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JEE Advanced · Mathematics · 2018

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1
Let f1:R→Rf_1:\mathbb{R}\to\mathbb{R}, f2:(−π2,π2)→Rf_2:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\to\mathbb{R}, f3:(−1,eπ/2−2)→Rf_3:\left(-1,e^{\pi/2}-2\right)\to\mathbb{R} and f4:R→Rf_4:\mathbb{R}\to\mathbb{R} be functions defined by: (i) f1(x)=sin⁡(1−e−x2)f_1(x)=\sin\left(\sqrt{1-e^{-x^2}}\right), (ii) f2(x)={∣sin⁡x∣tan⁡−1xif x≠01if x=0f_2(x)=\begin{cases}\frac{|\sin x|}{\tan^{-1}x} & \text{if } x\neq0 \\ 1 & \text{if } x=0\end{cases}, where the inverse trigonometric function tan⁡−1x\tan^{-1}x assumes values in (−π2,π2)\left(-\frac{\pi}{2},\frac{\pi}{2}\right), (iii) f3(x)=[sin⁡(log⁡e(x+2))]f_3(x)=\left[\sin(\log_e(x+2))\right], where, for t∈Rt\in\mathbb{R}, [t][t] denotes the greatest integer less than or equal to tt, (iv) f4(x)={x2sin⁡(1x)if x≠00if x=0f_4(x)=\begin{cases}x^2\sin\left(\frac1x\right) & \text{if } x\neq0 \\ 0 & \text{if } x=0\end{cases}. Match each entry in LIST-I to the correct entry in LIST-II. LIST-I\textbf{LIST-I} P. The function f1f_1 is Q. The function f2f_2 is R. The function f3f_3 is S. The function f4f_4 is LIST-II\textbf{LIST-II} 1. NOT continuous at x=0x=0 2. continuous at x=0x=0 and NOT differentiable at x=0x=0 3. differentiable at x=0x=0 and its derivative is NOT continuous at x=0x=0 4. differentiable at x=0x=0 and its derivative is continuous at x=0x=0
Not attempted
A
P→\to2; Q→\to3; R→\to1; S→\to4
B
P→\to4; Q→\to1; R→\to2; S→\to3
C
P→\to4; Q→\to2; R→\to1; S→\to3
D
P→\to2; Q→\to1; R→\to4; S→\to3
2
Let H:x2a2−y2b2=1H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1, where a>b>0a>b>0, be a hyperbola in the xyxy-plane whose conjugate axis LMLM subtends an angle of 60∘60^\circ at one of its vertices NN. Let the area of the triangle LMNLMN be 434\sqrt3. Match each entry in LIST-I to the correct entry in LIST-II. LIST-I\textbf{LIST-I} P. The length of the conjugate axis of HH is Q. The eccentricity of HH is R. The distance between the foci of HH is S. The length of the latus rectum of HH is LIST-II\textbf{LIST-II} 1. 88 2. 43\frac{4}{\sqrt3} 3. 23\frac{2}{\sqrt3} 4. 44
Not attempted
A
P→\to4; Q→\to2; R→\to1; S→\to3
B
P→\to4; Q→\to3; R→\to1; S→\to2
C
P→\to4; Q→\to1; R→\to3; S→\to2
D
P→\to3; Q→\to4; R→\to2; S→\to1
3
In a high school, a committee has to be formed from a group of 66 boys M1,M2,M3,M4,M5,M6M_1,M_2,M_3,M_4,M_5,M_6 and 55 girls G1,G2,G3,G4,G5G_1,G_2,G_3,G_4,G_5. (i) Let α1\alpha_1 be the total number of ways in which the committee can be formed such that the committee has 55 members, having exactly 33 boys and 22 girls. (ii) Let α2\alpha_2 be the total number of ways in which the committee can be formed such that the committee has at least 22 members, and having an equal number of boys and girls. (iii) Let α3\alpha_3 be the total number of ways in which the committee can be formed such