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JEE Advanced · Mathematics · 2024

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1
Let p⃗=2i^+j^+3k^\vec p=2\hat i+\hat j+3\hat k and q⃗=i^−j^+k^\vec q=\hat i-\hat j+\hat k. If for some real numbers α,β,\alpha,\beta, and γ\gamma, we have 15i^+10j^+6k^=α(2p⃗+q⃗)+β(p⃗−2q⃗)+γ(p⃗×q⃗)15\hat i+10\hat j+6\hat k=\alpha(2\vec p+\vec q)+\beta(\vec p-2\vec q)+\gamma(\vec p\times\vec q), then the value of γ\gamma is _______.
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Your answer: —
Correct answer: 2
2
A bag contains NN balls out of which 33 balls are white, 66 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For i=1,2,3i=1,2,3, let Wi,Gi,W_i,G_i, and BiB_i denote the events that the ball drawn in the ithi^{\text{th}} draw is a white ball, green ball, and blue ball, respectively. If the probability P(W1∩G2∩B3)=25NP(W_1\cap G_2\cap B_3)=\dfrac{2}{5N} and the conditional probability P(B3∣W1∩G2)=29P(B_3\mid W_1\cap G_2)=\dfrac{2}{9}, then NN equals ______.
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Your answer: —
Correct answer: 11
3
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a function such that f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y) for all x,y∈Rx,y\in\mathbb{R}, and g:R→(0,∞)g:\mathbb{R}\to(0,\infty) be a function such that g(x+y)=g(x)g(y)g(x+y)=g(x)g(y) for all x,y∈Rx,y\in\mathbb{R}. If f(−35)=12f\left(-\dfrac35\right)=12 and g(−13)=2g\left(-\dfrac13\right)=2, then the value of (f(14)+g(−2)−8)g(0)\left(f\left(\dfrac14\right)+g(-2)-8\right)g(0) is ________.
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Your answer: —
Correct answer: 51
4
Paragraph\textbf{Paragraph}: Let S={1,2,3,4,5,6}S=\{1,2,3,4,5,6\} and XX be the set of all relations RR from SS to SS that satisfy both the following properties: (i) RR has exactly 66 elements. (ii) For each (a,b)∈R(a,b)\in R, we have ∣a−b∣≥2|a-b|\ge2. If n(X)=mC6n(X)={}^{m}C_6, then the value of mm is __________.
Not attempted
Your answer: —
Correct answer: 20
5
Paragraph\textbf{Paragraph}: Let S={1,2,3,4,5,6}S=\{1,2,3,4,5,6\} and XX be the set of all relations RR from SS to SS that satisfy both the following properties: (i) RR has exactly 66 elements. (ii) For each (a,b)∈R(a,b)\in R, we have ∣a−b∣≥2|a-b|\ge2. Let Y={R∈X: the range of R has exactly one element}Y=\{R\in X:\text{ the range of }R\text{ has exactly one element}\} and Z={R∈X:R is a function from S to S}Z=\{R\in X: R\text{ is a function from }S\text{ to }S\}. Let n(A)n(A) denote the number of elements in a set AA. If the value of n(Y)+n(Z)n(Y)+n(Z) is k2k^2, then ∣k∣|k| is __________.
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Your answer: —
Correct answer: 36
6
Let A1,B1,C1A_1,B_1,C_1 be three points in the xyxy-plane. Suppose that the lines A1C1A_1C_1 and B1C1B_1C_1 are tangents to the curve y2=8xy^2=8x at A1A_1 and B1B_1, respectively. If O=(0,0)O=(0,0) and C1=(−4,0)C_1=(-4,0), then which of the following statements is (are) TRUE?
