JEE Main 2026, 28 Jan Shift 1, Q3] If g(x)=3x2+2x−3g(x)=3x^2+2x-3, f(0)=−3f(0)=-3 and 4g(f(x))=3x2−32x+724g(f(x))=3x^2-32x+72, then f(g(2))f(g(2)) is equal to:
Not attempted
A
−256-\frac{25}{6}
B
72\frac72
C
−72-\frac72
D
256\frac{25}{6}
276
[JEE Main 2026, 24 Jan Shift 2, Q23] The number of elements in the set {x∈[0,180∘]:tan⁡(x+100∘)=tan⁡(x+50∘)tan⁡xtan⁡(x−50∘)}\{x\in[0,180^\circ]:\tan(x+100^\circ)=\tan(x+50^\circ)\tan x\tan(x-50^\circ)\} is
Not attempted
Your answer: —
Correct answer: 4
277
[JEE Main 2026, 24 Jan Shift 2, Q10] Let X={x∈N:1≤x≤19}X=\{x\in\mathbb{N}:1\le x\le19\} and for some a,b∈Ra,b\in\mathbb{R}, define Y={ax+b:x∈X}Y=\{ax+b:x\in X\}. If the mean and variance of the elements of Y are 30 and 750 respectively, find the sum of all possible values of b.
Not attempted
A
2020
B
100100
C
8080
D
6060
278
[JEE Main 2026, 24 Jan Shift 2, Q25] Let S be a set of 5 elements and P(S) denote the power set of S. Let E be an event of choosing an ordered pair (A, B) from the set P(S)×P(S)P(S)\times P(S) such that A∩B=∅A\cap B=\emptyset. If the probability of the event E is 3p2q\frac{3^p}{2^q}, where p,q∈Np,q\in\mathbb{N}, then p+qp+q is equal to
Not attempted
Your answer: —
Correct answer: 15
279
[JEE Main 2026, 24 Jan Shift 2, Q7] Let a⃗=2i^−5j^+5k^\vec a=2\hat i-5\hat j+5\hat k and b⃗=i^−j^+3k^\vec b=\hat i-\hat j+3\hat k. If c⃗\vec c is a vector such that 2(a⃗×c⃗)+3(b⃗×c⃗)=0⃗2(\vec a\times\vec c)+3(\vec b\times\vec c)=\vec 0 and (a⃗−b⃗)⋅c⃗=−97(\vec a-\vec b)\cdot\vec c=-97, then ∣c⃗×k^∣2|\vec c\times\hat k|^2 is equal to
Not attempted
A
233233
B
205205
C
193193
D
218218
280
[JEE Main 2026, 24 Jan Shift 2, Q19] The sum of all values of α\alpha, for which the shortest distance between the lines x+1α=y−2−1=z−4−α\frac{x+1}{\alpha}=\frac{y-2}{-1}=\frac{z-4}{-\alpha} and xα=y−12=z−12α\frac{x}{\alpha}=\frac{y-1}{2}=\frac{z-1}{2\alpha} is 2\sqrt2, is
Not attempted
A
−8-8
B
66
C
−6-6
D
88
281
[JEE Main 2026, 24 Jan Shift 2, Q9] Let the image of parabola x2=4yx^2=4y, in the line x−y=1x-y=1 be (y+a)2=b(x−c)(y+a)^2=b(x-c), a,b,c∈Na,b,c\in\mathbb{N}. Then a+b+ca+b+c is equal to
Not attempted
A
44
B
88
C
1212
D
66
282
[JEE Main 2026, 24 Jan Shift 2, Q11] Let the length of the latus rectum of an ellipse x2a2+y2b2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, (a>b)(a>b), be 30. If its eccentricity is the maximum value of the function f(t)=−34+2t−t2f(t)=-\frac34+2t-t^2, then (a2+b2)(a^2+b^2) is equal to
Not attempted
A
516516
B
276276
C
256256
D
496496
283
[JEE Main 2026, 24 Jan Shift 2, Q20] Let the angles made with the positive x-axis by two straight lines drawn from the point P(2,3)P(2,3) and meeting the line x+y=6x+y=6 at a distance 23\sqrt{\frac23} from the point P be θ1\theta_1 and θ2\theta_2. Then the value of (θ1+θ2)(\theta_1+\theta_2) is:
Not attempted
A
π6\frac{\pi}{6}
B
π3\frac{\pi}{3}
C
π12\frac{\pi}{12}
D
π2\frac{\pi}{2}
284
[JEE Main 2026, 24 Jan Shift 2, Q21] Let (h,k)(h,k) lie on the circle C:x2+y2=4C:x^2+y^2=4 and the point (2h+1,3k+2)(2h+1,3k+2) lie on an ellipse with eccentricity e. Then the value of 5e2\frac{5}{e^2} is equal to
Not attempted
Your answer: —
Correct answer: 9
285
[JEE Main 2026, 24 Jan Shift 2, Q16] Let y=y(x)y=y(x) be a differentiable function in the interval (0,∞)(0,\infty) such that y(1)=2y(1)=2, and lim⁡t→x(t2y(x)−x2y(t)x−t)=3\lim_{t\to x}\left(\frac{t^2y(x)-x^2y(t)}{x-t}\right)=3 for each x>0x>0. Then 2y(2)2y(2) is equal to
Not attempted
A
1818
B
1212
C
2323
D
2727
286
[JEE Main 2026, 24 Jan Shift 2, Q15] Let f(α)f(\alpha) denote the area of the region in the first quadrant bounded by x=0x=0, x=1x=1, y2=xy^2=x and y=∣αx−5∣−∣1−αx∣+αx2y=|\alpha x-5|-|1-\alpha x|+\alpha x^2. Then (f(0)+f(1))(f(0)+f(1)) is equal to
Not attempted
A
1212
B
77
C
99
D
1414
287
[JEE Main 2026, 24 Jan Shift 2, Q22] If f(x)f(x) satisfies the relation f(x)=ex+∫01(y+xex)f(y) dyf(x)=e^x+\int_0^1(y+xe^x)f(y)\,dy, then e+f(0)e+f(0) is equal to
Not attempted
Your answer: —
Correct answer: 2
288
[JEE Main 2026, 24 Jan Shift 2, Q2] Consider the following three statements for the function f:(0,∞)→Rf:(0,\infty)\to\mathbb{R} defined by f(x)=∣log⁡ex∣−∣x−1∣f(x)=|\log_ex|-|x-1|: (I) f is differentiable at all x>0x>0. (II) f is increasing in (0,1)(0,1). (III) f is decreasing in (1,∞)(1,\infty). Then,
Not attempted
A
Only (I) and (III) are TRUE.
B
Only (II) and (III) are TRUE.
C
Only (I) is TRUE.
D
All (I), (II) and (III) are TRUE.
289
[JEE Main 2026, 24 Jan Shift 2, Q12] Let [t][t] denote the greatest integer less than or equal to t. If the function f(x)={b2sin⁡(π2[π2(cos⁡x+sin⁡x)cos⁡x]),x<0sin⁡x−12sin⁡2xx3,x>0a,x=0f(x)=\begin{cases}b^2\sin\left(\frac{\pi}{2}\left[\frac{\pi}{2}(\cos x+\sin x)\cos x\right]\right), & x<0\\ \frac{\sin x-\frac12\sin2x}{x^3}, & x>0\\ a, & x=0\end{cases} is continuous at x=0x=0, then a2+b2a^2+b^2 is equal to
Not attempted
A
12\frac12
B
916\frac{9}{16}
C
34\frac34
D
58\frac58
290
[JEE Main 2026, 24 Jan Shift 2, Q4] The value of (13+47)+(132+13×47+4272)+(133+132×47+13×4272+4373)+…\left(\frac13+\frac47\right)+\left(\frac{1}{3^2}+\frac13\times\frac47+\frac{4^2}{7^2}\right)+\left(\frac{1}{3^3}+\frac{1}{3^2}\times\frac47+\frac13\times\frac{4^2}{7^2}+\frac{4^3}{7^3}\right)+\ldots upto infinite terms, is equal to
Not attempted
A
43\frac43
B
74\frac74
C
65\frac65
D
52\frac52
291
[JEE Main 2026, 24 Jan Shift 2, Q8] Let a1,a2,a3,a4a_1,a_2,a_3,a_4 be an A.P. of four terms such that each term of the A.P. and its common difference ll are integers. If a1+a2+a3+a4=48a_1+a_2+a_3+a_4=48 and a1a2a3a4+l4=361a_1a_2a_3a_4+l^4=361, then the largest term of the A.P. is equal to
Not attempted
A
2323
B
2121
C
2424
D
2727
292
[JEE Main 2026, 24 Jan Shift 2, Q6] The letters of the word \"UDAYPUR\" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word \"UDAYPUR\" is
Not attempted
A
15801580
B
15811581
C
15791579
D
15781578
293
[JEE Main 2026, 24 Jan Shift 2, Q18] The largest value of n, for which 40n40^n divides 60!60!, is
Not attempted
A
1313
B
1414
C
1111
D
1212
294
[JEE Main 2026, 24 Jan Shift 2, Q5] Let P=[pij]P=[p_{ij}] and Q=[qij]Q=[q_{ij}] be two square matrices of order 3 such that qij=2(i+j−1)pijq_{ij}=2^{(i+j-1)}p_{ij} and det⁡(Q)=210\det(Q)=2^{10}. Then the value of det⁡(adj⁡(adj⁡P))\det(\operatorname{adj}(\operatorname{adj}P)) is:
Not attempted
A
3232
B
8181
C
124124
D
1616
295
[JEE Main 2026, 24 Jan Shift 2, Q13] Let f(x)=∫7x10+9x8(1+x2+2x9)2dxf(x)=\int\frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}dx, x>0x>0, lim⁡x→0f(x)=0\lim_{x\to0}f(x)=0 and f(1)=14f(1)=\frac14. If A=[00114f′(1)1α241]A=\begin{bmatrix}0&0&1\\\frac14&f'(1)&1\\\alpha^2&4&1\end{bmatrix} and B=adj⁡(adj⁡A)B=\operatorname{adj}(\operatorname{adj}A) be such that ∣B∣=81|B|=81, then α2\alpha^2 is equal to
Not attempted
A
22
B
11
C
44
D
33
296
[JEE Main 2026, 24 Jan Shift 2, Q1] The smallest positive integral value of aa, for which all the roots of x4−ax2+9=0x^4-ax^2+9=0 are real and distinct, is equal to
Not attempted
A
99
B
77
C
33
D
44
297
[JEE Main 2026, 24 Jan Shift 2, Q24] Let z=(1+i)(1+2i)(1+3i)…(1+ni)z=(1+i)(1+2i)(1+3i)\ldots(1+ni), where i=−1i=\sqrt{-1}. If ∣z∣2=44200|z|^2=44200, then n is equal to
Not attempted
Your answer: —
Correct answer: 5
298
[JEE Main 2026, 24 Jan Shift 2, Q3] Let f be a function such that 3f(x)+2f(m19x)=5x3f(x)+2f\left(\frac{m}{19x}\right)=5x, x≠0x\ne0, where m=∑i=19i2m=\sum_{i=1}^9i^2. Then f(5)−f(2)f(5)-f(2) is equal to
Not attempted
A
99
B
1818
C
3636
D
−9-9
299
[JEE Main 2026, 24 Jan Shift 2, Q17] If the domain of the function f(x)=sin⁡−1(1x2−2x−2)f(x)=\sin^{-1}\left(\frac{1}{x^2-2x-2}\right) is (−∞,α]∪[β,γ]∪[δ,∞)(-\infty,\alpha]\cup[\beta,\gamma]\cup[\delta,\infty), then α+β+γ+δ\alpha+\beta+\gamma+\delta is equal to
Not attempted
A
22
B
33
C
44
D
55
300
[JEE Main 2026, 24 Jan Shift 1, Q1] If cot⁡x=512\cot x=\frac{5}{12} for some x∈(π,3π2)x\in\left(\pi,\frac{3\pi}{2}\right), then sin⁡7x(cos⁡13x2+sin⁡13x2)+cos⁡7x(cos⁡13x2−sin⁡13x2)\sin7x\left(\cos\frac{13x}{2}+\sin\frac{13x}{2}\right)+\cos7x\left(\cos\frac{13x}{2}-\sin\frac{13x}{2}\right) is equal to
Not attempted
A
113\frac{1}{\sqrt{13}}
B
513\frac{5}{\sqrt{13}}
C
626\frac{6}{\sqrt{26}}
D
426\frac{4}{\sqrt{26}}
301
[JEE Main 2026, 24 Jan Shift 1, Q5] The value of 3cosec⁡20∘−sec⁡20∘cos⁡20∘cos⁡40∘cos⁡60∘cos⁡80∘\frac{\sqrt3\operatorname{cosec}20^\circ-\sec20^\circ}{\cos20^\circ\cos40^\circ\cos60^\circ\cos80^\circ} is equal to
Not attempted
A
6464
B
1616
C
1212
D
3232
302
[JEE Main 2026, 24 Jan Shift 1, Q11] From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is
Not attempted
A
7107\frac{7}{10^7}
B
81108\frac{81}{10^8}
C
67108\frac{67}{10^8}
D
73108\frac{73}{10^8}
303
[JEE Main 2026, 24 Jan Shift 1, Q20] The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observation α\alpha in this data is replaced by β\beta, then the mean and variance become 10.1 and 1.99, respectively. Then α+β\alpha+\beta equals
Not attempted
A
1515
B
55
C
2020
D
1010
304
[JEE Main 2026, 24 Jan Shift 1, Q14] Let a⃗=2i^+j^−2k^\vec a=2\hat i+\hat j-2\hat k, b⃗=i^+j^\vec b=\hat i+\hat j and c⃗=a⃗×b⃗\vec c=\vec a\times\vec b. Let d⃗\vec d be a vector such that ∣d⃗−a⃗∣=11|\vec d-\vec a|=\sqrt{11}, ∣c⃗×d⃗∣=3|\vec c\times\vec d|=3 and the angle between c⃗\vec c and d⃗\vec d is π4\frac{\pi}{4}. Then a⃗⋅d⃗\vec a\cdot\vec d is equal to
Not attempted
A
11
B
00
C
1111
D
33
305
[JEE Main 2026, 24 Jan Shift 1, Q13] Let the lines L1:r⃗=i^+2j^+3k^+λ(2i^+3j^+4k^)L_1:\vec r=\hat i+2\hat j+3\hat k+\lambda(2\hat i+3\hat j+4\hat k), λ∈R\lambda\in\mathbb{R} and L2:r⃗=(4i^+j^)+μ(5i^+2j^+k^)L_2:\vec r=(4\hat i+\hat j)+\mu(5\hat i+2\hat j+\hat k), μ∈R\mu\in\mathbb{R}, intersect at the point R. Let P and Q be the points lying on lines L1L_1 and L2L_2, respectively, such that ∣PR→∣=29|\overrightarrow{PR}|=\sqrt{29} and ∣PQ→∣=473|\overrightarrow{PQ}|=\sqrt{\frac{47}{3}}. If the point P lies in the first octant, then 27(QR)227(QR)^2 is equal to
Not attempted
A
320320
B
340340
C
360360
D
348348
306
[JEE Main 2026, 24 Jan Shift 1, Q23] Let a line L passing through the point P(1,1,1)P(1,1,1) be perpendicular to the lines x−44=y−11=z−11\frac{x-4}{4}=\frac{y-1}{1}=\frac{z-1}{1} and x−171=y−711=z0\frac{x-17}{1}=\frac{y-71}{1}=\frac z0. Let the line L intersect the yz plane at the point Q. Another line parallel to L and passing through the point S(1,0,−1)S(1,0,-1) intersects the yz-plane at the point R. Then the square of the area of the parallelogram PQRS is equal to ____ .
