Clear Notion
Login

Solve PYQs

JEE Advanced · Mathematics · 2015

Your Score

0 / 184

Correct

0 / 46

Attempted

0 / 46

1
Setup (shared with Q.60)\textbf{Setup (shared with Q.60)}: Let F:R→RF:\mathbb R\to\mathbb R be a thrice differentiable function. Suppose that F(1)=0, F(3)=−4F(1)=0,\ F(3)=-4 and F′(x)<0F'(x)<0 for all x∈(12,3)x\in\left(\dfrac12,3\right). Let f(x)=xF(x)f(x)=xF(x) for all x∈Rx\in\mathbb R. If ∫13x2F′(x) dx=−12\displaystyle\int_1^3x^2F'(x)\,dx=-12 and ∫13x3F′′(x) dx=40\displaystyle\int_1^3x^3F''(x)\,dx=40, then the correct expression(s) is(are)
Not attempted
A
9f′(3)+f′(1)−32=09f'(3)+f'(1)-32=0
B
∫13f(x) dx=12\displaystyle\int_1^3f(x)\,dx=12
C
9f′(3)−f′(1)+32=09f'(3)-f'(1)+32=0
D
∫13f(x) dx=−12\displaystyle\int_1^3f(x)\,dx=-12
2
Setup (shared with Q.60)\textbf{Setup (shared with Q.60)}: Let F:R→RF:\mathbb R\to\mathbb R be a thrice differentiable function. Suppose that F(1)=0, F(3)=−4F(1)=0,\ F(3)=-4 and F′(x)<0F'(x)<0 for all x∈(12,3)x\in\left(\dfrac12,3\right). Let f(x)=xF(x)f(x)=xF(x) for all x∈Rx\in\mathbb R. The correct statement(s) is(are)
Not attempted
A
f′(1)<0f'(1)<0
B
f(2)<0f(2)<0
C
f′(x)≠0f'(x)\ne0 for any x∈(1,3)x\in(1,3)
D
f′(x)=0f'(x)=0 for some x∈(1,3)x\in(1,3)
3
Setup (shared with Q.58)\textbf{Setup (shared with Q.58)}: Let n1n_1 and n2n_2 be the number of red and black balls, respectively, in box I. Let n3n_3 and n4n_4 be the number of red and black balls, respectively, in box II. One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is 13\dfrac13, then the correct option(s) with the possible values of n1,n2,n3n_1,n_2,n_3 and n4n_4 is(are)
Not attempted
A
n1=3, n2=3, n3=5, n4=15n_1=3,\ n_2=3,\ n_3=5,\ n_4=15
B
n1=3, n2=6, n3=10, n4=50n_1=3,\ n_2=6,\ n_3=10,\ n_4=50
C
n1=8, n2=6, n3=5, n4=20n_1=8,\ n_2=6,\ n_3=5,\ n_4=20
D
n1=6, n2=12, n3=5, n4=20n_1=6,\ n_2=12,\ n_3=5,\ n_4=20
4
Setup (shared with Q.58)\textbf{Setup (shared with Q.58)}: Let n1n_1 and n2n_2 be the number of red and black balls, respectively, in box I. Let n3n_3 and n4n_4 be the number of red and black balls, respectively, in box II. A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is 13\dfrac13, then the correct option(s) with the possible values of n1n_1 and n2n_2 is(are)
Not attempted
A
n1=4n_1=4 and n2=6n_2=6
B
n1=2n_1=2 and n2=3n_2=3
C
n1=10n_1=10 and n2=20n_2=20
D
n1=3n_1=3 and n2=6n_2=6
5
Let f(x)=7tan⁡8x+7tan⁡6x−3tan⁡4x−3tan⁡2xf(x)=7\tan^8x+7\tan^6x-3\tan^4x-3\tan^2x for all x∈(−π2,π2)x\in\left(-\dfrac\pi2,\dfrac\pi2\right). Then the correct expression(s) is(are)
Not attempted
A
∫0π/4xf(x) dx=112\displaystyle\int_0^{\pi/4}xf(x)\,dx=\dfrac1{12}
B
∫0π/4f(x) dx=0\displaystyle\int_0^{\pi/4}f(x)\,dx=0
C
∫0π/4xf(x) dx=16\displaystyle\int_0^{\pi/4}xf(x)\,dx=\dfrac16