Not attempted
A
The length of the line segment OA1OA_1 is 434\sqrt3
B
The length of the line segment A1B1A_1B_1 is 1616
C
The orthocenter of the triangle A1B1C1A_1B_1C_1 is (0,0)(0,0)
D
The orthocenter of the triangle A1B1C1A_1B_1C_1 is (1,0)(1,0)
7
A normal with slope 16\dfrac{1}{\sqrt6} is drawn from the point (0,−α)(0,-\alpha) to the parabola x2=−4ayx^2=-4ay, where a>0a>0. Let LL be the line passing through (0,−α)(0,-\alpha) and parallel to the directrix of the parabola. Suppose that LL intersects the parabola at two points AA and BB. Let rr denote the length of the latus rectum and ss denote the square of the length of the line segment ABAB. If r:s=1:16r:s=1:16, then the value of 24a24a is __________.
Not attempted
Your answer: —
Correct answer: 12
8
A straight line drawn from the point P(1,3,2)P(1,3,2), parallel to the line x−21=y−42=z−61\dfrac{x-2}{1}=\dfrac{y-4}{2}=\dfrac{z-6}{1}, intersects the plane L1:x−y+3z=6L_1: x-y+3z=6 at the point QQ. Another straight line which passes through QQ and is perpendicular to the plane L1L_1 intersects the plane L2:2x−y+z=−4L_2: 2x-y+z=-4 at the point RR. Then which of the following statements is (are) TRUE?
Not attempted
A
The length of the line segment PQPQ is 6\sqrt6
B
The coordinates of RR are (1,6,3)(1,6,3)
C
The centroid of the triangle PQRPQR is (43,143,53)\left(\dfrac43,\dfrac{14}3,\dfrac53\right)
D
The perimeter of the triangle PQRPQR is 2+6+11\sqrt2+\sqrt6+\sqrt{11}
9
Let k∈Rk\in\mathbb{R}. If lim⁡x→0+(sin⁡(sin⁡kx)+cos⁡x+x)2x=e6\displaystyle\lim_{x\to0^{+}}\left(\sin(\sin kx)+\cos x+x\right)^{\frac{2}{x}}=e^{6}, then the value of kk is
Not attempted
A
11
B
22
C
33
D
44
10
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a function defined by f(x)={x2sin⁡(πx2),if x≠0,0,if x=0.f(x)=\begin{cases} x^2\sin\left(\dfrac{\pi}{x^2}\right), & \text{if } x\ne0, \\ 0, & \text{if } x=0.\end{cases} Then which of the following statements is TRUE?
Not attempted
A
f(x)=0f(x)=0 has infinitely many solutions in the interval [11010,∞)\left[\dfrac{1}{10^{10}},\infty\right).
B
f(x)=0f(x)=0 has no solutions in the interval [1π,∞)\left[\dfrac{1}{\pi},\infty\right).
C
The set of solutions of f(x)=0f(x)=0 in the interval (0,11010)\left(0,\dfrac{1}{10^{10}}\right) is finite.
D
f(x)=0f(x)=0 has more than 2525 solutions in the interval (1π2,1π)\left(\dfrac{1}{\pi^2},\dfrac{1}{\pi}\right).
11
Let SS be the set of all (α,β)∈R×R(\alpha,\beta)\in\mathbb{R}\times\mathbb{R} such that lim⁡x→∞sin⁡(x2)(log⁡ex)αsin⁡(1x2)xαβ(log⁡e(1+x))β=0\displaystyle\lim_{x\to\infty}\frac{\sin(x^2)(\log_e x)^{\alpha}\sin\left(\dfrac{1}{x^2}\right)}{x^{\alpha\beta}(\log_e(1+x))^{\beta}}=0. Then which of the following is (are) correct?
Not attempted
A
(−1,3)∈S(-1,3)\in S
B
(−1,1)∈S(-1,1)\in S
C
(1,−1)∈S(1,-1)\in S
D
(1,−2)∈S(1,-2)\in S
12
Let the function f:R→Rf:\mathbb{R}\to\mathbb{R} be defined by f(x)=sin⁡xeπx⋅(x2023+2024x+2025)(x2−x+3)+2eπx⋅(x2023+2024x+2025)(x2−x+3)f(x)=\dfrac{\sin x}{e^{\pi x}}\cdot\dfrac{\left(x^{2023}+2024x+2025\right)}{\left(x^2-x+3\right)}+\dfrac{2}{e^{\pi x}}\cdot\dfrac{\left(x^{2023}+2024x+2025\right)}{\left(x^2-x+3\right)}. Then the number of solutions of f(x)=0f(x)=0 in R\mathbb{R} is ______.