Not attempted
Your answer: —
Correct answer: 6
307
[JEE Main 2026, 24 Jan Shift 1, Q3] Let each of the two ellipses E1:x2a2+y2b2=1 (a>b)E_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\ (a>b) and E2:x2A2+y2B2=1 (A<B)E_2:\frac{x^2}{A^2}+\frac{y^2}{B^2}=1\ (A<B) have eccentricity 45\frac45. Let the lengths of the latus recta of E1E_1 and E2E_2 be l1l_1 and l2l_2, respectively, such that 2l12=9l22l_1^2=9l_2. If the distance between the foci of E1E_1 is 8, then the distance between the foci of E2E_2 is
Not attempted
A
85\frac85
B
325\frac{32}{5}
C
165\frac{16}{5}
D
965\frac{96}{5}
308
[JEE Main 2026, 24 Jan Shift 1, Q6] Let A(1,0)A(1,0), B(2,−1)B(2,-1) and C(73,43)C\left(\frac73,\frac43\right) be three points. If the equation of the bisector of the angle ABC is αx+βy=5\alpha x+\beta y=5, then the value of α2+β2\alpha^2+\beta^2 is
Not attempted
A
55
B
1010
C
1313
D
88
309
[JEE Main 2026, 24 Jan Shift 1, Q19] Let a circle of radius 4 pass through the origin O, the points A(−3a,0)A(-\sqrt3a,0) and B(0,−2b)B(0,-\sqrt2b), where a and b are real parameters and ab≠0ab\ne0. Then the locus of the centroid of △OAB\triangle OAB is a circle of radius
Not attempted
A
73\frac73
B
53\frac53
C
83\frac83
D
113\frac{11}{3}
310
[JEE Main 2026, 24 Jan Shift 1, Q21] Let a differentiable function f satisfy the equation ∫036f(tx36)dt=4αf(x)\int_0^{36}f\left(\frac{tx}{36}\right)dt=4\alpha f(x). If y=f(x)y=f(x) is a standard parabola passing through the points (2,1)(2,1) and (−4,β)(-4,\beta), then βα\beta^\alpha is equal to ____ .
Not attempted
Your answer: —
Correct answer: 64
311
[JEE Main 2026, 24 Jan Shift 1, Q17] Let A1A_1 be the bounded area enclosed by the curves y=x2+2y=x^2+2, x+y=8x+y=8 and y-axis that lies in the first quadrant. Let A2A_2 be the bounded area enclosed by the curves y=x2+2y=x^2+2, y2=xy^2=x, x=2x=2, and y-axis that lies in the first quadrant. Then A1−A2A_1-A_2 is equal to
Not attempted
A
23(2+1)\frac23(\sqrt2+1)
B
23(32+1)\frac23(3\sqrt2+1)
C
23(42+1)\frac23(4\sqrt2+1)
D
23(22+1)\frac23(2\sqrt2+1)
312
[JEE Main 2026, 24 Jan Shift 1, Q18] Let f(t)=∫(1−sin⁡(log⁡et)1−cos⁡(log⁡et))dtf(t)=\int\left(\frac{1-\sin(\log_et)}{1-\cos(\log_et)}\right)dt, for t>1t>1. If f(eπ/2)=−eπ/2f(e^{\pi/2})=-e^{\pi/2} and f(eπ/4)=αeπ/4f(e^{\pi/4})=\alpha e^{\pi/4}, then find the value of α\alpha.
Not attempted
A
−1−22-1-2\sqrt2
B
−1+2-1+\sqrt2
C
1+21+\sqrt2
D
−1−2-1-\sqrt2
313
[JEE Main 2026, 24 Jan Shift 1, Q7] Let α,β∈R\alpha,\beta\in\mathbb{R} be such that the function f(x)={2α(x2−2)+2βx,x<1(α+3)x+(α−β),x≥1f(x)=\begin{cases}2\alpha(x^2-2)+2\beta x, & x<1\\(\alpha+3)x+(\alpha-\beta), & x\ge1\end{cases} be differentiable at all x∈Rx\in\mathbb{R}. Then 34(α+β)34(\alpha+\beta) is equal to
Not attempted
A
2424
B
8484
C
3636
D
4848
314
[JEE Main 2026, 24 Jan Shift 1, Q10] If the function f(x)=ex(etan⁡x−x−1)+log⁡e(sec⁡x+tan⁡x)−xtan⁡x−xf(x)=\frac{e^x\left(e^{\tan x-x}-1\right)+\log_e(\sec x+\tan x)-x}{\tan x-x} is continuous at x=0x=0, then the value of f(0)f(0) is equal to
Not attempted
A
12\frac12
B
22
C
32\frac32
D
23\frac23
315
[JEE Main 2026, 24 Jan Shift 1, Q22] Let (2α,α)(2\alpha,\alpha) be the largest interval in which the function f(t)=∣t+1∣t2f(t)=\frac{|t+1|}{t^2}, t<0t<0, is strictly decreasing. Then the local maximum value of the function g(x)=2log⁡e(x−2)+αx2+4x−αg(x)=2\log_e(x-2)+\alpha x^2+4x-\alpha, x>2x>2, is ____
Not attempted
Your answer: —
Correct answer: 4
316
[JEE Main 2026, 24 Jan Shift 1, Q9] Consider an A.P.: a1,a2,…,ana_1,a_2,\ldots,a_n; a1>0a_1>0. If a2−a1=−34a_2-a_1=-\frac34, an=14a1a_n=\frac14a_1, and ∑i=1nai=5252\sum_{i=1}^na_i=\frac{525}{2}, then ∑i=117ai\sum_{i=1}^{17}a_i is equal to
Not attempted
A
952952
B
476476
C
238238
D
136136
317
[JEE Main 2026, 24 Jan Shift 1, Q15] Let 729,81,9,1,…729,81,9,1,\ldots be a sequence and PnP_n denote the product of the first n terms of this sequence. If 2∑n=140(Pn)1n=3α−13β2\sum_{n=1}^{40}(P_n)^{\frac1n}=\frac{3^\alpha-1}{3^\beta} and gcd⁡(α,β)=1\gcd(\alpha,\beta)=1, then α+β\alpha+\beta is equal to
Not attempted
A
7676
B
7373
C
7474
D
7575
318
[JEE Main 2026, 24 Jan Shift 1, Q2] Let S=125!+13! 23!+15! 21!+…S=\frac{1}{25!}+\frac{1}{3!\,23!}+\frac{1}{5!\,21!}+\ldots up to 13 terms. If 13S=2kn!13S=\frac{2^k}{n!}, k∈Nk\in\mathbb{N}, then n+kn+k is equal to
Not attempted
A
4949
B
5151
C
5050
D
5252
319
[JEE Main 2026, 24 Jan Shift 1, Q24] The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ____
Not attempted
Your answer: —
Correct answer: 42
320
[JEE Main 2026, 24 Jan Shift 1, Q25] The number of 3×23\times2 matrices A, which can be formed using the elements of the set {−2,−1,0,1,2}\{-2,-1,0,1,2\} such that the sum of all the diagonal elements of ATAA^TA is 5, is
Not attempted
Your answer: —
Correct answer: 312
321
[JEE Main 2026, 24 Jan Shift 1, Q8] Let S={z∈C:∣z−6iz−2i∣=1 and ∣z−8+2iz+2i∣=35}S=\left\{z\in\mathbb{C}:\left|\frac{z-6i}{z-2i}\right|=1\text{ and }\left|\frac{z-8+2i}{z+2i}\right|=\frac35\right\}. Then ∑z∈S∣z∣2\sum_{z\in S}|z|^2 is equal to
Not attempted
A
413413
B
398398
C
423423
D
385385
322
[JEE Main 2026, 24 Jan Shift 1, Q12] The number of the real solutions of the equation x∣x+3∣+∣x−1∣−2=0x|x+3|+|x-1|-2=0 is
Not attempted
A
44
B
22
C
33
D
55
323
[JEE Main 2026, 24 Jan Shift 1, Q4] If the domain of the function f(x)=log⁡(10x2−17x+7)(18x2−11x+1)f(x)=\log_{(10x^2-17x+7)}(18x^2-11x+1) is (−∞,a)∪(b,c)∪(d,∞)−{e}(-\infty,a)\cup(b,c)\cup(d,\infty)-\{e\}, then 90(a+b+c+d+e)90(a+b+c+d+e) equals:
Not attempted
A
307307
B
177177
C
170170
D
316316
324
[JEE Main 2026, 24 Jan Shift 1, Q16] Let R be a relation defined on the set {1,2,3,4}×{1,2,3,4}\{1,2,3,4\}\times\{1,2,3,4\} by R={((a,b),(c,d)):2a+3b=3c+4d}R=\{((a,b),(c,d)):2a+3b=3c+4d\}. Then the number of elements in R is
Not attempted
A
66
B
1515
C
1818
D
1212
325
[JEE Main 2026, 23 Jan Shift 2, Q2] The least value of cos⁡2θ−6sin⁡θcos⁡θ+3sin⁡2θ+2\cos^2\theta-6\sin\theta\cos\theta+3\sin^2\theta+2 is
Not attempted
A
4+104+\sqrt{10}
B
−1-1
C
4−104-\sqrt{10}
D
11
326
[JEE Main 2026, 23 Jan Shift 2, Q18] Let π2<θ<π\frac{\pi}{2}<\theta<\pi and cot⁡θ=−122\cot\theta=-\frac{1}{2\sqrt2}. Then the value of sin⁡(15θ2)(cos⁡8θ+sin⁡8θ)+cos⁡(15θ2)(cos⁡8θ−sin⁡8θ)\sin\left(\frac{15\theta}{2}\right)(\cos8\theta+\sin8\theta)+\cos\left(\frac{15\theta}{2}\right)(\cos8\theta-\sin8\theta) is equal to
Not attempted
A
−23-\frac{\sqrt2}{\sqrt3}
B
2−13\frac{\sqrt2-1}{\sqrt3}
C
1−23\frac{1-\sqrt2}{\sqrt3}
D
23\frac{\sqrt2}{\sqrt3}
327
[JEE Main 2026, 23 Jan Shift 2, Q1] If the mean and the variance of the data Class4−88−1212−1616−20Frequency3λ47\displaystyle\begin{array}{|c|c|c|c|c|}\hline \text{Class} & 4-8 & 8-12 & 12-16 & 16-20 \\ \hline \text{Frequency} & 3 & \lambda & 4 & 7 \\ \hline \end{array} are μ\mu and 19 respectively, then the value of λ+μ\lambda+\mu is
Not attempted
A
1919
B
2020
C
1818
D
2121
328
[JEE Main 2026, 23 Jan Shift 2, Q11] Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability that the ball drawn is white is pq\frac pq, where gcd⁡(p,q)=1\gcd(p,q)=1, then what is the value of p+qp+q?