that the committee has 55 members, at least 22 of them being girls. (iv) Let α4\alpha_4 be the total number of ways in which the committee can be formed such that the committee has 44 members, having at least 22 girls and such that both M1M_1 and G1G_1 are NOT in the committee together. Match each entry in LIST-I to the correct entry in LIST-II. LIST-I\textbf{LIST-I} P. The value of α1\alpha_1 is Q. The value of α2\alpha_2 is R. The value of α3\alpha_3 is S. The value of α4\alpha_4 is LIST-II\textbf{LIST-II} 1. 136136 2. 189189 3. 192192 4. 200200 5. 381381 6. 461461
Not attempted
A
P→\to4; Q→\to6; R→\to2; S→\to1
B
P→\to1; Q→\to4; R→\to2; S→\to3
C
P→\to4; Q→\to6; R→\to5; S→\to2
D
P→\to4; Q→\to2; R→\to3; S→\to1
4
Let E1={x∈R:x≠1 and xx−1>0}E_1=\left\{x\in\mathbb{R}: x\neq1 \text{ and } \frac{x}{x-1}>0\right\} and E2={x∈E1:sin⁡−1(log⁡e(xx−1)) is a real number}E_2=\left\{x\in E_1: \sin^{-1}\left(\log_e\left(\frac{x}{x-1}\right)\right) \text{ is a real number}\right\}. (Here, the inverse trigonometric function sin⁡−1x\sin^{-1}x assumes values in [−π2,π2]\left[-\frac{\pi}{2},\frac{\pi}{2}\right].) Let f:E1→Rf:E_1\to\mathbb{R} be the function defined by f(x)=log⁡e(xx−1)f(x)=\log_e\left(\frac{x}{x-1}\right) and g:E2→Rg:E_2\to\mathbb{R} be the function defined by g(x)=sin⁡−1(log⁡e(xx−1))g(x)=\sin^{-1}\left(\log_e\left(\frac{x}{x-1}\right)\right). Match each entry in LIST-I to the correct entry in LIST-II. LIST-I\textbf{LIST-I} P. The range of ff is Q. The range of gg contains R. The domain of ff contains S. The domain of gg is LIST-II\textbf{LIST-II} 1. (−∞,11−e]∪[ee−1,∞)\left(-\infty,\frac{1}{1-e}\right]\cup\left[\frac{e}{e-1},\infty\right) 2. (0,1)(0,1) 3. [−12,12]\left[-\frac12,\frac12\right] 4. (−∞,0)∪(0,∞)(-\infty,0)\cup(0,\infty) 5. (−∞,ee−1]\left(-\infty,\frac{e}{e-1}\right] 6. (−∞,0)∪(12,ee−1](-\infty,0)\cup\left(\frac12,\frac{e}{e-1}\right]
Not attempted
A
P→\to4; Q→\to2; R→\to1; S→\to1
B
P→\to3; Q→\to3; R→\to6; S→\to5
C
P→\to4; Q→\to2; R→\to1; S→\to6
D
P→\to4; Q→\to3; R→\to6; S→\to5
5
Let X=(10C1)2+2(10C2)2+3(10C3)2+⋯+10(10C10)2X=\left({}^{10}C_1\right)^2+2\left({}^{10}C_2\right)^2+3\left({}^{10}C_3\right)^2+\cdots+10\left({}^{10}C_{10}\right)^2, where 10Cr{}^{10}C_r, r∈{1,2,…,10}r\in\{1,2,\ldots,10\} denote binomial coefficients. Then, the value of 11430X\frac{1}{1430}X is _____.
Not attempted
Your answer: —
Correct answer: 646
6
Consider the cube in the first octant with sides OPOP, OQOQ and OROR of length 11, along the xx-axis, yy-axis and zz-axis respectively, where O(0,0,0)O(0,0,0) is the origin. Let S(12,12,12)S\left(\frac12,\frac12,\frac12\right) be the centre of the cube and TT be the vertex of the cube opposite to the origin OO such that SS lies on the diagonal OTOT. If p⃗=SP⃗\vec p=\vec{SP}, q⃗=SQ⃗\vec q=\vec{SQ}, r⃗=RS⃗\vec r=\vec{RS} and t⃗=ST⃗\vec t=\vec{ST}, then the value of ∣(p⃗×q⃗)×(r⃗×t⃗)∣\left|(\vec p\times\vec q)\times(\vec r\times\vec t)\right| is _____.