Not attempted
Your answer: —
Correct answer: 1
13
Let S={(x,y)∈R×R:x≥0, y≥0, y2≤4x, y2≤12−2x, and 3y+8 x≤58}S=\left\{(x,y)\in\mathbb{R}\times\mathbb{R}: x\ge 0,\ y\ge 0,\ y^2\le 4x,\ y^2\le 12-2x,\ \text{and } 3y+\sqrt{8}\,x\le 5\sqrt{8}\right\}. If the area of the region SS is α2\alpha\sqrt{2}, then α\alpha is equal to
Not attempted
A
172\dfrac{17}{2}
B
173\dfrac{17}{3}
C
174\dfrac{17}{4}
D
175\dfrac{17}{5}
14
Let the function f:[1,∞)→Rf:[1,\infty)\to\mathbb{R} be defined by f(t)={(−1)n+12,if t=2n−1, n∈N,(2n+1−t)2f(2n−1)+(t−(2n−1))2f(2n+1),if 2n−1<t<2n+1, n∈N.f(t)=\begin{cases}(-1)^{n+1}2, & \text{if } t=2n-1,\ n\in\mathbb{N}, \\[4pt] \dfrac{(2n+1-t)}{2}f(2n-1)+\dfrac{(t-(2n-1))}{2}f(2n+1), & \text{if } 2n-1<t<2n+1,\ n\in\mathbb{N}.\end{cases} Define g(x)=∫1xf(t) dt, x∈(1,∞)g(x)=\displaystyle\int_1^x f(t)\,dt,\ x\in(1,\infty). Let α\alpha denote the number of solutions of the equation g(x)=0g(x)=0 in the interval (1,8](1,8] and β=lim⁡x→1+g(x)x−1\beta=\displaystyle\lim_{x\to1^+}\dfrac{g(x)}{x-1}. Then the value of α+β\alpha+\beta is equal to _____.
Not attempted
Your answer: —
Correct answer: 5
15
Paragraph\textbf{Paragraph}: Let f:[0,π2]→[0,1]f:\left[0,\dfrac{\pi}{2}\right]\to[0,1] be the function defined by f(x)=sin⁡2xf(x)=\sin^2x and let g:[0,π2]→[0,∞)g:\left[0,\dfrac{\pi}{2}\right]\to[0,\infty) be the function defined by g(x)=πx2−x2g(x)=\sqrt{\dfrac{\pi x}{2}-x^2}. The value of 2∫0π/2f(x)g(x) dx−∫0π/2g(x) dx2\displaystyle\int_0^{\pi/2}f(x)g(x)\,dx-\int_0^{\pi/2}g(x)\,dx is __________.
Not attempted
Your answer: —
Correct answer: 0
16
Paragraph\textbf{Paragraph}: Let f:[0,π2]→[0,1]f:\left[0,\dfrac{\pi}{2}\right]\to[0,1] be the function defined by f(x)=sin⁡2xf(x)=\sin^2x and let g:[0,π2]→[0,∞)g:\left[0,\dfrac{\pi}{2}\right]\to[0,\infty) be the function defined by g(x)=πx2−x2g(x)=\sqrt{\dfrac{\pi x}{2}-x^2}. The value of 16π3∫0π/2f(x)g(x) dx\dfrac{16}{\pi^3}\displaystyle\int_0^{\pi/2}f(x)g(x)\,dx is __________.