D
∫0π/4f(x) dx=1\displaystyle\int_0^{\pi/4}f(x)\,dx=1
6
Let f′(x)=192x32+sin⁡4πxf'(x)=\dfrac{192x^3}{2+\sin^4\pi x} for all x∈Rx\in\mathbb R with f ⁣(12)=0f\!\left(\dfrac12\right)=0. If m≤∫1/21f(x) dx≤Mm\le\displaystyle\int_{1/2}^{1}f(x)\,dx\le M, then the possible values of mm and MM are
Not attempted
A
m=13, M=24m=13,\ M=24
B
m=14, M=12m=\dfrac14,\ M=\dfrac12
C
m=−11, M=0m=-11,\ M=0
D
m=1, M=12m=1,\ M=12
7
Let f,g:[−1,2]→Rf,g:[-1,2]\to\mathbb R be continuous functions which are twice differentiable on the interval (−1,2)(-1,2). Let the values of ff and gg at the points −1,0-1,0 and 22 be given in the following table: x=−1x=0x=2f(x)360g(x)01−1\displaystyle\begin{array}{|c|c|c|c|}\hline & x=-1 & x=0 & x=2\\\hline f(x) & 3 & 6 & 0\\\hline g(x) & 0 & 1 & -1\\\hline\end{array}. In each of the intervals (−1,0)(-1,0) and (0,2)(0,2) the function (f−3g)′′(f-3g)'' never vanishes. Then the correct statement(s) is(are)
Not attempted
A
f′(x)−3g′(x)=0f'(x)-3g'(x)=0 has exactly three solutions in (−1,0)∪(0,2)(-1,0)\cup(0,2)
B
f′(x)−3g′(x)=0f'(x)-3g'(x)=0 has exactly one solution in (−1,0)(-1,0)
C
f′(x)−3g′(x)=0f'(x)-3g'(x)=0 has exactly one solution in (0,2)(0,2)
D
f′(x)−3g′(x)=0f'(x)-3g'(x)=0 has exactly two solutions in (−1,0)(-1,0) and exactly two solutions in (0,2)(0,2)
8
The option(s) with the values of aa and LL that satisfy the following equation is(are) ∫04πet(sin⁡6at+cos⁡4at)dt∫0πet(sin⁡6at+cos⁡4at)dt=L\dfrac{\displaystyle\int_0^{4\pi}e^t\left(\sin^6at+\cos^4at\right)dt}{\displaystyle\int_0^{\pi}e^t\left(\sin^6at+\cos^4at\right)dt}=L?
Not attempted
A
a=2, L=e4π−1eπ−1a=2,\ L=\dfrac{e^{4\pi}-1}{e^\pi-1}
B
a=2, L=e4π+1eπ+1a=2,\ L=\dfrac{e^{4\pi}+1}{e^\pi+1}
C
a=4, L=e4π−1eπ−1a=4,\ L=\dfrac{e^{4\pi}-1}{e^\pi-1}
D
a=4, L=e4π+1eπ+1a=4,\ L=\dfrac{e^{4\pi}+1}{e^\pi+1}
9
Let E1E_1 and E2E_2 be two ellipses whose centers are at the origin. The major axes of E1E_1 and E2E_2 lie along the xx-axis and the yy-axis, respectively. Let SS be the circle x2+(y−1)2=2x^2+(y-1)^2=2. The straight line x+y=3x+y=3 touches the curves S,E1,E2S,E_1,E_2 at P,QP,Q and RR, respectively. Suppose that PQ=PR=223PQ=PR=\dfrac{2\sqrt2}{3}. If e1e_1 and e2e_2 are the eccentricities of E1E_1 and E2E_2, respectively, then the correct expression(s) is(are)
Not attempted
A
e12+e22=4340e_1^2+e_2^2=\dfrac{43}{40}
B
e1e2=7210e_1e_2=\dfrac{\sqrt7}{2\sqrt{10}}
C
∣e12−e22∣=58\left|e_1^2-e_2^2\right|=\dfrac58
D
e1e2=34e_1e_2=\dfrac{\sqrt3}4
10
Consider the hyperbola H:x2−y2=1H:x^2-y^2=1 and a circle SS with center N(x2,0)N(x_2,0). Suppose that HH and SS touch each other at a point P(x1,y1)P(x_1,y_1) with x1>1x_1>1 and y1>0y_1>0. The common tangent to HH and SS at PP intersects the xx-axis at point MM. If (l,m)(l,m) is the centroid of the triangle △PMN\triangle PMN, then the correct expression(s) is(are)