Not attempted
Your answer: —
Correct answer: 0.5
7
Let PP be a point in the first octant, whose image QQ in the plane x+y=3x+y=3 (that is, the line segment PQPQ is perpendicular to the plane x+y=3x+y=3 and the mid-point of PQPQ lies in the plane x+y=3x+y=3) lies on the zz-axis. Let the distance of PP from the xx-axis be 55. If RR is the image of PP in the xyxy-plane, then the length of RPRP is _____.
Not attempted
Your answer: —
Correct answer: 8
8
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a differentiable function with f(0)=1f(0)=1 and satisfying the equation f(x+y)=f(x)f′(y)+f′(x)f(y)f(x+y)=f(x)f'(y)+f'(x)f(y) for all x,y∈Rx,y\in\mathbb{R}. Then, the value of log⁡e(f(4))\log_e(f(4)) is _____.
Not attempted
Your answer: —
Correct answer: 2
9
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a differentiable function with f(0)=0f(0)=0. If y=f(x)y=f(x) satisfies the differential equation dydx=(2+5y)(5y−2)\frac{dy}{dx}=(2+5y)(5y-2), then the value of lim⁡x→−∞f(x)\lim_{x\to-\infty}f(x) is _____.
Not attempted
Your answer: —
Correct answer: 0.4
10
Let XX be a set with exactly 55 elements and YY be a set with exactly 77 elements. If α\alpha is the number of one-one functions from XX to YY and β\beta is the number of onto functions from YY to XX, then the value of 15!(β−α)\frac{1}{5!}(\beta-\alpha) is _____.
Not attempted
Your answer: —
Correct answer: 119
11
Let PP be a matrix of order 3×33\times3 such that all the entries in PP are from the set {−1,0,1}\{-1,0,1\}. Then, the maximum possible value of the determinant of PP is _____.
Not attempted
Your answer: —
Correct answer: 4
12
The value of the integral ∫01/21+3((x+1)2(1−x)6)1/4 dx\int_{0}^{1/2}\frac{1+\sqrt3}{\left((x+1)^2(1-x)^6\right)^{1/4}}\,dx is _____.
Not attempted
Your answer: —
Correct answer: 2
13
Let f:(0,π)→Rf:(0,\pi)\to\mathbb{R} be a twice differentiable function such that lim⁡t→xf(x)sin⁡t−f(t)sin⁡xt−x=sin⁡2x\lim_{t\to x}\frac{f(x)\sin t - f(t)\sin x}{t-x}=\sin^2 x for all x∈(0,π)x\in(0,\pi). If f(π6)=−π12f\left(\frac{\pi}{6}\right)=-\frac{\pi}{12}, then which of the following statement(s) is (are) TRUE?
Not attempted
A
f(π4)=π42f\left(\frac{\pi}{4}\right) = \frac{\pi}{4\sqrt2}
B
f(x)<x46−x2f(x) < \frac{x^4}{6} - x^2 for all x∈(0,π)x\in(0,\pi)
C
There exists α∈(0,π)\alpha\in(0,\pi) such that f′(α)=0f'(\alpha)=0
D
f′′(π2)+f(π2)=0f''\left(\frac{\pi}{2}\right) + f\left(\frac{\pi}{2}\right) = 0
14
Let s,t,rs,t,r be non-zero complex numbers and LL be the set of solutions z=x+iyz=x+iy (x,y∈R,i=−1)(x,y\in\mathbb{R}, i=\sqrt{-1}) of the equation sz+tzˉ+r=0sz+t\bar z+r=0, where zˉ=x−iy\bar z=x-iy. Then, which of the following statement(s) is (are) TRUE?
Not attempted
A
If LL has exactly one element, then ∣s∣≠∣t∣|s|\neq|t|
B
If ∣s∣=∣t∣|s|=|t|, then LL has infinitely many elements
C
The number of elements in L∩{z:∣z−1+i∣=5}L\cap\{z:|z-1+i|=5\} is at most 22
D
If LL has more than one element, then LL has infinitely many elements
15
Consider two straight lines, each of which is tangent to both the circle x2+y2=12x^2+y^2=\frac{1}{2} and the parabola y2=4xy^2=4x. Let these lines intersect at the point QQ. Consider the ellipse whose center is at the origin O(0,0)O(0,0) and whose semi-major axis is OQOQ. If the length of the minor axis of this ellipse is 2\sqrt{2}, then which of the following statement(s) is (are) TRUE?