Not attempted
A
2121
B
2222
C
2424
D
2323
329
[JEE Main 2026, 23 Jan Shift 2, Q6] Let a⃗=i^−2j^+3k^\vec a=\hat i-2\hat j+3\hat k, b⃗=2i^+j^−k^\vec b=2\hat i+\hat j-\hat k, c⃗=λi^+j^+k^\vec c=\lambda\hat i+\hat j+\hat k and v⃗=a⃗×b⃗\vec v=\vec a\times\vec b. If v⃗⋅c⃗=11\vec v\cdot\vec c=11 and the length of the projection of b⃗\vec b on c⃗\vec c is pp, then 9p29p^2 is equal to
Not attempted
A
66
B
44
C
99
D
1212
330
[JEE Main 2026, 23 Jan Shift 2, Q13] Let a⃗,b⃗,c⃗\vec a,\vec b,\vec c be three vectors such that a⃗×b⃗=2(a⃗×c⃗)\vec a\times\vec b=2(\vec a\times\vec c). If ∣a⃗∣=1|\vec a|=1, ∣b⃗∣=4|\vec b|=4, ∣c⃗∣=2|\vec c|=2, and the angle between b⃗\vec b and c⃗\vec c is 60∘60^\circ, then ∣a⃗⋅c⃗∣|\vec a\cdot\vec c| is equal to
Not attempted
A
00
B
22
C
11
D
44
331
[JEE Main 2026, 23 Jan Shift 2, Q23] If the image of the point P(a,2,a)P(a,2,a) in the line x2=y+a1=z1\frac x2=\frac{y+a}{1}=\frac z1 is Q and the image of Q in the line x−2b2=y−a1=z+2b−5\frac{x-2b}{2}=\frac{y-a}{1}=\frac{z+2b}{-5} is P, then a+ba+b is equal to ____ .
Not attempted
Your answer: —
Correct answer: 3
332
[JEE Main 2026, 23 Jan Shift 2, Q4] An equilateral triangle OAB is inscribed in the parabola y2=4xy^2=4x with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is
Not attempted
A
2(8−33)2(8-3\sqrt3)
B
4(3−3)4(3-\sqrt3)
C
4(6+3)4(6+\sqrt3)
D
2(3+3)2(3+\sqrt3)
333
[JEE Main 2026, 23 Jan Shift 2, Q9] Let PQ be a chord of the hyperbola x24−y2b2=1\frac{x^2}{4}-\frac{y^2}{b^2}=1, perpendicular to the x-axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity of the hyperbola is 3\sqrt3, then the area of the triangle OPQ is
Not attempted
A
232\sqrt3
B
115\frac{11}{5}
C
95\frac95
D
835\frac{8\sqrt3}{5}
334
[JEE Main 2026, 23 Jan Shift 2, Q10] Let A(1,2)A(1,2) and C(−3,−6)C(-3,-6) be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line 7x−y=147x-y=14. If B(α,β)B(\alpha,\beta) and D(γ,δ)D(\gamma,\delta) are the other two vertices, then ∣α+β+γ+δ∣|\alpha+\beta+\gamma+\delta| is equal to
Not attempted
A
66
B
99
C
11
D
33
335
[JEE Main 2026, 23 Jan Shift 2, Q14] If the points of intersection of the ellipses x2+2y2−6x−12y+23=0x^2+2y^2-6x-12y+23=0 and 4x2+2y2−20x−12y+35=04x^2+2y^2-20x-12y+35=0 lie on a circle of radius rr and centre (a,b)(a,b), then the value of ab+18r2ab+18r^2 is
Not attempted
A
5555
B
5353
C
5151
D
5252
336
[JEE Main 2026, 23 Jan Shift 2, Q25] If the solution curve y=f(x)y=f(x) of the differential equation (x2−4)y′−2xy+2x(4−x2)2=0(x^2-4)y'-2xy+2x(4-x^2)^2=0, x>2x>2, passes through the point (3,15)(3,15), then the local maximum value of ff is ____
Not attempted
Your answer: —
Correct answer: 16
337
[JEE Main 2026, 23 Jan Shift 2, Q7] Let I(x)=∫3 dx(4x+6)(4x2+8x+3)I(x)=\int\frac{3\,dx}{(4x+6)\left(\sqrt{4x^2+8x+3}\right)} and I(0)=34+20I(0)=\frac{\sqrt3}{4}+20. If I(12)=a2b+cI\left(\frac12\right)=\frac{a\sqrt2}{b}+c, where a,b,c∈Na,b,c\in\mathbb{N}, gcd⁡(a,b)=1\gcd(a,b)=1, then a+b+ca+b+c is equal to
Not attempted
A
3131
B
2828
C
2929
D
3030
338
[JEE Main 2026, 23 Jan Shift 2, Q17] The area of the region enclosed between the circles x2+y2=4x^2+y^2=4 and x2+(y−2)2=4x^2+(y-2)^2=4 is:
Not attempted
A
23(4π−33)\frac23(4\pi-3\sqrt3)
B
23(2π−33)\frac23(2\pi-3\sqrt3)
C
43(2π−3)\frac43(2\pi-\sqrt3)
D
43(2π−33)\frac43(2\pi-3\sqrt3)
339
[JEE Main 2026, 23 Jan Shift 2, Q22] The number of elements in the set S={x:x∈[0,100] and ∫0xt2sin⁡(x−t) dt=x2}S=\{x:x\in[0,100]\text{ and }\int_0^xt^2\sin(x-t)\,dt=x^2\} is ____
Not attempted
Your answer: —
Correct answer: 16
340
[JEE Main 2026, 23 Jan Shift 2, Q15] If f(x)={a∣x∣+x2−2(sin⁡∣x∣)(cos⁡∣x∣)x,x≠0b,x=0f(x)=\begin{cases}\frac{a|x|+x^2-2(\sin|x|)(\cos|x|)}{x}, & x\ne0\\ b, & x=0\end{cases} is continuous at x=0x=0, then a+ba+b is equal to
Not attempted
A
44
B
11
C
00
D
22
341
[JEE Main 2026, 23 Jan Shift 2, Q16] Let ∑k=1nak=αn2+βn\sum_{k=1}^{n}a_k=\alpha n^2+\beta n. If a10=59a_{10}=59 and a6=7a1a_6=7a_1, then α+β\alpha+\beta is equal to
Not attempted
A
77
B
1212
C
33
D
55
342
[JEE Main 2026, 23 Jan Shift 2, Q19] The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is
Not attempted
A
384384
B
403403
C
429429
D
455455
343
[JEE Main 2026, 23 Jan Shift 2, Q21] Let S denote the set of 4-digit numbers abcd such that a>b>c>da>b>c>d and P denote the set of 5-digit numbers having product of its digits equal to 20. Then n(S)+n(P)n(S)+n(P) is equal to ____
Not attempted
Your answer: —
Correct answer: 260
344
[JEE Main 2026, 23 Jan Shift 2, Q3] The system of linear equations x+y+z=6x+y+z=6 2x+5y+az=362x+5y+az=36 x+2y+3z=bx+2y+3z=b has
Not attempted
A
unique solution for a=8a=8 and b=16b=16
B
infinitely many solutions for a=8a=8 and b=16b=16
C
unique solution for a=8a=8 and b=14b=14
D
infinitely many solutions for a=8a=8 and b=14b=14
345
[JEE Main 2026, 23 Jan Shift 2, Q24] Let A=[02−3−2013−10]A=\begin{bmatrix}0&2&-3\\-2&0&1\\3&-1&0\end{bmatrix} and B be a matrix such that B(I−A)=I+AB(I-A)=I+A. Then the sum of the diagonal elements of BTBB^TB is equal to ____
Not attempted
Your answer: —
Correct answer: 3
346
[JEE Main 2026, 23 Jan Shift 2, Q8] The sum of all the real solutions of the equation log⁡(x+3)(6x2+28x+30)=5−2log⁡(6x+10)(x2+6x+9)\log_{(x+3)}(6x^2+28x+30)=5-2\log_{(6x+10)}(x^2+6x+9) is equal to
Not attempted
A
00
B
22
C
11
D
44
347
[JEE Main 2026, 23 Jan Shift 2, Q20] If z=32+i2z=\frac{\sqrt3}{2}+\frac i2, i=−1i=\sqrt{-1}, then (z201−i)8(z^{201}-i)^8 is equal to
Not attempted
A
256256
B
11
C
00
D
−1-1
348
[JEE Main 2026, 23 Jan Shift 2, Q5] Let A={0,1,2,…,9}A=\{0,1,2,\ldots,9\}. Let R be a relation on A defined by (x,y)∈R(x,y)\in R if and only if ∣x−y∣|x-y| is a multiple of 3. Given below are two statements: Statement I: n(R)=36n(R)=36. Statement II: R is an equivalence relation. In the light of the above statements, choose the correct answer from the options given below.
Not attempted
A
Statement I is correct but Statement II is incorrect
B
Statement I is incorrect but Statement II is correct
C
Both Statement I and Statement II are correct
D
Both Statement I and Statement II are incorrect
349
[JEE Main 2026, 23 Jan Shift 2, Q12] Consider two sets A={x∈Z:∣(∣x−3∣−3)∣≤1}A=\{x\in\mathbb{Z}:|(|x-3|-3)|\le1\} and B={x∈R−{1,2}:(x−2)(x−4)x−1log⁡e(∣x−2∣)=0}B=\left\{x\in\mathbb{R}-\{1,2\}:\frac{(x-2)(x-4)}{x-1}\log_e(|x-2|)=0\right\}. Then the number of onto functions f:A→Bf:A\to B is equal to
Not attempted
A
8181
B
3232
C
7979
D
6262
350
[JEE Main 2026, 23 Jan Shift 1, Q3] Number of solutions of 3cos⁡2θ+8cos⁡θ+33=0\sqrt3\cos2\theta+8\cos\theta+3\sqrt3=0, θ∈[−3π,2π]\theta\in[-3\pi,2\pi] is:
Not attempted
A
00
B
33
C
44
D
55
351
[JEE Main 2026, 23 Jan Shift 1, Q16] Let α\alpha and β\beta respectively be the maximum and the minimum values of the function f(θ)=4(sin⁡4(7π2−θ)+sin⁡4(11π+θ))−2(sin⁡6(3π2−θ)+sin⁡6(9π−θ))f(\theta)=4\left(\sin^4\left(\frac{7\pi}{2}-\theta\right)+\sin^4(11\pi+\theta)\right)-2\left(\sin^6\left(\frac{3\pi}{2}-\theta\right)+\sin^6(9\pi-\theta)\right), θ∈R\theta\in\mathbb{R}. Then α+2β\alpha+2\beta is equal to:
Not attempted
A
33
B
44
C
55
D
66
352
[JEE Main 2026, 23 Jan Shift 1, Q2] Let the mean and variance of 8 numbers −10,−7,−1,x,y,9,2,16-10,-7,-1,x,y,9,2,16 be 72\frac72 and 2934\frac{293}{4}, respectively. Then the mean of 4 numbers x, y, x+y+1, ∣x−y∣x,\ y,\ x+y+1,\ |x-y| is :
Not attempted
A
1010
B
99
C
1111
D
1212
353
[JEE Main 2026, 23 Jan Shift 1, Q23] From the first 100 natural numbers, two numbers first aa and then bb are selected randomly without replacement. If the probability that a−b≥10a-b\ge10 is mn\frac mn, gcd⁡(m,n)=1\gcd(m,n)=1, then m+nm+n is equal to ____ .