Not attempted
A
dldx1=1−13x12\dfrac{dl}{dx_1}=1-\dfrac1{3x_1^2} for x1>1x_1>1
B
dmdx1=x13(x12−1)\dfrac{dm}{dx_1}=\dfrac{x_1}{3\left(\sqrt{x_1^2-1}\right)} for x1>1x_1>1
C
dldx1=1+13x12\dfrac{dl}{dx_1}=1+\dfrac1{3x_1^2} for x1>1x_1>1
D
dmdy1=13\dfrac{dm}{dy_1}=\dfrac13 for y1>0y_1>0
11
If α=3sin⁡−1(611)\alpha=3\sin^{-1}\left(\dfrac{6}{11}\right) and β=3cos⁡−1(49)\beta=3\cos^{-1}\left(\dfrac{4}{9}\right), where the inverse trigonometric functions take only the principal values, then the correct option(s) is(are)
Not attempted
A
cos⁡β>0\cos\beta>0
B
sin⁡β<0\sin\beta<0
C
cos⁡(α+β)>0\cos(\alpha+\beta)>0
D
cos⁡α<0\cos\alpha<0
12
Let SS be the set of all non-zero real numbers α\alpha such that the quadratic equation αx2−x+α=0\alpha x^2-x+\alpha=0 has two distinct real roots x1x_1 and x2x_2 satisfying the inequality ∣x1−x2∣<1|x_1-x_2|<1. Which of the following intervals is(are) a subset(s) of SS?
Not attempted
A
(−12,−15)\left(-\dfrac12,-\dfrac1{\sqrt5}\right)
B
(−15,0)\left(-\dfrac1{\sqrt5},0\right)
C
(0,15)\left(0,\dfrac1{\sqrt5}\right)
D
(15,12)\left(\dfrac1{\sqrt5},\dfrac12\right)
13
Suppose that p⃗,q⃗\vec p,\vec q and r⃗\vec r are three non-coplanar vectors in R3\mathbb R^3. Let the components of a vector s⃗\vec s along p⃗,q⃗\vec p,\vec q and r⃗\vec r be 4,34,3 and 55, respectively. If the components of this vector s⃗\vec s along (−p⃗+q⃗+r⃗)(-\vec p+\vec q+\vec r), (p⃗−q⃗+r⃗)(\vec p-\vec q+\vec r) and (−p⃗−q⃗+r⃗)(-\vec p-\vec q+\vec r) are x,yx,y and zz, respectively, then the value of 2x+y+z2x+y+z is
Not attempted
Your answer: —
Correct answer: 9
14
If α=∫01(e9x+3tan⁡−1x)(12+9x21+x2)dx\alpha=\displaystyle\int_0^1\left(e^{9x+3\tan^{-1}x}\right)\left(\dfrac{12+9x^2}{1+x^2}\right)dx, where tan⁡−1x\tan^{-1}x takes only principal values, then the value of (log⁡e∣1+α∣−3π4)\left(\log_e|1+\alpha|-\dfrac{3\pi}{4}\right) is
Not attempted
Your answer: —
Correct answer: 9
15
Let f:R→Rf:\mathbb R\to\mathbb R be a continuous odd function, which vanishes exactly at one point and f(1)=12f(1)=\dfrac12. Suppose that F(x)=∫−1xf(t) dtF(x)=\displaystyle\int_{-1}^{x}f(t)\,dt for all x∈[−1,2]x\in[-1,2] and G(x)=∫−1xt ∣f(f(t))∣dtG(x)=\displaystyle\int_{-1}^{x}t\,\left|f(f(t))\right|dt for all x∈[−1,2]x\in[-1,2]. If lim⁡x→1F(x)G(x)=114\displaystyle\lim_{x\to1}\dfrac{F(x)}{G(x)}=\dfrac1{14}, then the value of f ⁣(12)f\!\left(\dfrac12\right) is
Not attempted
Your answer: —
Correct answer: 7
16
Let mm and nn be two positive integers greater than 11. If lim⁡α→0(ecos⁡(αn)−eαm)=−e2\displaystyle\lim_{\alpha\to0}\left(\dfrac{e^{\cos(\alpha^n)}-e}{\alpha^m}\right)=-\dfrac e2, then the value of mn\dfrac mn is
Not attempted