Not attempted
A
For the ellipse, the eccentricity is 12\frac{1}{\sqrt2} and the length of the latus rectum is 11
B
For the ellipse, the eccentricity is 12\frac{1}{2} and the length of the latus rectum is 12\frac{1}{2}
C
The area of the region bounded by the ellipse between the lines x=12x=\frac{1}{\sqrt2} and x=1x=1 is 142(π−2)\frac{1}{4\sqrt2}(\pi-2)
D
The area of the region bounded by the ellipse between the lines x=12x=\frac{1}{\sqrt2} and x=1x=1 is 116(π−2)\frac{1}{16}(\pi-2)
16
Let SS be the set of all column matrices (b1b2b3)\begin{pmatrix}b_1\\b_2\\b_3\end{pmatrix} such that b1,b2,b3∈Rb_1,b_2,b_3\in\mathbb{R} and the system of equations (in real variables) −x+2y+5z=b1-x+2y+5z=b_1 2x−4y+3z=b22x-4y+3z=b_2 x−2y+2z=b3x-2y+2z=b_3 has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each (b1b2b3)∈S\begin{pmatrix}b_1\\b_2\\b_3\end{pmatrix}\in S?
Not attempted
A
x+2y+3z=b1, 4y+5z=b2, x+2y+6z=b3x+2y+3z=b_1,\ 4y+5z=b_2,\ x+2y+6z=b_3
B
x+y+3z=b1, 5x+2y+6z=b2, −2x−y−3z=b3x+y+3z=b_1,\ 5x+2y+6z=b_2,\ -2x-y-3z=b_3
C
−x+2y−5z=b1, 2x−4y+10z=b2, x−2y+5z=b3-x+2y-5z=b_1,\ 2x-4y+10z=b_2,\ x-2y+5z=b_3
D
x+2y+5z=b1, 2x+3z=b2, x+4y−5z=b3x+2y+5z=b_1,\ 2x+3z=b_2,\ x+4y-5z=b_3
17
Let TT be the line passing through the points P(−2,7)P(-2,7) and Q(2,−5)Q(2,-5). Let F1F_1 be the set of all pairs of circles (S1,S2)(S_1,S_2) such that TT is tangent to S1S_1 at PP and tangent to S2S_2 at QQ, and also such that S1S_1 and S2S_2 touch each other at a point, say, MM. Let E1E_1 be the set representing the locus of MM as the pair (S1,S2)(S_1,S_2) varies in F1F_1. Let the set of all straight line segments joining a pair of distinct points of E1E_1 and passing through the point R(1,1)R(1,1) be F2F_2. Let E2E_2 be the set of the mid-points of the line segments in the set F2F_2. Then, which of the following statement(s) is (are) TRUE?
Not attempted
A
The point (−2,7)(-2,7) lies in E1E_1
B
The point (45,75)\left(\frac{4}{5},\frac{7}{5}\right) does NOT lie in E2E_2
C
The point (12,1)\left(\frac{1}{2},1\right) lies in E2E_2
D
The point (0,32)\left(0,\frac{3}{2}\right) does NOT lie in E1E_1
18
For any positive integer nn, define fn:(0,∞)→Rf_n: (0, \infty) \to \mathbb{R} as fn(x)=∑j=1ntan⁡−1(11+(x+j)(x+j−1))f_n(x) = \sum_{j=1}^{n} \tan^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right) for all x∈(0,∞)x \in (0, \infty). (Here, the inverse trigonometric function tan⁡−1x\tan^{-1}x assumes values in (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right).) Then, which of the following statement(s) is (are) TRUE?