Not attempted
Your answer: —
Correct answer: 0.25
17
Considering only the principal values of the inverse trigonometric functions, the value of tan⁡(sin⁡−1(35)−2cos⁡−1(25))\tan\left(\sin^{-1}\left(\dfrac{3}{5}\right) - 2\cos^{-1}\left(\dfrac{2}{\sqrt{5}}\right)\right) is
Not attempted
A
724\dfrac{7}{24}
B
−724-\dfrac{7}{24}
C
−524-\dfrac{5}{24}
D
524\dfrac{5}{24}
18
Let OP→=α−1αi^+j^+k^\overrightarrow{OP}=\dfrac{\alpha-1}{\alpha}\hat i+\hat j+\hat k, OQ→=i^+β−1βj^+k^\overrightarrow{OQ}=\hat i+\dfrac{\beta-1}{\beta}\hat j+\hat k and OR→=i^+j^+12k^\overrightarrow{OR}=\hat i+\hat j+\dfrac12\hat k be three vectors, where α,β∈R−{0}\alpha,\beta\in\mathbb R-\{0\} and OO denotes the origin. If (OP→×OQ→)⋅OR→=0\left(\overrightarrow{OP}\times\overrightarrow{OQ}\right)\cdot\overrightarrow{OR}=0 and the point (α,β,2)(\alpha,\beta,2) lies on the plane 3x+3y−z+l=03x+3y-z+l=0, then the value of ll is ________.
Not attempted
Your answer: —
Correct answer: 5
19
A group of 99 students, s1,s2,…,s9s_1,s_2,\ldots,s_9, is to be divided to form three teams X,Y,ZX,Y,Z of sizes 2,3,2,3, and 44, respectively. Suppose that s1s_1 cannot be selected for the team XX, and s2s_2 cannot be selected for the team YY. Then the number of ways to form such teams, is _______.
Not attempted
Your answer: —
Correct answer: 665
20
Let f(x)=x4+ax3+bx2+cf(x)=x^4+ax^3+bx^2+c be a polynomial with real coefficients such that f(1)=−9f(1)=-9. Suppose that i3i\sqrt3 is a root of the equation 4x3+3ax2+2bx=04x^3+3ax^2+2bx=0, where i=−1i=\sqrt{-1}. If α1,α2,α3,α4\alpha_1,\alpha_2,\alpha_3,\alpha_4 are all the roots of the equation f(x)=0f(x)=0, then ∣α1∣2+∣α2∣2+∣α3∣2+∣α4∣2|\alpha_1|^2+|\alpha_2|^2+|\alpha_3|^2+|\alpha_4|^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 20
21
Let R3\mathbb R^3 denote the three-dimensional space. Take two points P=(1,2,3)P=(1,2,3) and Q=(4,2,7)Q=(4,2,7). Let dist(X,Y)\text{dist}(X,Y) denote the distance between two points XX and YY in R3\mathbb R^3. Let S={X∈R3:(dist(X,P))2−(dist(X,Q))2=50}S=\{X\in\mathbb R^3 : (\text{dist}(X,P))^2-(\text{dist}(X,Q))^2=50\} and T={Y∈R3:(dist(Y,Q))2−(dist(Y,P))2=50}T=\{Y\in\mathbb R^3 : (\text{dist}(Y,Q))^2-(\text{dist}(Y,P))^2=50\}. Then which of the following statements is (are) TRUE?
Not attempted
A
There is a triangle whose area is 11 and all of whose vertices are from SS.
B
There are two distinct points LL and MM in TT such that each point on segment LMLM is also in TT.
C
There are infinitely many rectangles of perimeter 4848, two of whose vertices are from SS and the other two from TT.
D
There is a square of perimeter 4848, two of whose vertices are from SS and the other two from TT.