Your answer: —
Correct answer: 2
17
Suppose that the foci of the ellipse x29+y25=1\dfrac{x^2}{9}+\dfrac{y^2}{5}=1 are (f1,0)(f_1,0) and (f2,0)(f_2,0), where f1>0f_1>0 and f2<0f_2<0. Let P1P_1 and P2P_2 be two parabolas with a common vertex at (0,0)(0,0) and with foci at (f1,0)(f_1,0) and (2f2,0)(2f_2,0), respectively. Let T1T_1 be a tangent to P1P_1 which passes through (2f2,0)(2f_2,0) and T2T_2 be a tangent to P2P_2 which passes through (f1,0)(f_1,0). If m1m_1 is the slope of T1T_1 and m2m_2 is the slope of T2T_2, then the value of (1m12+m22)\left(\dfrac{1}{m_1^2}+m_2^2\right) is
Not attempted
Your answer: —
Correct answer: 4
18
The coefficient of x9x^9 in the expansion of (1+x)(1+x2)(1+x3)…(1+x100)(1+x)(1+x^2)(1+x^3)\ldots(1+x^{100}) is
Not attempted
Your answer: —
Correct answer: 8
19
Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:116:11 and the seventh term lies between 130130 and 140140, then the common difference of this A.P. is
Not attempted
Your answer: —
Correct answer: 9
20
For any integer kk, let αk=cos⁡(kπ7)+isin⁡(kπ7)\alpha_k=\cos\left(\dfrac{k\pi}{7}\right)+i\sin\left(\dfrac{k\pi}{7}\right), where i=−1i=\sqrt{-1}. The value of the expression ∑k=112∣αk+1−αk∣∑k=13∣α4k−1−α4k−2∣\dfrac{\displaystyle\sum_{k=1}^{12}\left|\alpha_{k+1}-\alpha_k\right|}{\displaystyle\sum_{k=1}^{3}\left|\alpha_{4k-1}-\alpha_{4k-2}\right|} is
Not attempted
Your answer: —
Correct answer: 4
21
This question is part of a Column-I to Column-II matching set (JEE Advanced 2015 Section 3, Q.60). Column II lists the values 1,2,3,5,61,2,3,5,6; select every value that correctly matches this Column-I entry. Entry (D)\textbf{Entry (D)}: Suppose that F(α)F(\alpha) denotes the area of the region bounded by x=0, x=2, y2=4xx=0,\ x=2,\ y^2=4x and y=∣αx−1∣+∣αx−2∣+αxy=|\alpha x-1|+|\alpha x-2|+\alpha x, where α∈{0,1}\alpha\in\{0,1\}. Then the value(s) of F(α)+832F(\alpha)+\dfrac83\sqrt2, when α=0\alpha=0 and α=1\alpha=1, is (are)
Not attempted
A
11
B
22
C
33
D
55
E
66
22
This question is part of a Column-I to Column-II matching set (JEE Advanced 2015 Section 3, Q.60). Column II lists the values 1,2,3,5,61,2,3,5,6; select every value that correctly matches this Column-I entry. Entry (C)\textbf{Entry (C)}: In R2\mathbb{R}^2, let 3i^+j^, i^+3j^\sqrt3\hat i+\hat j,\ \hat i+\sqrt3\hat j and βi^+(1−β)j^\beta\hat i+(1-\beta)\hat j be the position vectors of X,YX,Y and ZZ with respect to the origin OO, respectively. If the distance of ZZ from the bisector of the acute angle of OX⃗\vec{OX} with OY⃗\vec{OY} is 32\dfrac{3}{\sqrt2}, then possible value(s) of ∣β∣|\beta| is (are)
Not attempted
A
11
B
22
C
33
D
55
E
66
23