Not attempted
A
∑j=15tan⁡2(fj(0))=55\sum_{j=1}^{5} \tan^2(f_j(0)) = 55
B
∑j=110(1+fj′(0))sec⁡2(fj(0))=10\sum_{j=1}^{10} \left(1+f_j'(0)\right)\sec^2(f_j(0)) = 10
C
For any fixed positive integer nn, lim⁡x→∞tan⁡(fn(x))=1n\lim_{x\to\infty}\tan(f_n(x)) = \frac{1}{n}
D
For any fixed positive integer nn, lim⁡x→∞sec⁡2(fn(x))=1\lim_{x\to\infty}\sec^2(f_n(x)) = 1
19
There are five students S1,S2,S3,S4S_1, S_2, S_3, S_4 and S5S_5 in a music class and for them there are five seats R1,R2,R3,R4R_1, R_2, R_3, R_4 and R5R_5 arranged in a row, where initially the seat RiR_i is allotted to the student SiS_i, i=1,2,3,4,5i=1,2,3,4,5. But, on the examination day, the five students are randomly allotted the five seats. The probability that, on the examination day, the student S1S_1 gets the previously allotted seat R1R_1, and NONE of the remaining students gets the seat previously allotted to him/her is
Not attempted
A
340\dfrac{3}{40}
B
18\dfrac{1}{8}
C
740\dfrac{7}{40}
D
15\dfrac{1}{5}
20
There are five students S1,S2,S3,S4S_1, S_2, S_3, S_4 and S5S_5 in a music class and for them there are five seats R1,R2,R3,R4R_1, R_2, R_3, R_4 and R5R_5 arranged in a row, where initially the seat RiR_i is allotted to the student SiS_i, i=1,2,3,4,5i=1,2,3,4,5. But, on the examination day, the five students are randomly allotted the five seats. For i=1,2,3,4i=1,2,3,4, let TiT_i denote the event that the students SiS_i and Si+1S_{i+1} do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event T1∩T2∩T3∩T4T_1\cap T_2\cap T_3\cap T_4 is
Not attempted
A
115\dfrac{1}{15}
B
110\dfrac{1}{10}
C
760\dfrac{7}{60}
D
15\dfrac{1}{5}
21
Let SS be the circle in the xyxy-plane defined by the equation x2+y2=4x^2+y^2=4. Let E1E2E_1E_2 and F1F2F_1F_2 be the chords of SS passing through the point P0(1,1)P_0(1,1) and parallel to the xx-axis and the yy-axis, respectively. Let G1G2G_1G_2 be the chord of SS passing through P0P_0 and having slope −1-1. Let the tangents to SS at E1E_1 and E2E_2 meet at E3E_3, the tangents to SS at F1F_1 and F2F_2 meet at F3F_3, and the tangents to SS at G1G_1 and G2G_2 meet at G3G_3. Then, the points E3E_3, F3F_3, and G3G_3 lie on the curve
Not attempted
A
x+y=4x+y=4
B
(x−4)2+(y−4)2=16(x-4)^2+(y-4)^2=16
C
(x−4)(y−4)=4(x-4)(y-4)=4
D
xy=4xy=4
22
Let SS be the circle in the xyxy-plane defined by the equation x2+y2=4x^2+y^2=4. Let PP be a point on the circle SS with both coordinates being positive. Let the tangent to SS at PP intersect the coordinate axes at the points MM and NN. Then, the mid-point of the line segment MNMN must lie on the curve
Not attempted
A
(x+y)2=3xy(x+y)^2=3xy
B
x2/3+y2/3=24/3x^{2/3}+y^{2/3}=2^{4/3}
C
x2+y2=2xyx^2+y^2=2xy
D
x2+y2=x2y2x^2+y^2=x^2y^2
23
A farmer F1F_1 has a land in the shape of a triangle with vertices at P(0,0)P(0,0), Q(1,1)Q(1,1) and R(2,0)R(2,0). From this land, a neighbouring farmer F2F_2 takes away the region which lies between the side PQPQ and a curve of the form y=xny=x^n (n>1)(n>1). If the area of the region taken away by the farmer F2F_2 is exactly 30% of the area of △PQR\triangle PQR, then the value of nn is ______.
Not attempted
Your answer: —
Correct answer: 4
24
Let a,b,ca,b,c be three non-zero real numbers such that the equation 3acos⁡x+2bsin⁡x=c\sqrt3 a\cos x + 2b\sin x = c, x∈[−π2,π2]x\in\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right], has two distinct real roots α\alpha and β\beta with α+β=π3\alpha+\beta=\dfrac{\pi}{3}. Then, the value of ba\dfrac ba is ______.