Not attempted
Your answer: —
Correct answer: 311
354
[JEE Main 2026, 23 Jan Shift 1, Q14] Let a⃗=−i^+j^+2k^\vec a=-\hat i+\hat j+2\hat k, b⃗=i^−j^−3k^\vec b=\hat i-\hat j-3\hat k, c⃗=a⃗×b⃗\vec c=\vec a\times\vec b and d⃗=c⃗×a⃗\vec d=\vec c\times\vec a. Then (a⃗−b⃗)⋅d⃗(\vec a-\vec b)\cdot\vec d is equal to:
Not attempted
A
44
B
22
C
−2-2
D
−4-4
355
[JEE Main 2026, 23 Jan Shift 1, Q8] The vertices B and C of a triangle ABC lie on the line x1=1−y−2=z−23\frac x1=\frac{1-y}{-2}=\frac{z-2}{3}. The coordinates of A and B are (1,6,3)(1,6,3) and (4,9,α)(4,9,\alpha) respectively and C is at a distance of 10 units from B. The area (in sq. units) of △ABC\triangle ABC is :
Not attempted
A
151315\sqrt{13}
B
201320\sqrt{13}
C
5135\sqrt{13}
D
101310\sqrt{13}
356
[JEE Main 2026, 23 Jan Shift 1, Q15] Let the direction cosines of two lines satisfy the equations: 4l+m−n=04l+m-n=0 and 2mn+10nl+3lm=02mn+10nl+3lm=0. Then the cosine of the acute angle between these lines is :
Not attempted
A
10338\frac{10}{3\sqrt{38}}
B
10738\frac{10}{7\sqrt{38}}
C
1038\frac{10}{\sqrt{38}}
D
20338\frac{20}{3\sqrt{38}}
357
[JEE Main 2026, 23 Jan Shift 1, Q10] Let the line y−x=1y-x=1 intersect the ellipse x22+y21=1\frac{x^2}{2}+\frac{y^2}{1}=1 at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is:
Not attempted
A
π2−tan⁡−1(14)\frac{\pi}{2}-\tan^{-1}\left(\frac14\right)
B
π2+tan⁡−1(14)\frac{\pi}{2}+\tan^{-1}\left(\frac14\right)
C
π−tan⁡−1(14)\pi-\tan^{-1}\left(\frac14\right)
D
π2+2tan⁡−1(14)\frac{\pi}{2}+2\tan^{-1}\left(\frac14\right)
358
[JEE Main 2026, 23 Jan Shift 1, Q12] A rectangle is formed by the lines x=0x=0, y=0y=0, x=3x=3 and y=4y=4. Let the line L be perpendicular to 3x+y+6=03x+y+6=0 and divide the area of the rectangle into two equal parts. Then the distance of the point (12,−5)\left(\frac12,-5\right) from the line L is equal to :
Not attempted
A
3103\sqrt{10}
B
10\sqrt{10}
C
252\sqrt5
D
2102\sqrt{10}
359
[JEE Main 2026, 23 Jan Shift 1, Q20] Let the domain of the function f(x)=log⁡3log⁡5log⁡7(9x−x2−13)f(x)=\log_3\log_5\log_7(9x-x^2-13) be the interval (m,n)(m,n). Let the hyperbola x2a2−y2b2=1\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 have eccentricity n3\frac n3 and the length of the latus rectum 8m3\frac{8m}{3}. Then b2−a2b^2-a^2 is equal to :
Not attempted
A
99
B
77
C
55
D
1111
360
[JEE Main 2026, 23 Jan Shift 1, Q18] Let y=y(x)y=y(x) be the solution of the differential equation x4dy+(4x3y+2sin⁡x)dx=0x^4dy+(4x^3y+2\sin x)dx=0, x>0x>0, y(π2)=0y\left(\frac{\pi}{2}\right)=0. Then π4y(π3)\pi^4y\left(\frac{\pi}{3}\right) is equal to:
Not attempted
A
6464
B
9292
C
7272
D
8181
361
[JEE Main 2026, 23 Jan Shift 1, Q25] Let ff be a twice differentiable non-negative function such that (f(x))2=25+∫0x((f(t))2+(f′(t))2)dt(f(x))^2=25+\int_0^x\left((f(t))^2+(f'(t))^2\right)dt. Then the mean of f(log⁡e(1)),f(log⁡e(2)),…,f(log⁡e(625))f(\log_e(1)),f(\log_e(2)),\ldots,f(\log_e(625)) is equal to ____ .
Not attempted
Your answer: —
Correct answer: 1565
362
[JEE Main 2026, 23 Jan Shift 1, Q7] Let f(x)=∫(2−x2)ex1+x (1−x)3/2dxf(x)=\int\frac{(2-x^2)e^x}{\sqrt{1+x}\,(1-x)^{3/2}}dx. If f(0)=0f(0)=0, then f(12)f\left(\frac12\right) is equal to:
Not attempted
A
3e−1\sqrt{3e}-1
B
2e−1\sqrt{2e}-1
C
2e+1\sqrt{2e}+1
D
3e+1\sqrt{3e}+1
363
[JEE Main 2026, 23 Jan Shift 1, Q11] The value of the integral ∫π/245π/24dx1+tan⁡2x3\int_{\pi/24}^{5\pi/24}\frac{dx}{1+\sqrt[3]{\tan2x}} is :
Not attempted
A
π3\frac{\pi}{3}
B
π18\frac{\pi}{18}
C
π6\frac{\pi}{6}
D
π12\frac{\pi}{12}
364
[JEE Main 2026, 23 Jan Shift 1, Q22] Let the area of the region bounded by the curve y=max⁡{sin⁡x,cos⁡x}y=\max\{\sin x,\cos x\}, lines x=0x=0, x=3π2x=\frac{3\pi}{2}, and the xx-axis be A. Then, A+A2A+A^2 is equal to ____ .
Not attempted
Your answer: —
Correct answer: 12
365
[JEE Main 2026, 23 Jan Shift 1, Q9] Let f(x)={ax2+2ax+34x2+4x−3,x≠−32,12b,x=−32,12f(x)=\begin{cases}\frac{ax^2+2ax+3}{4x^2+4x-3}, & x\ne-\frac32,\frac12\\ b, & x=-\frac32,\frac12\end{cases} be continuous at x=−32x=-\frac32. If f∘f(x)=75f\circ f(x)=\frac75, then xx is equal to:
Not attempted
A
11
B
22
C
1.41.4
D
00
366
[JEE Main 2026, 23 Jan Shift 1, Q5] The value of 100C5051+100C5152+…+100C100101\frac{^{100}C_{50}}{51}+\frac{^{100}C_{51}}{52}+\ldots+\frac{^{100}C_{100}}{101} is :
Not attempted
A
2100100\frac{2^{100}}{100}
B
2100101\frac{2^{100}}{101}
C
2101100\frac{2^{101}}{100}
D
2101101\frac{2^{101}}{101}
367
[JEE Main 2026, 23 Jan Shift 1, Q13] The sum of all possible values of n∈Nn\in\mathbb{N}, so that the coefficients of xx, x2x^2 and x3x^3 in the expansion of (1+x2)2(1+x)n(1+x^2)^2(1+x)^n, are in arithmetic progression is :
Not attempted
A
77
B
1212
C
99
D
33
368
[JEE Main 2026, 23 Jan Shift 1, Q24] The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is ____ .
Not attempted
Your answer: —
Correct answer: 1422
369
[JEE Main 2026, 23 Jan Shift 1, Q17] Among the statements : I: If ∣1cos⁡αcos⁡βcos⁡α1cos⁡γcos⁡βcos⁡γ1∣=∣0cos⁡αcos⁡βcos⁡α0cos⁡γcos⁡βcos⁡γ0∣\begin{vmatrix}1&\cos\alpha&\cos\beta\\\cos\alpha&1&\cos\gamma\\\cos\beta&\cos\gamma&1\end{vmatrix}=\begin{vmatrix}0&\cos\alpha&\cos\beta\\\cos\alpha&0&\cos\gamma\\\cos\beta&\cos\gamma&0\end{vmatrix}, then cos⁡2α+cos⁡2β+cos⁡2γ=32\cos^2\alpha+\cos^2\beta+\cos^2\gamma=\frac32, and II : If ∣x2+xx+1x−22x2+3x−13x3x−3x2+2x+32x−12x−1∣=px+q\begin{vmatrix}x^2+x&x+1&x-2\\2x^2+3x-1&3x&3x-3\\x^2+2x+3&2x-1&2x-1\end{vmatrix}=px+q, then p2=196q2p^2=196q^2,
Not attempted
A
both are false
B
only I is true
C
both are true
D
only II is true
370
[JEE Main 2026, 23 Jan Shift 1, Q21] Let ∣A∣=6|A|=6, where A is a 3×33\times3 matrix. If ∣adj⁡(3adj⁡(A2⋅adj⁡(2A)))∣=2m⋅3n\left|\operatorname{adj}\left(3\operatorname{adj}\left(A^2\cdot\operatorname{adj}(2A)\right)\right)\right|=2^m\cdot3^n, m,n∈Nm,n\in\mathbb{N}, then m+nm+n is equal to ____ .
Not attempted
Your answer: —
Correct answer: 62
371
[JEE Main 2026, 23 Jan Shift 1, Q1] Let S={z:3≤∣2z−3(1+i)∣≤7}S=\{z:3\le|2z-3(1+i)|\le7\} be a set of complex numbers. Then min⁡z∈S∣z+12(5+3i)∣\min_{z\in S}\left|z+\frac12(5+3i)\right| is equal to:
Not attempted
A
12\frac12
B
52\frac52
C
22
D
32\frac32
372
[JEE Main 2026, 23 Jan Shift 1, Q4] If α\alpha and β\beta (α<β)(\alpha<\beta) are the roots of the equation (−2+3)(∣x−3∣)+(x−6x)+(9−23)=0(-2+\sqrt3)(|\sqrt x-3|)+(x-6\sqrt x)+(9-2\sqrt3)=0, x≥0x\ge0, then βα+αβ\sqrt{\frac{\beta}{\alpha}}+\sqrt{\alpha\beta} is equal to:
Not attempted
A
1010
B
1111
C
88
D
99
373
[JEE Main 2026, 23 Jan Shift 1, Q6] A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :
Not attempted
A
36 days
B
42 days
C
24 days
D
30 days
374
[JEE Main 2026, 23 Jan Shift 1, Q19] Let A={−2,−1,0,1,2,3,4}A=\{-2,-1,0,1,2,3,4\}. Let R be a relation on A defined by xRyxRy if and only if 2x+y≤22x+y\le2. Let ll be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then l+m+nl+m+n is equal to :
Not attempted
A
3535
B
3232
C
3434
D
3333
375
[JEE Main 2026, 22 Jan Shift 2, Q21] Let cos⁡(α+β)=−110\cos(\alpha+\beta)=-\frac{1}{10} and sin⁡(α−β)=38\sin(\alpha-\beta)=\frac38, where 0<α<π30<\alpha<\frac{\pi}{3} and 0<β<π40<\beta<\frac{\pi}{4}. If tan⁡2α=3(1−r5)11(s+5)\tan2\alpha=\frac{3(1-r\sqrt5)}{\sqrt{11}(s+\sqrt5)}, r,s∈Nr,s\in\mathbb{N}, then r+sr+s is equal to ____ .
Not attempted
Your answer: —
Correct answer: 20
376
[JEE Main 2026, 22 Jan Shift 2, Q18] If the mean deviation about the median of the numbers k,2k,3k,…,1000kk,2k,3k,\ldots,1000k is 500, then k2k^2 is equal to:
Not attempted
A
44
B
11
C
99
D
1616
377
[JEE Main 2026, 22 Jan Shift 2, Q20] Let a⃗=2i^−j^+k^\vec a=2\hat i-\hat j+\hat k and b⃗=λj^+2k^\vec b=\lambda\hat j+2\hat k, λ∈Z\lambda\in\mathbb{Z} be two vectors. Let c⃗=a⃗×b⃗\vec c=\vec a\times\vec b and d⃗\vec d be a vector of magnitude 2 in yzyz-plane. If ∣c⃗∣=53|\vec c|=\sqrt{53}, then the maximum possible value of (c⃗⋅d⃗)2(\vec c\cdot\vec d)^2 is equal to:
Not attempted
A
208208
B
2626
C
5252
D
104104
378
[JEE Main 2026, 22 Jan Shift 2, Q25] Let a vector a⃗=2i^−j^+λk^\vec a=\sqrt2\hat i-\hat j+\lambda\hat k, λ>0\lambda>0, make an obtuse angle with the vector b⃗=−λ2i^+42j^+42k^\vec b=-\lambda^2\hat i+4\sqrt2\hat j+4\sqrt2\hat k and an angle θ\theta, π6<θ<π2\frac{\pi}{6}<\theta<\frac{\pi}{2} with the positive zz-axis. If the set of all possible values of λ\lambda is (α,β)−{γ}(\alpha,\beta)-\{\gamma\}, then α+β+γ\alpha+\beta+\gamma is equal to ____ .