This question is part of a Column-I to Column-II matching set (JEE Advanced 2015 Section 3, Q.60). Column II lists the values 1,2,3,5,61,2,3,5,6; select every value that correctly matches this Column-I entry. Entry (A)\textbf{Entry (A)}: In a triangle △XYZ\triangle XYZ, let a,b,ca,b,c be the lengths of the sides opposite to the angles X,Y,ZX,Y,Z respectively. If 2(a2−b2)=c22(a^2-b^2)=c^2 and λ=sin⁡(X−Y)sin⁡Z\lambda=\dfrac{\sin(X-Y)}{\sin Z}, then possible values of nn for which cos⁡(nπλ)=0\cos(n\pi\lambda)=0 is (are)
Not attempted
A
11
B
22
C
33
D
55
E
66
24
This question is part of a Column-I to Column-II matching set (JEE Advanced 2015 Section 3, Q.60). Column II lists the values 1,2,3,5,61,2,3,5,6; select every value that correctly matches this Column-I entry. Entry (B)\textbf{Entry (B)}: In a triangle △XYZ\triangle XYZ, let a,b,ca,b,c be the lengths of the sides opposite to the angles X,Y,ZX,Y,Z respectively. If 1+cos⁡2X−2cos⁡2Y=2sin⁡Xsin⁡Y1+\cos2X-2\cos2Y=2\sin X\sin Y, then possible value(s) of ab\dfrac{a}{b} is (are)
Not attempted
A
11
B
22
C
33
D
55
E
66
25
This question is part of a Column-I to Column-II matching set (JEE Advanced 2015 Section 3, Q.59). Column II lists the values 1,2,3,4,51,2,3,4,5; select every value that correctly matches this Column-I entry. Entry (D)\textbf{Entry (D)}: Let the harmonic mean of two positive real numbers aa and bb be 44. If qq is a positive real number such that a,5,q,ba,5,q,b is an arithmetic progression, then the value(s) of ∣q−a∣|q-a| is (are)
Not attempted
A
11
B
22
C
33
D
44
E
55
26
This question is part of a Column-I to Column-II matching set (JEE Advanced 2015 Section 3, Q.59). Column II lists the values 1,2,3,4,51,2,3,4,5; select every value that correctly matches this Column-I entry. Entry (C)\textbf{Entry (C)}: Let ω≠1\omega\ne1 be a complex cube root of unity. If (3−3ω+2ω2)4n+3+(2+3ω−3ω2)4n+3+(−3+2ω+3ω2)4n+3=0(3-3\omega+2\omega^2)^{4n+3}+(2+3\omega-3\omega^2)^{4n+3}+(-3+2\omega+3\omega^2)^{4n+3}=0, then possible value(s) of nn is (are)
Not attempted
A
11
B
22
C
33
D
44
E
55
27
This question is part of a Column-I to Column-II matching set (JEE Advanced 2015 Section 3, Q.59). Column II lists the values 1,2,3,4,51,2,3,4,5; select every value that correctly matches this Column-I entry. Entry (B)\textbf{Entry (B)}: Let aa and bb be real numbers such that the function f(x)={−3ax2−2,x<1bx+a2,x≥1f(x)=\begin{cases}-3ax^2-2, & x<1\\ bx+a^2, & x\ge1\end{cases} is differentiable for all x∈Rx\in\mathbb{R}. Then possible value(s) of aa is (are)
Not attempted
A
11
B
22
C
33
D
44
E
55
28
Let △PQR\triangle PQR be a triangle. Let a⃗=QR⃗, b⃗=RP⃗\vec a=\vec{QR},\ \vec b=\vec{RP} and c⃗=PQ⃗\vec c=\vec{PQ}. If ∣a⃗∣=12, ∣b⃗∣=43|\vec a|=12,\ |\vec b|=4\sqrt3 and b⃗⋅c⃗=24\vec b\cdot\vec c=24, then which of the following is (are) true?