Not attempted
Your answer: —
Correct answer: 0.5
25
Let a⃗\vec a and b⃗\vec b be two unit vectors such that a⃗⋅b⃗=0\vec a\cdot\vec b=0. For some x,y∈Rx,y\in\mathbb{R}, let c⃗=xa⃗+yb⃗+(a⃗×b⃗)\vec c = x\vec a+y\vec b+(\vec a\times\vec b). If ∣c⃗∣=2|\vec c|=2 and the vector c⃗\vec c is inclined at the same angle α\alpha to both a⃗\vec a and b⃗\vec b, then the value of 8cos⁡2α8\cos^2\alpha is ______.
Not attempted
Your answer: —
Correct answer: 3
26
For each positive integer nn, let yn=1n((n+1)(n+2)⋯(n+n))1/ny_n = \dfrac{1}{n}\Big((n+1)(n+2)\cdots(n+n)\Big)^{1/n}. For x∈Rx\in\mathbb{R}, let [x][x] be the greatest integer less than or equal to xx. If lim⁡n→∞yn=L\displaystyle\lim_{n\to\infty} y_n = L, then the value of [L][L] is ______.
Not attempted
Your answer: —
Correct answer: 1
27
The number of real solutions of the equation sin⁡−1(∑i=1∞xi+1−x∑i=1∞(x2)i)=π2−cos⁡−1(∑i=1∞(−x2)i−∑i=1∞(−x)i)\sin^{-1}\left(\displaystyle\sum_{i=1}^{\infty}x^{i+1} - x\sum_{i=1}^{\infty}\left(\dfrac{x}{2}\right)^{i}\right) = \dfrac{\pi}{2} - \cos^{-1}\left(\displaystyle\sum_{i=1}^{\infty}\left(-\dfrac{x}{2}\right)^{i} - \sum_{i=1}^{\infty}(-x)^{i}\right) lying in the interval (−12,12)\left(-\dfrac12,\dfrac12\right) is ______. (Here, sin⁡−1x\sin^{-1}x and cos⁡−1x\cos^{-1}x assume values in [−π2,π2]\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right] and [0,π][0,\pi], respectively.)
Not attempted
Your answer: —
Correct answer: 2
28
Let XX be the set consisting of the first 2018 terms of the arithmetic progression 1,6,11,…1,6,11,\ldots, and YY be the set consisting of the first 2018 terms of the arithmetic progression 9,16,23,…9,16,23,\ldots. Then, the number of elements in the set X∪YX\cup Y is ______.
Not attempted
Your answer: —
Correct answer: 3748
29
The number of 5 digit numbers which are divisible by 4, with digits from the set {1,2,3,4,5}\{1,2,3,4,5\} and the repetition of digits is allowed, is ______.
Not attempted
Your answer: —
Correct answer: 625
30
The value of ((log⁡29)2)1log⁡2(log⁡29)×(7)1log⁡47\left((\log_2 9)^2\right)^{\frac{1}{\log_2(\log_2 9)}} \times (\sqrt7)^{\frac{1}{\log_4 7}} is ______.
Not attempted
Your answer: —
Correct answer: 8
31
Let f:R→Rf:\mathbb{R}\to\mathbb{R} and g:R→Rg:\mathbb{R}\to\mathbb{R} be two non-constant differentiable functions. If f′(x)=(e(f(x)−g(x)))g′(x)f'(x) = \left(e^{(f(x)-g(x))}\right) g'(x) for all x∈Rx\in\mathbb{R}, and f(1)=g(2)=1f(1)=g(2)=1, then which of the following statement(s) is (are) TRUE?
Not attempted
A
f(2)<1−log⁡e2f(2) < 1-\log_e 2
B
f(2)>1−log⁡e2f(2) > 1-\log_e 2
C
g(1)>1−log⁡e2g(1) > 1-\log_e 2
D
g(1)<1−log⁡e2g(1) < 1-\log_e 2
32
Let f:[0,∞)→Rf:[0,\infty)\to\mathbb{R} be a continuous function such that f(x)=1−2x+∫0xex−tf(t) dtf(x) = 1-2x+\displaystyle\int_0^x e^{x-t}f(t)\,dt for all x∈[0,∞)x\in[0,\infty). Then, which of the following statement(s) is (are) TRUE?