Not attempted
Your answer: —
Correct answer: 5
379
[JEE Main 2026, 22 Jan Shift 2, Q16] Let L be the line x+12=y+13=z+36\frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6} and let S be the set of all points (a,b,c)(a,b,c) on L, whose distance from the line x+12=y+13=z−90\frac{x+1}{2}=\frac{y+1}{3}=\frac{z-9}{0} along the line L is 7. Then ∑(a,b,c)∈S(a+b+c)\sum_{(a,b,c)\in S}(a+b+c) is equal to:
Not attempted
A
4040
B
2828
C
3434
D
66
380
[JEE Main 2026, 22 Jan Shift 2, Q2] Among the statements (S1)(S_1) : If A(5,−1)A(5,-1) and B(−2,3)B(-2,3) are two vertices of a triangle, whose orthocentre is (0,0)(0,0), then its third vertex is (−4,−7)(-4,-7) and (S2)(S_2) : If positive numbers 2a2a, bb, cc are three consecutive terms of an A.P., then the lines ax+by+c=0ax+by+c=0 are concurrent at (2,−2)(2,-2).
Not attempted
A
only (S2)(S_2) is correct
B
only (S1)(S_1) is correct
C
both are incorrect
D
both are correct
381
[JEE Main 2026, 22 Jan Shift 2, Q6] Let P(10,215)P(10,2\sqrt{15}) be a point on the hyperbola x2a2−y2b2=1\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, whose foci are S and S'. If the length of its latus rectum is 8, then the square of the area of ΔPSS′\Delta PSS' is equal to:
Not attempted
A
900900
B
42004200
C
27002700
D
14621462
382
[JEE Main 2026, 22 Jan Shift 2, Q8] Let S and S' be the foci of the ellipse x225+y29=1\frac{x^2}{25}+\frac{y^2}{9}=1 and P(α,β)P(\alpha,\beta) be a point on the ellipse in the first quadrant. If (SP)2+(S′P)2−SP⋅S′P=37(SP)^2+(S'P)^2-SP\cdot S'P=37, then α2+β2\alpha^2+\beta^2 is equal to:
Not attempted
A
1313
B
1515
C
1717
D
1111
383
[JEE Main 2026, 22 Jan Shift 2, Q12] Let the locus of the mid-point of the chord through the origin O of the parabola y2=4xy^2=4x be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3:1, is:
Not attempted
A
2x2=3y2x^2=3y
B
2y2=3x2y^2=3x
C
3y2=2x3y^2=2x
D
3x2=2y3x^2=2y
384
[JEE Main 2026, 22 Jan Shift 2, Q5] If y=y(x)y=y(x) satisfies the differential equation 16(xx+9x)(4+9+x)cos⁡y dy=(1+2sin⁡y) dx16\left(\sqrt{x\sqrt x+9x}\right)\left(4+\sqrt{9+\sqrt x}\right)\cos y\,dy=(1+2\sin y)\,dx, x>0x>0 and y(256)=π2y(256)=\frac{\pi}{2}, y(49)=αy(49)=\alpha, then 2sin⁡α2\sin\alpha is equal to:
Not attempted
A
2(2−1)2(\sqrt2-1)
B
2−1\sqrt2-1
C
3(2−1)3(\sqrt2-1)
D
22−12\sqrt2-1
385
[JEE Main 2026, 22 Jan Shift 2, Q17] The area of the region A={(x,y):4x2+y2≤8 and y2≤4x}A=\{(x,y):4x^2+y^2\le8\text{ and }y^2\le4x\} is:
Not attempted
A
π2+2\frac{\pi}{2}+2
B
π+4\pi+4
C
π+23\pi+\frac23
D
π2+13\frac{\pi}{2}+\frac13
386
[JEE Main 2026, 22 Jan Shift 2, Q23] Let [⋅][\cdot] be the greatest integer function. If α=∫064(x1/3−[x1/3])dx\alpha=\int_0^{64}\left(x^{1/3}-[x^{1/3}]\right)dx, then 1π∫0απ(sin⁡2θsin⁡6θ+cos⁡6θ)dθ\frac1\pi\int_0^{\alpha\pi}\left(\frac{\sin^2\theta}{\sin^6\theta+\cos^6\theta}\right)d\theta is equal to ____ .
Not attempted
Your answer: —
Correct answer: 36
387
[JEE Main 2026, 22 Jan Shift 2, Q4] If lim⁡x→0e(a−1)x+2cos⁡(bx)+(c−2)e−xxcos⁡x−log⁡e(1+x)=2\lim_{x\to0}\frac{e^{(a-1)x}+2\cos(bx)+(c-2)e^{-x}}{x\cos x-\log_e(1+x)}=2, then a2+b2+c2a^2+b^2+c^2 is equal to:
Not attempted
A
77
B
99
C
33
D
55
388
[JEE Main 2026, 22 Jan Shift 2, Q14] Let [⋅][\cdot] denote the greatest integer function, and let f(x)=min⁡{2x, x2}f(x)=\min\{\sqrt2x,\,x^2\}. Let S be the set of all x∈(−2,2)x\in(-2,2) at which the function g(x)=∣x∣[x2]g(x)=|x|[x^2] is discontinuous. Then ∑x∈Sf(x)\sum_{x\in S}f(x) equals
Not attempted
A
2−22-\sqrt2
B
1−21-\sqrt2
C
26−322\sqrt6-3\sqrt2
D
6−22\sqrt6-2\sqrt2
389
[JEE Main 2026, 22 Jan Shift 2, Q7] Let ff and gg be functions satisfying f(x+y)=f(x)f(y)f(x+y)=f(x)f(y), f(1)=7f(1)=7 and g(x+y)=g(xy)g(x+y)=g(xy), g(1)=1g(1)=1, for all x,y∈Nx,y\in\mathbb{N}. If ∑x=1n(f(x)g(x))=19607\sum_{x=1}^{n}\left(\frac{f(x)}{g(x)}\right)=19607, then nn is equal to:
Not attempted
A
77
B
66
C
55
D
44
390
[JEE Main 2026, 22 Jan Shift 2, Q24] Suppose a,b,ca,b,c are in A.P. and a2,2b2,c2a^2,2b^2,c^2 are in G.P. If a<b<ca<b<c and a+b+c=1a+b+c=1, then 9(a2+b2+c2)9(a^2+b^2+c^2) is equal to ____.
Not attempted
Your answer: —
Correct answer: 9
391
[JEE Main 2026, 22 Jan Shift 2, Q11] Let CrC_r denote the coefficient of xrx^r in the binomial expansion of (1+x)n(1+x)^n, n∈Nn\in\mathbb{N}, 0≤r≤n0\le r\le n. If Pn=C0−C1+223C2−234C3+…+(−2)nn+1CnP_n=C_0-C_1+\frac{2^2}{3}C_2-\frac{2^3}{4}C_3+\ldots+\frac{(-2)^n}{n+1}C_n, then the value of ∑n=1251P2n\sum_{n=1}^{25}\frac{1}{P_{2n}} equals.
Not attempted
A
580580
B
675675
C
525525
D
650650
392
[JEE Main 2026, 22 Jan Shift 2, Q22] Let S be the set of the first 11 natural numbers. Then the number of elements in A={B⊆S:n(B)≥2 and the product of all elements of B is even}A=\{B\subseteq S:n(B)\ge2\text{ and the product of all elements of }B\text{ is even}\} is ____.
Not attempted
Your answer: —
Correct answer: 1979
393
[JEE Main 2026, 22 Jan Shift 2, Q1] Let nn be the number obtained on rolling a fair die. If the probability that the system x−ny+z=6x-ny+z=6 x+(n−2)y+(n+1)z=8x+(n-2)y+(n+1)z=8 (n−1)y+z=1(n-1)y+z=1 has a unique solution is k6\frac{k}{6}, then the sum of kk and all possible values of nn is:
Not attempted
A
2121
B
2424
C
2020
D
2222
394
[JEE Main 2026, 22 Jan Shift 2, Q13] If X=[xyz]X=\begin{bmatrix}x\\y\\z\end{bmatrix} is a solution of the system of equations AX=BAX=B, where adj⁡A=[422−5051−23]\operatorname{adj}A=\begin{bmatrix}4&2&2\\-5&0&5\\1&-2&3\end{bmatrix} and B=[402]B=\begin{bmatrix}4\\0\\2\end{bmatrix}, then ∣x+y+z∣|x+y+z| is equal to :
Not attempted
A
22
B
11
C
33
D
32\frac32
395
[JEE Main 2026, 22 Jan Shift 2, Q3] Let α,β\alpha,\beta be the roots of the quadratic equation 12x2−20x+3λ=012x^2-20x+3\lambda=0, λ∈Z\lambda\in\mathbb{Z}. If 12≤∣β−α∣≤32\frac12\le|\beta-\alpha|\le\frac32, then the sum of all possible values of λ\lambda is :
Not attempted
A
11
B
33
C
66
D
44
396
[JEE Main 2026, 22 Jan Shift 2, Q9] Let S={z∈C:4z2+zˉ=0}S=\{z\in\mathbb{C}:4z^2+\bar z=0\}. Then ∑z∈S∣z∣2\sum_{z\in S}|z|^2 is equal to :
Not attempted
A
764\frac{7}{64}
B
116\frac{1}{16}
C
564\frac{5}{64}
D
316\frac{3}{16}
397
[JEE Main 2026, 22 Jan Shift 2, Q10] The number of elements in the relation R={(x,y):4x2+y2<52, x,y∈Z}R=\{(x,y):4x^2+y^2<52,\ x,y\in\mathbb{Z}\} is
Not attempted
A
6767
B
7777
C
8686
D
8989
398
[JEE Main 2026, 22 Jan Shift 2, Q15] Let f(x)=[x]2−[x+3]−3f(x)=[x]^2-[x+3]-3, x∈Rx\in\mathbb{R}, where [⋅][\cdot] is the greatest integer function. Then
Not attempted
A
∫02f(x) dx=−6\int_0^2f(x)\,dx=-6
B
f(x)=0f(x)=0 for finitely many values of xx
C
f(x)>0f(x)>0 only for x∈[4,∞)x\in[4,\infty)
D
f(x)<0f(x)<0 only for x∈[−1,3)x\in[-1,3)
399
[JEE Main 2026, 22 Jan Shift 2, Q19] Let the domain of the function f(x)=log⁡3log⁡5(7−log⁡2(x2−10x+85))+sin⁡−1(∣3x−717−x∣)f(x)=\log_3\log_5\left(7-\log_2(x^2-10x+85)\right)+\sin^{-1}\left(\left|\frac{3x-7}{17-x}\right|\right) be (α,β](\alpha,\beta]. Then α+β\alpha+\beta is equal to :
Not attempted
A
88
B
99
C
1212
D
1010
400
[JEE Main 2026, 22 Jan Shift 1, Q19] The number of solutions of tan⁡−14x+tan⁡−16x=π6\tan^{-1}4x+\tan^{-1}6x=\frac{\pi}{6}, where −126<x<126-\frac{1}{2\sqrt6}<x<\frac{1}{2\sqrt6}, is equal to
Not attempted
A
22
B
33
C
11
D
00
401
[JEE Main 2026, 22 Jan Shift 1, Q24] If cos⁡248∘−sin⁡212∘sin⁡224∘−sin⁡26∘=α+β52\frac{\cos^248^\circ-\sin^212^\circ}{\sin^224^\circ-\sin^26^\circ}=\frac{\alpha+\beta\sqrt5}{2}, where α,β∈N\alpha,\beta\in\mathbb{N}, then α+β\alpha+\beta is equal to
Not attempted
Your answer: —
Correct answer: 4
402
[JEE Main 2026, 22 Jan Shift 1, Q9] If a random variable xx has the probability distribution x01234567P(x)02kk3k2k22kk2+k7k2\displaystyle\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline P(x) & 0 & 2k & k & 3k & 2k^2 & 2k & k^2+k & 7k^2 \\ \hline \end{array} then P(3<x≤6)P(3<x\le6) is equal to