Not attempted
A
∣c⃗∣22−∣a⃗∣=12\dfrac{|\vec c|^2}{2}-|\vec a|=12
B
∣c⃗∣22+∣a⃗∣=30\dfrac{|\vec c|^2}{2}+|\vec a|=30
C
∣a⃗×b⃗+c⃗×a⃗∣=483|\vec a\times\vec b+\vec c\times\vec a|=48\sqrt3
D
a⃗⋅b⃗=−72\vec a\cdot\vec b=-72
29
This question is part of a Column-I to Column-II matching set (JEE Advanced 2015 Section 3, Q.59). Column II lists the values 1,2,3,4,51,2,3,4,5; select every value that correctly matches this Column-I entry. Entry (A)\textbf{Entry (A)}: In R2\mathbb{R}^2, if the magnitude of the projection vector of the vector αi^+βj^\alpha\hat i+\beta\hat j on 3i^+j^\sqrt3\hat i+\hat j is 3\sqrt3, and if α=2+3β\alpha=2+\sqrt3\beta, then the possible value(s) of ∣α∣|\alpha| is (are)
Not attempted
A
11
B
22
C
33
D
44
E
55
30
Let g:R→Rg:\mathbb{R}\to\mathbb{R} be a differentiable function with g(0)=0, g′(0)=0g(0)=0,\ g'(0)=0 and g′(1)≠0g'(1)\ne0. Let f(x)={x∣x∣g(x),x≠00,x=0f(x)=\begin{cases}\dfrac{x}{|x|}g(x), & x\ne0\\ 0, & x=0\end{cases} and h(x)=e∣x∣h(x)=e^{|x|} for all x∈Rx\in\mathbb{R}. Let (f∘h)(x)(f\circ h)(x) denote f(h(x))f(h(x)) and (h∘f)(x)(h\circ f)(x) denote h(f(x))h(f(x)). Then which of the following is (are) true?
Not attempted
A
ff is differentiable at x=0x=0
B
hh is differentiable at x=0x=0
C
f∘hf\circ h is differentiable at x=0x=0
D
h∘fh\circ f is differentiable at x=0x=0
31
Let f(x)=sin⁡(π6sin⁡(π2sin⁡x))f(x)=\sin\left(\dfrac{\pi}{6}\sin\left(\dfrac{\pi}{2}\sin x\right)\right) for all x∈Rx\in\mathbb{R} and g(x)=π2sin⁡xg(x)=\dfrac{\pi}{2}\sin x for all x∈Rx\in\mathbb{R}. Let (f∘g)(x)(f\circ g)(x) denote f(g(x))f(g(x)) and (g∘f)(x)(g\circ f)(x) denote g(f(x))g(f(x)). Then which of the following is (are) true?
Not attempted
A
Range of ff is [−12,12]\left[-\dfrac12,\dfrac12\right]
B
Range of f∘gf\circ g is [−12,12]\left[-\dfrac12,\dfrac12\right]
C
lim⁡x→0f(x)g(x)=π6\displaystyle\lim_{x\to0}\dfrac{f(x)}{g(x)}=\dfrac{\pi}{6}
D
There is an x∈Rx\in\mathbb{R} such that (g∘f)(x)=1(g\circ f)(x)=1
32
Let y(x)y(x) be a solution of the differential equation (1+ex)y′+yex=1(1+e^x)y'+ye^x=1. If y(0)=2y(0)=2, then which of the following statements is (are) true?
Not attempted
A
y(−4)=0y(-4)=0
B
y(−2)=0y(-2)=0
C
y(x)y(x) has a critical point in the interval (−1,0)(-1,0)
D
y(x)y(x) has no critical point in the interval (−1,0)(-1,0)
33
Consider the family of all circles whose centers lie on the straight line y=xy=x. If this family of circles is represented by the differential equation Py′′+Qy′+1=0Py''+Qy'+1=0, where P,QP,Q are functions of x,yx,y and y′y' (here y′=dydx, y′′=d2ydx2y'=\dfrac{dy}{dx},\ y''=\dfrac{d^2y}{dx^2}), then which of the following statements is (are) true?
Not attempted
A
P=y+xP=y+x
B
P=y−xP=y-x
C
P+Q=1−x+y+y′+(y′)2P+Q=1-x+y+y'+(y')^2
D
P−Q=x+y−y′−(y′)2P-Q=x+y-y'-(y')^2
34
Let PP and QQ be distinct points on the parabola y2=2xy^2=2x such that a circle with PQPQ as diameter passes through the vertex OO of the parabola. If PP lies in the first quadrant and the area of the triangle △OPQ\triangle OPQ is 323\sqrt2, then which of the following is (are) the coordinates of PP?