Not attempted
A
The curve y=f(x)y=f(x) passes through the point (1,2)(1,2)
B
The curve y=f(x)y=f(x) passes through the point (2,−1)(2,-1)
C
The area of the region {(x,y)∈[0,1]×R:f(x)≤y≤1−x2}\{(x,y)\in[0,1]\times\mathbb{R} : f(x)\le y\le\sqrt{1-x^2}\} is π−24\dfrac{\pi-2}{4}
D
The area of the region {(x,y)∈[0,1]×R:f(x)≤y≤1−x2}\{(x,y)\in[0,1]\times\mathbb{R} : f(x)\le y\le\sqrt{1-x^2}\} is π−14\dfrac{\pi-1}{4}
33
For every twice differentiable function f:R→[−2,2]f:\mathbb{R}\to[-2,2] with (f(0))2+(f′(0))2=85(f(0))^2+(f'(0))^2=85, which of the following statement(s) is (are) TRUE?
Not attempted
A
There exist r,s∈Rr,s\in\mathbb{R}, where r<sr<s, such that ff is one-one on the open interval (r,s)(r,s)
B
There exists x0∈(−4,0)x_0\in(-4,0) such that ∣f′(x0)∣≤1|f'(x_0)|\le 1
C
lim⁡x→∞f(x)=1\displaystyle\lim_{x\to\infty} f(x)=1
D
There exists α∈(−4,4)\alpha\in(-4,4) such that f(α)+f′′(α)=0f(\alpha)+f''(\alpha)=0 and f′(α)≠0f'(\alpha)\ne0
34
Let P1:2x+y−z=3P_1: 2x+y-z=3 and P2:x+2y+z=2P_2: x+2y+z=2 be two planes. Then, which of the following statement(s) is (are) TRUE?
Not attempted
A
The line of intersection of P1P_1 and P2P_2 has direction ratios 1,2,−11,2,-1
B
The line 3x−49=1−3y9=z3\dfrac{3x-4}{9}=\dfrac{1-3y}{9}=\dfrac{z}{3} is perpendicular to the line of intersection of P1P_1 and P2P_2
C
The acute angle between P1P_1 and P2P_2 is 60∘60^\circ
D
If P3P_3 is the plane passing through the point (4,2,−2)(4,2,-2) and perpendicular to the line of intersection of P1P_1 and P2P_2, then the distance of the point (2,1,1)(2,1,1) from the plane P3P_3 is 23\dfrac{2}{\sqrt3}
35
In a triangle PQRPQR, let ∠PQR=30∘\angle PQR = 30^\circ and the sides PQPQ and QRQR have lengths 10310\sqrt{3} and 1010, respectively. Then, which of the following statement(s) is (are) TRUE?
Not attempted
A
∠QPR=45∘\angle QPR = 45^\circ
B
The area of the triangle PQRPQR is 25325\sqrt{3} and ∠QRP=120∘\angle QRP = 120^\circ
C
The radius of the incircle of the triangle PQRPQR is 103−1510\sqrt{3}-15
D
The area of the circumcircle of the triangle PQRPQR is 100π100\pi
36
For a non-zero complex number zz, let arg⁡(z)\arg(z) denote the principal argument with −π<arg⁡(z)≤π-\pi < \arg(z) \le \pi. Then, which of the following statement(s) is (are) FALSE?
Not attempted
A
arg⁡(−1−i)=π4\arg(-1-i) = \dfrac{\pi}{4}, where i=−1i=\sqrt{-1}
B
The function f:R→(−π,π]f:\mathbb{R}\to(-\pi,\pi], defined by f(t)=arg⁡(−1+it)f(t)=\arg(-1+it) for all t∈Rt\in\mathbb{R}, is continuous at all points of R\mathbb{R}, where i=−1i=\sqrt{-1}
C
For any two non-zero complex numbers z1z_1 and z2z_2, arg⁡(z1z2)−arg⁡(z1)+arg⁡(z2)\arg\left(\dfrac{z_1}{z_2}\right)-\arg(z_1)+\arg(z_2) is an integer multiple of 2π2\pi
D
For any three given distinct complex numbers z1,z2,z3z_1,z_2,z_3, the locus of the point zz satisfying arg⁡((z−z1)(z2−z3)(z−z3)(z2−z1))=π\arg\left(\dfrac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right)=\pi lies on a straight line
Compute the principal argument directly for (A). For (B) check the left and right limits of arg⁡(−1+it)\arg(-1+it) as t→0t \to 0. For (C) use the identity arg⁡(z1/z2)=arg⁡(z1)−arg⁡(z2)+2kπ\arg(z_1/z_2) = \arg(z_1) - \arg(z_2) + 2k\pi. For (D), recognize the expression as a cross-ratio condition for concyclicity.