22
Let γ∈R\gamma\in\mathbb R be such that the lines L1:x+111=y+212=z+293L_1:\dfrac{x+11}{1}=\dfrac{y+21}{2}=\dfrac{z+29}{3} and L2:x+163=y+112=z+4γL_2:\dfrac{x+16}{3}=\dfrac{y+11}{2}=\dfrac{z+4}{\gamma} intersect. Let R1R_1 be the point of intersection of L1L_1 and L2L_2. Let O=(0,0,0)O=(0,0,0), and n^\hat n denote a unit normal vector to the plane containing both lines L1L_1 and L2L_2. Match each entry in List-I to the correct entry in List-II. List-I: (P) γ\gamma equals (Q) A possible choice for n^\hat n is (R) OR1→\overrightarrow{OR_1} equals (S) A possible value of OR1→⋅n^\overrightarrow{OR_1}\cdot\hat n is List-II: (1) −i^−j^+k^-\hat i-\hat j+\hat k (2) 3/2\sqrt{3/2} (3) 11 (4) 16i^−26j^+16k^\dfrac1{\sqrt6}\hat i-\dfrac2{\sqrt6}\hat j+\dfrac1{\sqrt6}\hat k (5) 2/3\sqrt{2/3} The correct option is
Not attempted
A
(P)→\to(3) (Q)→\to(4) (R)→\to(1) (S)→\to(2)
B
(P)→\to(5) (Q)→\to(4) (R)→\to(1) (S)→\to(2)
C
(P)→\to(3) (Q)→\to(4) (R)→\to(1) (S)→\to(5)
D
(P)→\to(3) (Q)→\to(1) (R)→\to(4) (S)→\to(5)
23
Let R2\mathbb R^2 denote R×R\mathbb R\times\mathbb R. Let S={(a,b,c):a,b,c∈R and ax2+2bxy+cy2>0 for all (x,y)∈R2−{(0,0)}}S=\{(a,b,c) : a,b,c\in\mathbb R \text{ and } ax^2+2bxy+cy^2>0 \text{ for all } (x,y)\in\mathbb R^2-\{(0,0)\}\}. Then which of the following statements is (are) TRUE?
Not attempted
A
(2,72,6)∈S\left(2,\dfrac72,6\right)\in S
B
If (3,b,112)∈S\left(3,b,\dfrac1{12}\right)\in S, then ∣2b∣<1|2b|<1.
C
For any given (a,b,c)∈S(a,b,c)\in S, the system ax+by=1, bx+cy=−1ax+by=1,\ bx+cy=-1 has a unique solution.
D
For any given (a,b,c)∈S(a,b,c)\in S, the system (a+1)x+by=0, bx+(c+1)y=0(a+1)x+by=0,\ bx+(c+1)y=0 has a unique solution.
24
Let S={A=(01c1ad1be):a,b,c,d,e∈{0,1} and ∣A∣∈{−1,1}}S=\left\{A=\begin{pmatrix}0&1&c\\1&a&d\\1&b&e\end{pmatrix} : a,b,c,d,e\in\{0,1\} \text{ and } |A|\in\{-1,1\}\right\}, where ∣A∣|A| denotes the determinant of AA. Then the number of elements in SS is _______.
Not attempted
Your answer: —
Correct answer: 16
25
Let α\alpha and β\beta be the distinct roots of the equation x2+x−1=0x^2+x-1=0. Consider the set T={1,α,β}T=\{1,\alpha,\beta\}. For a 3×33\times3 matrix M=(aij)3×3M=(a_{ij})_{3\times3}, define Ri=ai1+ai2+ai3R_i=a_{i1}+a_{i2}+a_{i3} and Cj=a1j+a2j+a3jC_j=a_{1j}+a_{2j}+a_{3j} for i=1,2,3i=1,2,3 and j=1,2,3j=1,2,3. Match each entry in List-I to the correct entry in List-II. List-I: (P) The number of matrices M=(aij)3×3M=(a_{ij})_{3\times3} with all entries in TT such that Ri=Cj=0R_i=C_j=0 for all i,ji,j is (Q) The number of symmetric matrices M=(aij)3×3M=(a_{ij})_{3\times3} with all entries in TT such that Cj=0C_j=0 for all jj is (R) Let M=(aij)3×3M=(a_{ij})_{3\times3} be a skew symmetric matrix such that aij∈Ta_{ij}\in T for i>ji>j. Then the number of elements in the set {(xyz):x,y,z∈R, M(xyz)=(a120−a23)}\left\{\begin{pmatrix}x\\y\\z\end{pmatrix} : x,y,z\in\mathbb R,\ M\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}a_{12}\\0\\-a_{23}\end{pmatrix}\right\} is (S) Let M=(aij)3×3M=(a_{ij})_{3\times3} be a matrix with all entries in TT such that Ri=0R_i=0 for all ii. Then the absolute value of the determinant of MM is List-II: (1) 11 (2) 1212 (3) infinite (4) 66 (5) 00 The correct option is
Not attempted
A
(P)→\to(4) (Q)→\to(2) (R)→\to(5) (S)→\to(1)
B
(P)→\to(2) (Q)→\to(4) (R)→\to(1) (S)→\to(5)
C
(P)→\to(2) (Q)→\to(4) (R)→\to(3) (S)→\to(5)
D
(P)→\to(1) (Q)→\to(5) (R)→\to(3) (S)→\to(4)
26
Let S={a+b2:a,b∈Z}S=\{a+b\sqrt2 : a,b\in\mathbb Z\}, T1={(−1+2)n:n∈N}T_1=\{(-1+\sqrt2)^n : n\in\mathbb N\}, and T2={(1+2)n:n∈N}T_2=\{(1+\sqrt2)^n : n\in\mathbb N\}. Then which of the following statements is (are) TRUE?