Not attempted
A
(4,22)(4,2\sqrt2)
B
(9,32)(9,3\sqrt2)
C
(14,12)\left(\dfrac14,\dfrac{1}{\sqrt2}\right)
D
(1,2)(1,\sqrt2)
35
In R3\mathbb{R}^3, consider the planes P1:y=0P_1: y=0 and P2:x+z=1P_2: x+z=1. Let P3P_3 be a plane, different from P1P_1 and P2P_2, which passes through the intersection of P1P_1 and P2P_2. If the distance of the point (0,1,0)(0,1,0) from P3P_3 is 11 and the distance of a point (α,β,γ)(\alpha,\beta,\gamma) from P3P_3 is 22, then which of the following relations is (are) true?
Not attempted
A
2α+β+2γ+2=02\alpha+\beta+2\gamma+2=0
B
2α−β+2γ+4=02\alpha-\beta+2\gamma+4=0
C
2α+β−2γ−10=02\alpha+\beta-2\gamma-10=0
D
2α−β+2γ−8=02\alpha-\beta+2\gamma-8=0
36
In R3\mathbb{R}^3, let LL be a straight line passing through the origin. Suppose that all the points on LL are at a constant distance from the two planes P1:x+2y−z+1=0P_1: x+2y-z+1=0 and P2:2x−y+z−1=0P_2: 2x-y+z-1=0. Let MM be the locus of the feet of the perpendiculars drawn from the points on LL to the plane P1P_1. Which of the following points lie(s) on MM?
Not attempted
A
(0,−56,−23)\left(0,-\dfrac56,-\dfrac23\right)
B
(−16,−13,16)\left(-\dfrac16,-\dfrac13,\dfrac16\right)
C
(−56,0,16)\left(-\dfrac56,0,\dfrac16\right)
D
(−13,0,23)\left(-\dfrac13,0,\dfrac23\right)
37
Let XX and YY be two arbitrary, 3×33\times3, non-zero, skew-symmetric matrices and ZZ be an arbitrary 3×33\times3, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?
Not attempted
A
Y3Z4−Z4Y3Y^3Z^4-Z^4Y^3
B
X44+Y44X^{44}+Y^{44}
C
X4Z3−Z3X4X^4Z^3-Z^3X^4
D
X23+Y23X^{23}+Y^{23}
38
Which of the following values of α\alpha satisfy the equation ∣(1+α)2(1+2α)2(1+3α)2(2+α)2(2+2α)2(2+3α)2(3+α)2(3+2α)2(3+3α)2∣=−648α\begin{vmatrix}(1+\alpha)^2 & (1+2\alpha)^2 & (1+3\alpha)^2\\ (2+\alpha)^2 & (2+2\alpha)^2 & (2+3\alpha)^2\\ (3+\alpha)^2 & (3+2\alpha)^2 & (3+3\alpha)^2\end{vmatrix}=-648\alpha?
Not attempted
A
−4-4
B
99
C
−9-9
D
44
39
Let F(x)=∫xx2+π62cos⁡2t dtF(x)=\displaystyle\int_{x}^{x^2+\frac{\pi}{6}}2\cos^2t\,dt for all x∈Rx\in\mathbb{R} and f:[0,12]→[0,∞)f:\left[0,\dfrac12\right]\to[0,\infty) be a continuous function. For a∈[0,12]a\in\left[0,\dfrac12\right], if F′(a)+2F'(a)+2 is the area of the region bounded by x=0x=0, y=0y=0, y=f(x)y=f(x) and x=ax=a, then f(0)f(0) is
Not attempted
Your answer: —
Correct answer: 3
40
A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of V mm3V\ \text{mm}^3, has a 22 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 22 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 1010 mm, then the value of V250π\dfrac{V}{250\pi} is
Not attempted
Your answer: —
Correct answer: 4
41
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a function defined by f(x)={[x],x≤20,x>2f(x)=\begin{cases}[x], & x\le 2\\ 0, & x>2\end{cases}, where [x][x] is the greatest integer less than or equal to xx. If I=∫−12xf(x2)2+f(x+1) dxI=\displaystyle\int_{-1}^{2}\dfrac{xf(x^2)}{2+f(x+1)}\,dx, then the value of (4I−1)(4I-1) is