Not attempted
A
0.330.33
B
0.340.34
C
0.220.22
D
0.640.64
403
[JEE Main 2026, 22 Jan Shift 1, Q10] Two distinct numbers aa and bb are selected at random from 1,2,3,…,501,2,3,\ldots,50. The probability that their product abab is divisible by 3 is
Not attempted
A
2721225\frac{272}{1225}
B
825\frac{8}{25}
C
6641225\frac{664}{1225}
D
5611225\frac{561}{1225}
404
[JEE Main 2026, 22 Jan Shift 1, Q2] Let AB→=2i^+4j^−5k^\overrightarrow{AB}=2\hat i+4\hat j-5\hat k and AD→=i^+2j^+λk^\overrightarrow{AD}=\hat i+2\hat j+\lambda\hat k, λ∈R\lambda\in\mathbb{R}. Let the projection of the vector v⃗=i^+j^+k^\vec v=\hat i+\hat j+\hat k on the diagonal AC→\overrightarrow{AC} of the parallelogram ABCD be of length one unit. If α,β\alpha,\beta, where α>β\alpha>\beta, be the roots of the equation λ2x2−6λx+5=0\lambda^2x^2-6\lambda x+5=0, then 2α−β2\alpha-\beta is equal to:
Not attempted
A
33
B
66
C
11
D
44
405
[JEE Main 2026, 22 Jan Shift 1, Q1] If the image of the point P(1,2,a)P(1,2,a) in the line x−63=y−72=7−z2\frac{x-6}{3}=\frac{y-7}{2}=\frac{7-z}{2} is Q(5,b,c)Q(5,b,c), then a2+b2+c2a^2+b^2+c^2 is equal to
Not attempted
A
264264
B
293293
C
298298
D
283283
406
[JEE Main 2026, 22 Jan Shift 1, Q11] Let P(α,β,γ)P(\alpha,\beta,\gamma) be the point on the line x−12=y+1−3=z\frac{x-1}{2}=\frac{y+1}{-3}=z at a distance 4144\sqrt{14} from the point (1,−1,0)(1,-1,0) and nearer to the origin. Then the shortest distance, between the lines x−α1=y−β2=z−γ3\frac{x-\alpha}{1}=\frac{y-\beta}{2}=\frac{z-\gamma}{3} and x+52=y−101=z−31\frac{x+5}{2}=\frac{y-10}{1}=\frac{z-3}{1}, is equal to
Not attempted
A
4754\sqrt{\frac75}
B
4574\sqrt{\frac57}
C
7547\sqrt{\frac54}
D
2742\sqrt{\frac74}
407
[JEE Main 2026, 22 Jan Shift 1, Q4] If the line αx+2y=1\alpha x+2y=1, where α∈R\alpha\in\mathbb{R}, does not meet the hyperbola x2−9y2=9x^2-9y^2=9, then a possible value of α\alpha is:
Not attempted
A
0.70.7
B
0.60.6
C
0.50.5
D
0.80.8
408
[JEE Main 2026, 22 Jan Shift 1, Q5] If the chord joining the points P1(x1,y1)P_1(x_1,y_1) and P2(x2,y2)P_2(x_2,y_2) on the parabola y2=12xy^2=12x subtends a right angle at the vertex of the parabola, then x1x2−y1y2x_1x_2-y_1y_2 is equal to
Not attempted
A
292292
B
288288
C
280280
D
284284
409
[JEE Main 2026, 22 Jan Shift 1, Q15] Let the set of all values of rr, for which the circles (x+1)2+(y+4)2=r2(x+1)^2+(y+4)^2=r^2 and x2+y2−4x−2y−4=0x^2+y^2-4x-2y-4=0 intersect at two distinct points be the interval (α,β)(\alpha,\beta). Then αβ\alpha\beta is equal to
Not attempted
A
2525
B
2121
C
2424
D
2020
410
[JEE Main 2026, 22 Jan Shift 1, Q13] Let the solution curve of the differential equation x dy−y dx=x2+y2 dxx\,dy-y\,dx=\sqrt{x^2+y^2}\,dx, x>0x>0, y(1)=0y(1)=0, be y=y(x)y=y(x). Then y(3)y(3) is equal to
Not attempted
A
66
B
11
C
22
D
44
411
[JEE Main 2026, 22 Jan Shift 1, Q16] Let f:[1,∞)→Rf:[1,\infty)\to\mathbb{R} be a differentiable function. If 6∫1xf(t) dt=3xf(x)+x3−46\int_1^x f(t)\,dt=3xf(x)+x^3-4 for all x≥1x\ge1, then the value of f(2)−f(3)f(2)-f(3) is
Not attempted
A
33
B
44
C
−3-3
D
−4-4
412
[JEE Main 2026, 22 Jan Shift 1, Q6] Let the line x=−1x=-1 divide the area of the region {(x,y):1+x2≤y≤3−x}\{(x,y):1+x^2\le y\le3-x\} in the ratio m:nm:n, gcd⁡(m,n)=1\gcd(m,n)=1. Then m+nm+n is equal to
Not attempted
A
2626
B
2525
C
2727
D
2828
413
[JEE Main 2026, 22 Jan Shift 1, Q8] The value of ∫−π/2π/2(1[x]+4)dx\int_{-\pi/2}^{\pi/2}\left(\frac{1}{[x]+4}\right)dx, where [⋅][\cdot] denotes the greatest integer function, is
Not attempted
A
160(π−7)\frac{1}{60}(\pi-7)
B
160(21π−1)\frac{1}{60}(21\pi-1)
C
760(3π−1)\frac{7}{60}(3\pi-1)
D
760(π−3)\frac{7}{60}(\pi-3)
414
[JEE Main 2026, 22 Jan Shift 1, Q25] If ∫(sin⁡x)−112(cos⁡x)−52dx=−p1q1(cot⁡x)92−p2q2(cot⁡x)52−p3q3(cot⁡x)12+p4q4(cot⁡x)−32+C\int(\sin x)^{-\frac{11}{2}}(\cos x)^{-\frac52}dx=-\frac{p_1}{q_1}(\cot x)^{\frac92}-\frac{p_2}{q_2}(\cot x)^{\frac52}-\frac{p_3}{q_3}(\cot x)^{\frac12}+\frac{p_4}{q_4}(\cot x)^{-\frac32}+C, where pip_i and qiq_i are positive integers with gcd⁡(pi,qi)=1\gcd(p_i,q_i)=1 for i=1,2,3,4i=1,2,3,4 and C is the constant of integration, then 15p1p2p3p4q1q2q3q4\frac{15p_1p_2p_3p_4}{q_1q_2q_3q_4} is equal to ____
Not attempted
Your answer: —
Correct answer: 16
415
[JEE Main 2026, 22 Jan Shift 1, Q14] Let f(x)=x2025−x2000f(x)=x^{2025}-x^{2000}, x∈[0,1]x\in[0,1] and the minimum value of the function f(x)f(x) in the interval [0,1][0,1] be (80)80(n)−81(80)^{80}(n)^{-81}. Then nn is equal to
Not attempted
A
−41-41
B
−81-81
C
−80-80
D
−40-40
416
[JEE Main 2026, 22 Jan Shift 1, Q3] If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is
Not attempted
A
−22-22
B
−26-26
C
−20-20
D
−24-24
417
[JEE Main 2026, 22 Jan Shift 1, Q18] The coefficient of x48x^{48} in (1+x)+2(1+x)2+3(1+x)3+…+100(1+x)100(1+x)+2(1+x)^2+3(1+x)^3+\ldots+100(1+x)^{100} is equal to
Not attempted
A
100⋅100C49−100C50100\cdot{}^{100}C_{49}-{}^{100}C_{50}
B
100C50+101C49{}^{100}C_{50}+{}^{101}C_{49}
C
100⋅101C49−101C50100\cdot{}^{101}C_{49}-{}^{101}C_{50}
D
100⋅100C49−100C48100\cdot{}^{100}C_{49}-{}^{100}C_{48}
418
[JEE Main 2026, 22 Jan Shift 1, Q22] Let ABC be a triangle. Consider four points p1,p2,p3,p4p_1,p_2,p_3,p_4 on the side AB, five points p5,p6,p7,p8,p9p_5,p_6,p_7,p_8,p_9 on the side BC, and four points p10,p11,p12,p13p_{10},p_{11},p_{12},p_{13} on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons that can be formed by taking all the vertices from the points p1,p2,…,p13p_1,p_2,\ldots,p_{13} is ____.
Not attempted
Your answer: —
Correct answer: 660
419
[JEE Main 2026, 22 Jan Shift 1, Q12] If A=[2335]A=\begin{bmatrix}2&3\\3&5\end{bmatrix}, then the determinant of the matrix (A2025−3A2024+A2023)(A^{2025}-3A^{2024}+A^{2023}) is
Not attempted
A
2828
B
2424
C
1616
D
1212
420
[JEE Main 2026, 22 Jan Shift 1, Q21] Let A be a 3×33\times3 matrix such that A+AT=OA+A^T=O. If A[1−10]=[332]A\begin{bmatrix}1\\-1\\0\end{bmatrix}=\begin{bmatrix}3\\3\\2\end{bmatrix}, A2[1−10]=[−319−24]A^2\begin{bmatrix}1\\-1\\0\end{bmatrix}=\begin{bmatrix}-3\\19\\-24\end{bmatrix} and det⁡(adj⁡(2adj⁡(A+I)))=(2)α⋅(3)β⋅(11)γ\det(\operatorname{adj}(2\operatorname{adj}(A+I)))=(2)^\alpha\cdot(3)^\beta\cdot(11)^\gamma, α,β,γ\alpha,\beta,\gamma are non-negative integers, then α+β+γ\alpha+\beta+\gamma is equal to ____
Not attempted
Your answer: —
Correct answer: 18
421
[JEE Main 2026, 22 Jan Shift 1, Q20] The number of distinct real solutions of the equation x∣x+4∣+3∣x+2∣+10=0x|x+4|+3|x+2|+10=0 is
Not attempted
A
33
B
11
C
22
D
00
422
[JEE Main 2026, 22 Jan Shift 1, Q23] Let α=−1+i32\alpha=\frac{-1+i\sqrt3}{2} and β=−1−i32\beta=\frac{-1-i\sqrt3}{2}, i=−1i=\sqrt{-1}. If (7−7α+9β)20+(9+7α−7β)20+(−7+9α+7β)20+(14+7α+7β)20=m10(7-7\alpha+9\beta)^{20}+(9+7\alpha-7\beta)^{20}+(-7+9\alpha+7\beta)^{20}+(14+7\alpha+7\beta)^{20}=m^{10}, then mm is ____.
Not attempted
Your answer: —
Correct answer: 49
423
[JEE Main 2026, 22 Jan Shift 1, Q7] Let the relation R on the set M={1,2,3,…,16}M=\{1,2,3,\ldots,16\} be given by R={(x,y):4y=5x−3, x,y∈M}R=\{(x,y):4y=5x-3,\ x,y\in M\}. Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to
Not attempted
A
44
B
11
C
33
D
22
424
[JEE Main 2026, 22 Jan Shift 1, Q17] If the domain of the function f(x)=sin⁡−1(5−x3+2x)+1log⁡e(10−x)f(x)=\sin^{-1}\left(\frac{5-x}{3+2x}\right)+\frac{1}{\log_e(10-x)} is (−∞,α]∪[β,γ)−{δ}(-\infty,\alpha]\cup[\beta,\gamma)-\{\delta\}, then 6(α+β+γ+δ)6(\alpha+\beta+\gamma+\delta) is equal to
Not attempted
A
7070
B
6767
C
6868
D
6666
425
[JEE Main 2026, 21 Jan Shift 2, Q24] Let the maximum value of (sin⁡−1x)2+(cos⁡−1x)2(\sin^{-1}x)^2+(\cos^{-1}x)^2 for x∈[−32,12]x\in\left[-\frac{\sqrt3}{2},\frac{1}{\sqrt2}\right] be mnπ2\frac{m}{n}\pi^2, where gcd⁡(m,n)=1\gcd(m,n)=1. Then m+nm+n is equal to ____.