Not attempted
A
Z∪T1∪T2⊂S\mathbb Z\cup T_1\cup T_2\subset S
B
T1∩(0,12024)=ϕT_1\cap\left(0,\dfrac1{2024}\right)=\phi
C
T2∩(2024,∞)≠ϕT_2\cap(2024,\infty)\ne\phi
D
For any a,b∈Za,b\in\mathbb Z, cos⁡(π(a+b2))+isin⁡(π(a+b2))∈Z\cos(\pi(a+b\sqrt2))+i\sin(\pi(a+b\sqrt2))\in\mathbb Z if and only if b=0b=0
27
Let a=32a=3\sqrt2 and b=151/66b=\dfrac{1}{5^{1/6}\sqrt6}. If x,y∈Rx,y\in\mathbb R are such that 3x+2y=log⁡a(18)5/43x+2y=\log_a\left(18\right)^{5/4} and 2x−y=log⁡b(1080)2x-y=\log_b\left(\sqrt{1080}\right), then 4x+5y4x+5y is equal to ________.
Not attempted
Your answer: —
Correct answer: 8
28
Let f:R→Rf:\mathbb R\to\mathbb R and g:R→Rg:\mathbb R\to\mathbb R be functions defined by f(x)={x∣x∣sin⁡(1x),x≠00,x=0f(x)=\begin{cases}x|x|\sin\left(\dfrac1x\right), & x\ne0\\ 0, & x=0\end{cases} and g(x)={1−2x,0≤x≤120,otherwiseg(x)=\begin{cases}1-2x, & 0\le x\le\dfrac12\\ 0, & \text{otherwise}\end{cases} Let a,b,c,d∈Ra,b,c,d\in\mathbb R. Define the function h:R→Rh:\mathbb R\to\mathbb R by h(x)=a f(x)+b(g(x)+g(12−x))+c(x−g(x))+d g(x), x∈Rh(x)=a\,f(x)+b\left(g(x)+g\left(\dfrac12-x\right)\right)+c\left(x-g(x)\right)+d\,g(x),\ x\in\mathbb R. Match each entry in List-I to the correct entry in List-II. List-I: (P) If a=0,b=1,c=0,d=0a=0,b=1,c=0,d=0, then (Q) If a=1,b=0,c=0,d=0a=1,b=0,c=0,d=0, then (R) If a=0,b=0,c=1,d=0a=0,b=0,c=1,d=0, then (S) If a=0,b=0,c=0,d=1a=0,b=0,c=0,d=1, then List-II: (1) hh is one-one. (2) hh is onto. (3) hh is differentiable on R\mathbb R. (4) the range of hh is [0,1][0,1]. (5) the range of hh is {0,1}\{0,1\}. The correct option is
Not attempted
A
(P)→\to(4) (Q)→\to(3) (R)→\to(1) (S)→\to(2)
B
(P)→\to(5) (Q)→\to(2) (R)→\to(4) (S)→\to(3)
C
(P)→\to(5) (Q)→\to(3) (R)→\to(2) (S)→\to(4)
D
(P)→\to(4) (Q)→\to(2) (R)→\to(1) (S)→\to(3)
29
Consider the ellipse x29+y24=1\dfrac{x^2}{9}+\dfrac{y^2}{4}=1. Let S(p,q)S(p,q) be a point in the first quadrant such that p29+q24>1\dfrac{p^2}{9}+\dfrac{q^2}{4}>1. Two tangents are drawn from SS to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point TT in the fourth quadrant. Let RR be the vertex of the ellipse with positive xx-coordinate and OO be the center of the ellipse. If the area of the triangle △ORT\triangle ORT is 32\dfrac32, then which of the following options is correct?