Deconstruction\textbf{Deconstruction}: We must identify which of the four statements about principal arguments are FALSE. Strategy\textbf{Strategy}: Evaluate each statement directly using the definition −π<arg⁡(z)≤π-\pi < \arg(z) \le \pi, checking limits for continuity and using the standard argument-of-quotient identity for (C), and cross-ratio geometry for (D). Execution\textbf{Execution}: (A)\textbf{(A)} −1−i-1-i lies in the third quadrant. Its reference angle is π/4\pi/4, so the principal argument is arg⁡(−1−i)=−π+π/4=−3π/4\arg(-1-i) = -\pi+\pi/4 = -3\pi/4, not π/4\pi/4. So (A) is FALSE. (B)\textbf{(B)} At t=0t=0: f(0)=arg⁡(−1)=πf(0)=\arg(-1)=\pi. As t→0+t\to 0^+, the point −1+it-1+it sits just above the negative real axis (second quadrant), so f(t)→π−f(t)\to \pi^-. As t→0−t\to 0^-, the point −1+it-1+it sits just below the negative real axis (third quadrant), so its principal argument approaches −π-\pi (not π\pi), since arguments near the negative real axis from below jump to values near −π-\pi. So lim⁡t→0−f(t)=−π≠π=f(0)\lim_{t\to 0^-} f(t) = -\pi \ne \pi = f(0): ff has a jump discontinuity at t=0t=0. So (B), which claims continuity everywhere, is FALSE. (C)\textbf{(C)} For non-zero complex z1,z2z_1,z_2: arg⁡(z1)−arg⁡(z2)\arg(z_1)-\arg(z_2) and arg⁡(z1/z2)\arg(z_1/z_2) can differ only by an integer multiple of 2π2\pi (since both are arguments of the same complex number modulo 2π2\pi). Hence arg⁡(z1/z2)−arg⁡(z1)+arg⁡(z2)\arg(z_1/z_2)-\arg(z_1)+\arg(z_2) is always an integer multiple of 2π2\pi. This statement is TRUE, so it is not one of the false ones. (D)\textbf{(D)} Writing the condition as arg⁡(z−z1z−z3)=π+arg⁡(z2−z1z2−z3)(mod2π)\arg\left(\dfrac{z-z_1}{z-z_3}\right) = \pi + \arg\left(\dfrac{z_2-z_1}{z_2-z_3}\right) \pmod{2\pi}, a fixed constant depending on z1,z2,z3z_1,z_2,z_3: this says the segment z1z3z_1z_3 subtends a constant angle at zz, which is the classical locus of an arc of a circle through z1z_1 and z3z_3 (the inscribed-angle circle), not a straight line in general (a straight line only arises in the degenerate case where that constant angle is 00 or π\pi, which need not hold for arbitrary distinct z1,z2,z3z_1,z_2,z_3). So the claim "lies on a straight line" is FALSE in general. Verification\textbf{Verification}: For (A), a quick sketch of −1−i-1-i in the Argand plane confirms it is in the third quadrant, ruling out the first-quadrant-looking value π/4\pi/4. For (B), plugging in t=−0.001t=-0.001 numerically gives arg⁡(−1−0.001i)≈−π+0.001≈−3.1406\arg(-1-0.001i) \approx -\pi + 0.001 \approx -3.1406, far from f(0)=π≈3.1416f(0)=\pi\approx3.1416, confirming the jump. For (D), the standard cross-ratio-real criterion for four points being concyclic or collinear confirms that a constant non-0,π0,\pi argument value describes a genuine circular arc. Final Answer\textbf{Final Answer}: The FALSE statements are (A), (B), (D).