Not attempted
Your answer: —
Correct answer: 65
426
[JEE Main 2026, 21 Jan Shift 2, Q17] A random variable X takes values 0, 1, 2, 3 with probabilities 2a+130\frac{2a+1}{30}, 8a−130\frac{8a-1}{30}, 4a+130\frac{4a+1}{30}, bb respectively, where a,b∈Ra,b\in\mathbb{R}. Let μ\mu and σ\sigma respectively be the mean and standard deviation of X such that σ2+μ2=2\sigma^2+\mu^2=2. Then ab\frac{a}{b} is equal to :
Not attempted
A
3030
B
6060
C
1212
D
33
427
[JEE Main 2026, 21 Jan Shift 2, Q13] For a triangle ABC, let p⃗=BC→\vec p=\overrightarrow{BC}, q⃗=CA→\vec q=\overrightarrow{CA} and r⃗=BA→\vec r=\overrightarrow{BA}. If ∣p⃗∣=23|\vec p|=2\sqrt3, ∣q⃗∣=2|\vec q|=2 and cos⁡θ=13\cos\theta=\frac{1}{\sqrt3}, where θ\theta is the angle between p⃗\vec p and q⃗\vec q, then ∣p⃗×(q⃗−3r⃗)∣2+3∣r⃗∣2|\vec p\times(\vec q-3\vec r)|^2+3|\vec r|^2 is equal to:
Not attempted
A
410410
B
340340
C
220220
D
200200
428
[JEE Main 2026, 21 Jan Shift 2, Q3] Let the line L1L_1 be parallel to the vector −3i^+2j^+4k^-3\hat i+2\hat j+4\hat k and pass through the point (2,6,7)(2,6,7), and the line L2L_2 be parallel to the vector 2i^+j^+3k^2\hat i+\hat j+3\hat k and pass through the point (4,3,5)(4,3,5). If the line L3L_3 is parallel to the vector −3i^+5j^+16k^-3\hat i+5\hat j+16\hat k and intersects the lines L1L_1 and L2L_2 at the points C and D, respectively, then ∣CD→∣2|\overrightarrow{CD}|^2 is equal to:
Not attempted
A
312312
B
290290
C
171171
D
8989
429
[JEE Main 2026, 21 Jan Shift 2, Q10] Let the line LL pass through the point (−3,5,2)(-3,5,2) and make equal angles with the positive coordinate axes. If the distance of LL from the point (−2,r,1)(-2,r,1) is 143\sqrt{\frac{14}{3}}, then the sum of all possible values of rr is:
Not attempted
A
66
B
1010
C
1212
D
1616
430
[JEE Main 2026, 21 Jan Shift 2, Q4] Let one end of a focal chord of the parabola y2=16xy^2=16x be (16,16)(16,16). If P(α,β)P(\alpha,\beta) divides this focal chord internally in the ratio 5:25:2, then the minimum value of α+β\alpha+\beta is equal to :
Not attempted
A
77
B
55
C
1616
D
2222
431
[JEE Main 2026, 21 Jan Shift 2, Q7] Let y2=12xy^2=12x be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the xx-axis such that ∠OPA=90∘\angle OPA=90^\circ. Then the locus of the centroid of such triangles OPA is :
Not attempted
A
y2−4x+8=0y^2-4x+8=0
B
y2−9x+6=0y^2-9x+6=0
C
y2−6x+4=0y^2-6x+4=0
D
y2−2x+8=0y^2-2x+8=0
432
[JEE Main 2026, 21 Jan Shift 2, Q11] If the line αx+4y=7\alpha x+4y=\sqrt7, where α∈R\alpha\in\mathbb{R}, touches the ellipse 3x2+4y2=13x^2+4y^2=1 at the point P in the first quadrant, then one of the focal distances of P is:
Not attempted
A
13+127\frac{1}{\sqrt3}+\frac{1}{2\sqrt7}
B
13−1211\frac{1}{\sqrt3}-\frac{1}{2\sqrt{11}}
C
13+125\frac{1}{\sqrt3}+\frac{1}{2\sqrt5}
D
13−125\frac{1}{\sqrt3}-\frac{1}{2\sqrt5}
433
[JEE Main 2026, 21 Jan Shift 2, Q25] If P is a point on the circle x2+y2=4x^2+y^2=4, Q is a point on the straight line 5x+y+2=05x+y+2=0 and x−y+1=0x-y+1=0 is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is
Not attempted
Your answer: —
Correct answer: 2
434
[JEE Main 2026, 21 Jan Shift 2, Q1] Let y=y(x)y=y(x) be the solution of the differential equation sec⁡xdydx−2y=2+3sin⁡x\sec x\frac{dy}{dx}-2y=2+3\sin x, x∈(−π2,π2)x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right), y(0)=−74y(0)=-\frac74. Then y(π6)y\left(\frac{\pi}{6}\right) is equal to :
Not attempted
A
−32−7-3\sqrt2-7
B
−33−7-3\sqrt3-7
C
−54-\frac54
D
−52-\frac52
435
[JEE Main 2026, 21 Jan Shift 2, Q18] If the area of the region {(x,y):1−2x≤y≤4−x2, x≥0, y≥0}\{(x,y):1-2x\le y\le4-x^2,\ x\ge0,\ y\ge0\} is αβ\frac{\alpha}{\beta}, α,β∈N\alpha,\beta\in\mathbb{N}, gcd⁡(α,β)=1\gcd(\alpha,\beta)=1, then the value of (α+β)(\alpha+\beta) is:
Not attempted
A
6767
B
7373
C
9191
D
8585
436
[JEE Main 2026, 21 Jan Shift 2, Q22] If ∫014cot⁡−1(1−2x+4x2) dx=atan⁡−1(2)−blog⁡e(5)\int_0^1 4\cot^{-1}(1-2x+4x^2)\,dx=a\tan^{-1}(2)-b\log_e(5), where a,b∈Na,b\in\mathbb{N}, then (2a+b)(2a+b) is equal to ____ .
Not attempted
Your answer: —
Correct answer: 9
437
[JEE Main 2026, 21 Jan Shift 2, Q8] Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a twice differentiable function such that f′′(x)>0f''(x)>0 for all x∈Rx\in\mathbb{R} and f′(a−1)=0f'(a-1)=0, where aa is a real number. Let g(x)=f(tan⁡2x−2tan⁡x+a)g(x)=f(\tan^2x-2\tan x+a), 0<x<π20<x<\frac{\pi}{2}. Consider the following two statements: (I) gg is increasing in (0,π4)\left(0,\frac{\pi}{4}\right) (II) gg is decreasing in (π4,π2)\left(\frac{\pi}{4},\frac{\pi}{2}\right) Then,
Not attempted
A
Neither (I) nor (II) is True
B
Only (II) is True
C
Both (I) and (II) are True
D
Only (I) is True
438
[JEE Main 2026, 21 Jan Shift 2, Q19] Let f(x)=x3+x2f′(1)+2xf′′(2)+f′′′(3)f(x)=x^3+x^2f'(1)+2xf''(2)+f'''(3), x∈Rx\in\mathbb{R}. Then the value of f′(5)f'(5) is:
Not attempted
A
1175\frac{117}{5}
B
6575\frac{657}{5}
C
25\frac25
D
625\frac{62}{5}
439
[JEE Main 2026, 21 Jan Shift 2, Q23] Let [⋅][\cdot] denote the greatest integer function and f(x)=lim⁡n→∞1n3∑k=1n[k23x]f(x)=\lim_{n\to\infty}\frac{1}{n^3}\sum_{k=1}^{n}\left[\frac{k^2}{3^x}\right]. Then 12∑j=1∞f(j)12\sum_{j=1}^{\infty}f(j) is equal to ______.
Not attempted
Your answer: —
Correct answer: 2
440
[JEE Main 2026, 21 Jan Shift 2, Q14] Let a1,a22,a322,…,a1029a_1,\frac{a_2}{2},\frac{a_3}{2^2},\ldots,\frac{a_{10}}{2^9} be a G.P. of common ratio 12\frac{1}{\sqrt2}. If a1+a2+…+a10=62a_1+a_2+\ldots+a_{10}=62, then a1a_1 is equal to :
Not attempted
A
2(2−2)2(2-\sqrt2)
B
2−1\sqrt2-1
C
2−22-\sqrt2
D
2(2−1)2(\sqrt2-1)
441
[JEE Main 2026, 21 Jan Shift 2, Q20] The positive integer nn, for which the solutions of the equation x(x+2)+(x+2)(x+4)+⋯+(x+2n−2)(x+2n)=8n3x(x+2)+(x+2)(x+4)+\cdots+(x+2n-2)(x+2n)=\frac{8n}{3} are two consecutive even integers, is :
Not attempted
A
33
B
99
C
66
D
1212
442
[JEE Main 2026, 21 Jan Shift 2, Q21] If (115C0+115C1)(115C1+115C2)⋯(115C12+115C13)=α1314C0 14C1⋯14C12\left(\frac{1}{{}^{15}C_0}+\frac{1}{{}^{15}C_1}\right)\left(\frac{1}{{}^{15}C_1}+\frac{1}{{}^{15}C_2}\right)\cdots\left(\frac{1}{{}^{15}C_{12}}+\frac{1}{{}^{15}C_{13}}\right)=\frac{\alpha^{13}}{{}^{14}C_0\,{}^{14}C_1\cdots{}^{14}C_{12}}, then 30α30\alpha is equal to
Not attempted
Your answer: —
Correct answer: 32
443
[JEE Main 2026, 21 Jan Shift 2, Q15] The largest n∈Nn\in\mathbb{N}, for which 7n7^n divides 101!101!, is :
Not attempted
A
1919
B
1515
C
1616
D
1818
444
[JEE Main 2026, 21 Jan Shift 2, Q6] For the matrices A=[3−41−1]A=\begin{bmatrix}3&-4\\1&-1\end{bmatrix} and B=[−2949−1318]B=\begin{bmatrix}-29&49\\-13&18\end{bmatrix}, if (A15+B)[xy]=[00](A^{15}+B)\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}, then among the following which one is true?