Not attempted
A
q=2, p=33q=2,\ p=3\sqrt3
B
q=2, p=43q=2,\ p=4\sqrt3
C
q=1, p=53q=1,\ p=5\sqrt3
D
q=1, p=63q=1,\ p=6\sqrt3
30
Let the straight line y=2xy=2x touch a circle with center (0,α)(0,\alpha), α>0\alpha>0, and radius rr at a point A1A_1. Let B1B_1 be the point on the circle such that the line segment A1B1A_1B_1 is a diameter of the circle. Let α+r=5+5\alpha+r=5+\sqrt5. Match each entry in List-I to the correct entry in List-II. List-I: (P) α\alpha equals (Q) rr equals (R) A1A_1 equals (S) B1B_1 equals List-II: (1) (−2,4)(-2,4) (2) 5\sqrt5 (3) (−2,6)(-2,6) (4) 55 (5) (2,4)(2,4) The correct option is
Not attempted
A
(P)→\to(4) (Q)→\to(2) (R)→\to(1) (S)→\to(3)
B
(P)→\to(2) (Q)→\to(4) (R)→\to(1) (S)→\to(3)
C
(P)→\to(4) (Q)→\to(2) (R)→\to(5) (S)→\to(3)
D
(P)→\to(2) (Q)→\to(4) (R)→\to(3) (S)→\to(5)
31
Let π2<x<π\dfrac{\pi}{2}<x<\pi be such that cot⁡x=−511\cot x=\dfrac{-5}{\sqrt{11}}. Then (sin⁡11x2)(sin⁡6x−cos⁡6x)+(cos⁡11x2)(sin⁡6x+cos⁡6x)\left(\sin\dfrac{11x}{2}\right)\left(\sin6x-\cos6x\right)+\left(\cos\dfrac{11x}{2}\right)\left(\sin6x+\cos6x\right) is equal to
Not attempted
A
11−123\dfrac{\sqrt{11}-1}{2\sqrt3}
B
11+123\dfrac{\sqrt{11}+1}{2\sqrt3}
C
11+132\dfrac{\sqrt{11}+1}{3\sqrt2}
D
11−132\dfrac{\sqrt{11}-1}{3\sqrt2}
32
A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is 12\dfrac12. Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is 16\dfrac16. Then the probability that the student knows the answer of a randomly chosen question is
Not attempted
A
112\dfrac{1}{12}
B
17\dfrac{1}{7}
C
57\dfrac{5}{7}
D
512\dfrac{5}{12}
33
Let XX be a random variable, and let P(X=x)P(X=x) denote the probability that XX takes the value xx. Suppose that the points (x,P(X=x))(x,P(X=x)), x=0,1,2,3,4x=0,1,2,3,4, lie on a fixed straight line in the xyxy-plane, and P(X=x)=0P(X=x)=0 for all x∈R−{0,1,2,3,4}x\in\mathbb R-\{0,1,2,3,4\}. If the mean of XX is 52\dfrac52, and the variance of XX is α\alpha, then the value of 24α24\alpha is ______.