Not attempted
Your answer: —
Correct answer: 0
42
If the normals of the parabola y2=4xy^2=4x drawn at the end points of its latus rectum are tangents to the circle (x−3)2+(y+2)2=r2(x-3)^2+(y+2)^2=r^2, then the value of r2r^2 is
Not attempted
Your answer: —
Correct answer: 2
43
Let nn be the number of ways in which 55 boys and 55 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let mm be the number of ways in which 55 boys and 55 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of mn\dfrac{m}{n} is
Not attempted
Your answer: —
Correct answer: 5
44
The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.960.96, is
Not attempted
Your answer: —
Correct answer: 8
45
Let the curve CC be the mirror image of the parabola y2=4xy^2=4x with respect to the line x+y+4=0x+y+4=0. If AA and BB are the points of intersection of CC with the line y=−5y=-5, then the distance between AA and BB is
Not attempted
Your answer: —
Correct answer: 4
46
The number of distinct solutions of the equation 54cos⁡22x+cos⁡4x+sin⁡4x+cos⁡6x+sin⁡6x=2\dfrac{5}{4}\cos^2 2x+\cos^4 x+\sin^4 x+\cos^6 x+\sin^6 x=2 in the interval [0,2π][0,2\pi] is
Not attempted
Your answer: —
Correct answer: 8
Strategy\textbf{Strategy}: Write cos⁡4x+sin⁡4x=1−2cs\cos^4x+\sin^4x=1-2cs and cos⁡6x+sin⁡6x=1−3cs\cos^6x+\sin^6x=1-3cs where c=cos⁡2x, s=sin⁡2xc=\cos^2x,\ s=\sin^2x, then use cs=14sin⁡22xcs=\tfrac14\sin^2 2x to reduce everything to a single condition on cos⁡4x\cos 4x.
Deconstruction\textbf{Deconstruction}: Let c=cos⁡2x, s=sin⁡2xc=\cos^2x,\ s=\sin^2x, so c+s=1c+s=1. Then cos⁡4x+sin⁡4x=(c+s)2−2cs=1−2cs\cos^4x+\sin^4x=(c+s)^2-2cs=1-2cs, and cos⁡6x+sin⁡6x=(c+s)3−3cs(c+s)=1−3cs\cos^6x+\sin^6x=(c+s)^3-3cs(c+s)=1-3cs.\n\nStrategy\textbf{Strategy}: Substitute these into the given equation.\n\nExecution\textbf{Execution}: 54cos⁡22x+(1−2cs)+(1−3cs)=2⇒54cos⁡22x+2−5cs=2⇒54cos⁡22x=5cs⇒cos⁡22x=4cs\dfrac54\cos^2 2x+(1-2cs)+(1-3cs)=2 \Rightarrow \dfrac54\cos^2 2x+2-5cs=2 \Rightarrow \dfrac54\cos^2 2x=5cs \Rightarrow \cos^2 2x=4cs.\n\nNow cs=sin⁡2xcos⁡2x=14sin⁡22xcs=\sin^2x\cos^2x=\dfrac14\sin^2 2x, so cos⁡22x=4⋅14sin⁡22x=sin⁡22x\cos^2 2x=4\cdot\dfrac14\sin^2 2x=\sin^2 2x.\n\nSo cos⁡22x−sin⁡22x=0⇒cos⁡4x=0⇒4x=π2+kπ⇒x=π8+kπ4\cos^2 2x-\sin^2 2x=0 \Rightarrow \cos 4x=0 \Rightarrow 4x=\dfrac{\pi}{2}+k\pi \Rightarrow x=\dfrac{\pi}{8}+\dfrac{k\pi}{4}.\n\nOver x∈[0,2π]x\in[0,2\pi], k=0,1,…,7k=0,1,\ldots,7 gives 88 distinct values, all strictly between 00 and 2π2\pi: π8,3π8,…,15π8\dfrac{\pi}{8},\dfrac{3\pi}{8},\ldots,\dfrac{15\pi}{8}.\n\nVerification\textbf{Verification}: cos⁡4x=0\cos 4x=0 has period π2\dfrac{\pi}{2} in xx (since 4x4x has period 2π2\pi), and [0,2π][0,2\pi] spans exactly 44 such periods, each containing 22 zeros of cosine, giving 4×2=84\times2=8 solutions — consistent with the direct count.\n\nFinal Answer\textbf{Final Answer}: 88.