Not attempted
A
x=5, y=7x=5,\ y=7
B
x=16, y=3x=16,\ y=3
C
x=18, y=11x=18,\ y=11
D
x=11, y=2x=11,\ y=2
445
[JEE Main 2026, 21 Jan Shift 2, Q9] If the system of equations 3x+y+4z=33x+y+4z=3 2x+αy−z=−32x+\alpha y-z=-3 x+2y+z=4x+2y+z=4 has no solution, then the value of α\alpha is equal to:
Not attempted
A
1313
B
1919
C
44
D
2323
446
[JEE Main 2026, 21 Jan Shift 2, Q5] Let α\alpha and β\beta be the roots of the equation x2+2ax+(3a+10)=0x^2+2ax+(3a+10)=0 such that α<1<β\alpha<1<\beta. Then the set of all possible values of aa is :
Not attempted
A
(−∞,−115)∪(5,∞)\left(-\infty,-\frac{11}{5}\right)\cup(5,\infty)
B
(−∞,−2)∪(5,∞)(-\infty,-2)\cup(5,\infty)
C
(−∞,−3)(-\infty,-3)
D
(−∞,−115)\left(-\infty,-\frac{11}{5}\right)
447
[JEE Main 2026, 21 Jan Shift 2, Q16] Let zz be the complex number satisfying ∣z−5∣≤3|z-5|\le3 and having maximum positive principal argument. Then 34∣5z−125iz+16∣234\left|\frac{5z-12}{5iz+16}\right|^2 is equal to :
Not attempted
A
1212
B
1616
C
2626
D
2020
448
[JEE Main 2026, 21 Jan Shift 2, Q2] Let A={2,3,5,7,9}A=\{2,3,5,7,9\}. Let R be the relation on A defined by xRyxRy if and only if 2x≤3y2x\le3y. Let ll be the number of elements in R, and mm be the minimum number of elements required to be added in R to make it a symmetric relation. Then l+ml+m is equal to:
Not attempted
A
2727
B
2121
C
2525
D
2323
449
[JEE Main 2026, 21 Jan Shift 2, Q12] Let A={x:∣x2−10∣≤6}A=\{x:|x^2-10|\le6\} and B={x:∣x−2∣>1}B=\{x:|x-2|>1\}. Then
Not attempted
A
A∩B=[−4,−2]∪[3,4]A\cap B=[-4,-2]\cup[3,4]
B
A∪B=(−∞,1]∪(2,∞)A\cup B=(-\infty,1]\cup(2,\infty)
C
B−A=(−∞,−4)∪(−2,1)∪(4,∞)B-A=(-\infty,-4)\cup(-2,1)\cup(4,\infty)
D
A−B=[2,3)A-B=[2,3)
450
[JEE Main 2026, 21 Jan Shift 1, Q11] The value of cosec⁡10∘−3sec⁡10∘\operatorname{cosec}10^\circ-\sqrt3\sec10^\circ is equal to :
Not attempted
A
66
B
88
C
44
D
22
451
[JEE Main 2026, 21 Jan Shift 1, Q1] Let the mean and variance of 7 observations 2,4,10,x,12,14,y2,4,10,x,12,14,y, x>yx>y, be 8 and 16 respectively. Two numbers are chosen from {1,2,3,x−4,y,5}\{1,2,3,x-4,y,5\} one after another without replacement. Then the probability that the smaller number among the two chosen numbers is less than 4 is :
Not attempted
A
25\frac{2}{5}
B
13\frac{1}{3}
C
45\frac{4}{5}
D
35\frac{3}{5}
452
[JEE Main 2026, 21 Jan Shift 1, Q3] Let c⃗\vec c and d⃗\vec d be vectors such that ∣c⃗+d⃗∣=29|\vec c+\vec d|=\sqrt{29} and c⃗×(2i^+3j^+4k^)=(2i^+3j^+4k^)×d⃗\vec c\times(2\hat i+3\hat j+4\hat k)=(2\hat i+3\hat j+4\hat k)\times\vec d. If λ1,λ2 (λ1>λ2)\lambda_1,\lambda_2\ (\lambda_1>\lambda_2) are the possible values of (c⃗+d⃗)⋅(−7i^+2j^+3k^)(\vec c+\vec d)\cdot(-7\hat i+2\hat j+3\hat k), then the equation K2x2+(K2−5K+λ1)xy+(3K+λ22)y2−8x+12y+λ2=0K^2x^2+(K^2-5K+\lambda_1)xy+\left(3K+\frac{\lambda_2}{2}\right)y^2-8x+12y+\lambda_2=0 represents a circle, for KK equal to :
Not attempted
A
−1-1
B
11
C
44
D
22
453
[JEE Main 2026, 21 Jan Shift 1, Q20] Let a⃗=−i^+2j^+2k^\vec a=-\hat i+2\hat j+2\hat k, b⃗=8i^+7j^−3k^\vec b=8\hat i+7\hat j-3\hat k and c⃗\vec c be a vector such that a⃗×c⃗=b⃗\vec a\times\vec c=\vec b. If c⃗⋅(i^+j^+k^)=4\vec c\cdot(\hat i+\hat j+\hat k)=4, then ∣a⃗+c⃗∣2|\vec a+\vec c|^2 is equal to :
Not attempted
A
3333
B
3030
C
3535
D
2727
454
[JEE Main 2026, 21 Jan Shift 1, Q2] Let (α,β,γ)(\alpha,\beta,\gamma) be the coordinates of the foot of the perpendicular drawn from the point (5,4,2)(5,4,2) on the line r⃗=(−i^+3j^+k^)+λ(2i^+3j^−k^)\vec r=(-\hat i+3\hat j+\hat k)+\lambda(2\hat i+3\hat j-\hat k). Then the length of the projection of the vector αi^+βj^+γk^\alpha\hat i+\beta\hat j+\gamma\hat k on the vector 6i^+2j^+3k^6\hat i+2\hat j+3\hat k is :
Not attempted
A
33
B
44
C
187\frac{18}{7}
D
157\frac{15}{7}
455
[JEE Main 2026, 21 Jan Shift 1, Q7] Let O be the vertex of the parabola x2=4yx^2=4y and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2:32:3 be the conic CC. Then the equation of the chord of CC, which is bisected at the point (1,2)(1,2), is :
Not attempted
A
4x−5y+6=04x-5y+6=0
B
x−2y+3=0x-2y+3=0
C
5x−4y+3=05x-4y+3=0
D
5x−y−3=05x-y-3=0
456
[JEE Main 2026, 21 Jan Shift 1, Q8] Let the foci of a hyperbola coincide with the foci of the ellipse x236+y216=1\frac{x^2}{36}+\frac{y^2}{16}=1. If the eccentricity of the hyperbola is 5, then the length of its latus rectum is :
Not attempted
A
24524\sqrt5
B
1616
C
965\frac{96}{\sqrt5}
D
1212
457
[JEE Main 2026, 21 Jan Shift 1, Q9] Let PQ and MN be two straight lines touching the circle x2+y2−4x−6y−3=0x^2+y^2-4x-6y-3=0 at the points A and B respectively. Let O be the centre of the circle and ∠AOB=π3\angle AOB=\frac{\pi}{3}. Then the locus of the point of intersection of the lines PQ and MN is :
Not attempted
A
x2+y2−18x−12y−25=0x^2+y^2-18x-12y-25=0
B
3(x2+y2)−18x−12y+25=03(x^2+y^2)-18x-12y+25=0
C
3(x2+y2)−12x−18y−25=03(x^2+y^2)-12x-18y-25=0
D
x2+y2−12x−18y−25=0x^2+y^2-12x-18y-25=0
458
[JEE Main 2026, 21 Jan Shift 1, Q17] Let a point A lie between the parallel lines L1L_1 and L2L_2 such that its distances from L1L_1 and L2L_2 are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC, where the points B and C lie on the lines L1L_1 and L2L_2, respectively, is :
Not attempted
A
12212\sqrt2
B
21321\sqrt3
C
2727
D
15615\sqrt6
459
[JEE Main 2026, 21 Jan Shift 1, Q15] Let y=y(x)y=y(x) be the solution curve of the differential equation (1+x2) dy+(y−tan⁡−1x) dx=0(1+x^2)\,dy+(y-\tan^{-1}x)\,dx=0, with initial condition y(0)=1y(0)=1. Then find the value of y(1)y(1).
Not attempted
A
2eπ/4+π4−1\frac{2}{e^{\pi/4}}+\frac{\pi}{4}-1
B
2eπ/4−π4−1\frac{2}{e^{\pi/4}}-\frac{\pi}{4}-1
C
4eπ/4−π2−1\frac{4}{e^{\pi/4}}-\frac{\pi}{2}-1
D
4eπ/4+π2−1\frac{4}{e^{\pi/4}}+\frac{\pi}{2}-1
460
[JEE Main 2026, 21 Jan Shift 1, Q21] Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a twice differentiable function such that the quadratic equation f(x)m2−2f′(x)m+f′′(x)=0f(x)m^2-2f'(x)m+f''(x)=0 in mm, has two equal roots for every x∈Rx\in\mathbb{R}. If f(0)=1f(0)=1, f′(0)=2f'(0)=2, and (α,β)(\alpha,\beta) is the largest interval in which the function f(log⁡ex−x)f(\log_e x-x) is increasing, then α+β\alpha+\beta is equal to
Not attempted
Your answer: —
Correct answer: 1
461
[JEE Main 2026, 21 Jan Shift 1, Q4] The area of the region, inside the ellipse x2+4y2=4x^2+4y^2=4 and outside the region bounded by the curves y=∣x∣−1y=|x|-1 and y=1−∣x∣y=1-|x|, is :
Not attempted
A
2(π−1)2(\pi-1)
B
2π−122\pi-\frac{1}{2}
C
3(π−1)3(\pi-1)
D
2π−12\pi-1
462
[JEE Main 2026, 21 Jan Shift 1, Q13] The value of ∫−π/6π/6(π+4x111−sin⁡(∣x∣+π/6))dx\int_{-\pi/6}^{\pi/6}\left(\frac{\pi+4x^{11}}{1-\sin(|x|+\pi/6)}\right)dx is equal to :
Not attempted
A
8π8\pi
B
2π2\pi
C
6π6\pi
D
4π4\pi
463
[JEE Main 2026, 21 Jan Shift 1, Q24] 6∫0π∣sin⁡3x+sin⁡2x+sin⁡x∣dx6\int_0^{\pi}\left|\sin3x+\sin2x+\sin x\right|dx is equal to ____ .
Not attempted
Your answer: —
Correct answer: 17
464
[JEE Main 2026, 21 Jan Shift 1, Q10] Let f:R→(0,∞)f:\mathbb{R}\to(0,\infty) be a twice differentiable function such that f(3)=18f(3)=18, f′(3)=0f'(3)=0 and f′′(3)=4f''(3)=4. Then evaluate: lim⁡x→1 log⁡e((f(2+x)f(3))18(x−1)2)\lim_{x\to1}\ \log_e\left(\left(\frac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^2}}\right)
Not attempted
A
22
B
99
C
1818
D
11
465
[JEE Main 2026, 21 Jan Shift 1, Q14] Let a1,a2,a3,…a_1,a_2,a_3,\ldots be a G.P. of increasing positive terms such that a2⋅a3⋅a4=64a_2\cdot a_3\cdot a_4=64 and a1+a3+a5=8137a_1+a_3+a_5=\frac{813}{7}. Then a3+a5+a7a_3+a_5+a_7 is equal to :
Not attempted
A
32443244
B
32563256
C
32483248
D
32523252
466
[JEE Main 2026, 21 Jan Shift 1, Q22] Let a1=1a_1=1 and for n≥1n\ge1, an+1=12an+n2−2n−1n2(n+1)2a_{n+1}=\frac12a_n+\frac{n^2-2n-1}{n^2(n+1)^2}. Then ∣∑n=1∞(an−2n2)∣\left|\sum_{n=1}^{\infty}\left(a_n-\frac{2}{n^2}\right)\right| is equal to ____ .
Not attempted
Your answer: —
Correct answer: 2
467
[JEE Main 2026, 21 Jan Shift 1, Q5] If the coefficient of xx in the expansion of (ax2+bx+c)(1−2x)26(ax^2+bx+c)(1-2x)^{26} is −56-56 and the coefficients of x2x^2 and x3x^3 are both zero, then a+b+ca+b+c is equal to :
Not attempted
A
14831483
B
13001300
C
15001500
D
14031403
468
[JEE Main 2026, 21 Jan Shift 1, Q6] The number of strictly increasing functions ff from the set {1,2,3,4,5,6}\{1,2,3,4,5,6\} to the set {1,2,3,…,9}\{1,2,3,\ldots,9\} such that f(i)≠if(i)\neq i for 1≤i≤61\le i\le6, is equal to :
Not attempted
A
2121
B
2727
C
2828
D
2222
469
[JEE Main 2026, 21 Jan Shift 1, Q25] Let S={(m,n):m,n∈{1,2,3,…,50}}S=\{(m,n):m,n\in\{1,2,3,\ldots,50\}\}. If the number of elements (m,n)(m,n) in SS such that 6m+9n6^m+9^n is a multiple of 5 is pp and the number of elements (m,n)(m,n) in SS such that m+nm+n is a square of a prime number is qq, then p+qp+q is equal to ______.
Not attempted
Your answer: —
Correct answer: 1333
470
[JEE Main 2026, 21 Jan Shift 1, Q23] For some α,β∈R\alpha,\beta\in\mathbb{R}, let A=[α212]A=\begin{bmatrix}\alpha&2\\1&2\end{bmatrix} and B=[111β]B=\begin{bmatrix}1&1\\1&\beta\end{bmatrix} be such that A2−4A+2I=B2−3B+I=OA^2-4A+2I=B^2-3B+I=O. Then (det⁡(adj⁡(A3−B3)))2\left(\det\left(\operatorname{adj}(A^3-B^3)\right)\right)^2 is equal to ____ .
Not attempted
Your answer: —
Correct answer: 225
471
[JEE Main 2026, 21 Jan Shift 1, Q12] The sum of all the roots of the equation (x−1)2−5∣x−1∣+6=0(x-1)^2-5|x-1|+6=0, is :
Not attempted
A
55
B
44
C
11
D
33
472
[JEE Main 2026, 21 Jan Shift 1, Q18] If x2+x+1=0x^2+x+1=0, then the value of (x+1x)4+(x2+1x2)4+(x3+1x3)4+…+(x25+1x25)4\left(x+\frac1x\right)^4+\left(x^2+\frac{1}{x^2}\right)^4+\left(x^3+\frac{1}{x^3}\right)^4+\ldots+\left(x^{25}+\frac{1}{x^{25}}\right)^4 is :
Not attempted
A
145145
B
175175
C
128128
D
162162
473
[JEE Main 2026, 21 Jan Shift 1, Q16] If the domain of the function f(x)=cos⁡−1(2x−511−3x)+sin⁡−1(2x2−3x+1)f(x)=\cos^{-1}\left(\frac{2x-5}{11-3x}\right)+\sin^{-1}(2x^2-3x+1) is the interval [α,β][\alpha,\beta], then α+2β\alpha+2\beta is equal to :
Not attempted
A
11
B
22
C
55
D
33
474
[JEE Main 2026, 21 Jan Shift 1, Q19] The number of relations, defined on the set {a,b,c,d}\{a,b,c,d\}, which are both reflexive and symmetric, is equal to :
Not attempted
A
1616
B
256256
C
6464
D
10241024
Diagonal pairs are forced; each unordered pair {x,y}\{x,y\} with x≠yx\ne y is either in (both orders) or out.
Deconstruction\textbf{Deconstruction}: Count reflexive and symmetric relations on a 4-element set. Strategy\textbf{Strategy}: Reflexive fixes all (x,x)(x,x); symmetry pairs (x,y)(x,y) with (y,x)(y,x). Execution\textbf{Execution}: Number of unordered pairs of distinct elements: (42)=6\binom42=6. Each can be included or not: 26=642^6=64. Verification\textbf{Verification}: For n=2n=2 the formula gives 21=22^1=2: {(a,a),(b,b)}\{(a,a),(b,b)\} and with (a,b),(b,a)(a,b),(b,a) added ✓\checkmark. Final Answer\textbf{Final Answer}: (C) 6464