Not attempted
Your answer: —
Correct answer: 42
34
Let f(x)f(x) be a continuously differentiable function on the interval (0,∞)(0,\infty) such that f(1)=2f(1)=2 and lim⁡t→xt10f(x)−x10f(t)t9−x9=1\displaystyle\lim_{t\to x}\frac{t^{10}f(x)-x^{10}f(t)}{t^9-x^9}=1 for each x>0x>0. Then, for all x>0x>0, f(x)f(x) is equal to
Not attempted
A
3111x−911x10\dfrac{31}{11x}-\dfrac{9}{11}x^{10}
B
911x+1311x10\dfrac{9}{11x}+\dfrac{13}{11}x^{10}
C
−911x+3111x10\dfrac{-9}{11x}+\dfrac{31}{11}x^{10}
D
1311x+911x10\dfrac{13}{11x}+\dfrac{9}{11}x^{10}
The limit is a 0/00/0 form in tt as t→xt\to x. Apply L'Hopital's rule differentiating with respect to tt (treating xx as fixed) to convert the limit condition into a differential equation in xx, then solve the resulting linear ODE using an integrating factor.
Deconstruction\textbf{Deconstruction}: We are given a limit condition that must reduce to a first-order linear ODE satisfied by ff, plus the initial condition f(1)=2f(1)=2. This is a single-correct question. Strategy\textbf{Strategy}: As t→xt\to x, numerator t10f(x)−x10f(t)→x10f(x)−x10f(x)=0t^{10}f(x)-x^{10}f(t)\to x^{10}f(x)-x^{10}f(x)=0 and denominator t9−x9→0t^9-x^9\to0, a 0/00/0 form. Apply L'Hopital's rule in the variable tt. Execution\textbf{Execution}: Differentiating numerator and denominator with respect to tt: lim⁡t→x10t9f(x)−x10f′(t)9t8=1\displaystyle\lim_{t\to x}\frac{10t^9f(x)-x^{10}f'(t)}{9t^8}=1 Substituting t=xt=x: 10x9f(x)−x10f′(x)9x8=1\dfrac{10x^9f(x)-x^{10}f'(x)}{9x^8}=1, so 10x9f(x)−x10f′(x)=9x810x^9f(x)-x^{10}f'(x)=9x^8. Dividing by x8x^8 (valid since x>0x>0): x2f′(x)−10xf(x)=−9x^2f'(x)-10xf(x)=-9, i.e. f′(x)−10xf(x)=−9x2f'(x)-\dfrac{10}{x}f(x)=-\dfrac{9}{x^2}. This is linear in ff. Integrating factor μ(x)=e∫−10/x dx=x−10\mu(x)=e^{\int -10/x\,dx}=x^{-10}. ddx[x−10f(x)]=−9x−12\dfrac{d}{dx}\left[x^{-10}f(x)\right]=-9x^{-12} Integrating: x−10f(x)=911x−11+Cx^{-10}f(x)=\dfrac{9}{11}x^{-11}+C, so f(x)=911x+Cx10f(x)=\dfrac{9}{11x}+Cx^{10}. Using f(1)=2f(1)=2: 911+C=2⇒C=1311\dfrac{9}{11}+C=2 \Rightarrow C=\dfrac{13}{11}. So f(x)=911x+1311x10f(x)=\dfrac{9}{11x}+\dfrac{13}{11}x^{10}. Verification\textbf{Verification}: Substitute back into the ODE: f′(x)=−911x2+13011x9f'(x)=-\dfrac{9}{11x^2}+\dfrac{130}{11}x^9. Check x2f′(x)−10xf(x)=x2(−911x2+13011x9)−10x(911x+1311x10)=−911+13011x11−9011−13011x11=−9911=−9x^2f'(x)-10xf(x)=x^2\left(-\dfrac{9}{11x^2}+\dfrac{130}{11}x^9\right)-10x\left(\dfrac{9}{11x}+\dfrac{13}{11}x^{10}\right)=-\dfrac{9}{11}+\dfrac{130}{11}x^{11}-\dfrac{90}{11}-\dfrac{130}{11}x^{11}=-\dfrac{99}{11}=-9. This matches exactly, confirming the derivation. Also f(1)=9/11+13/11=22/11=2f(1)=9/11+13/11=22/11=2, matching the given condition. Final Answer\textbf{Final Answer}: Option (B): f(x)=911x+1311x10f(x)=\dfrac{9}{11x}+\dfrac{13}{11}x^{10}.