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JEE Advanced · Mathematics · 2017

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1
Paragraph\textbf{Paragraph}: Let pp, qq be integers and let α\alpha, β\beta be the roots of the equation x2−x−1=0x^2 - x - 1 = 0, where α≠β\alpha \ne \beta. For n=0,1,2,…n = 0, 1, 2, \ldots, let an=pαn+qβna_n = p\alpha^n + q\beta^n. FACT\textbf{FACT}: If aa and bb are rational numbers and a+b5=0a + b\sqrt{5} = 0, then a=0=ba = 0 = b. a12=a_{12} =
Not attempted
A
a11−a10a_{11} - a_{10}
B
a11+a10a_{11} + a_{10}
C
2a11+a102a_{11} + a_{10}
D
a11+2a10a_{11} + 2a_{10}
2
Paragraph\textbf{Paragraph}: Let pp, qq be integers and let α\alpha, β\beta be the roots of the equation x2−x−1=0x^2 - x - 1 = 0, where α≠β\alpha \ne \beta. For n=0,1,2,…n = 0, 1, 2, \ldots, let an=pαn+qβna_n = p\alpha^n + q\beta^n. FACT\textbf{FACT}: If aa and bb are rational numbers and a+b5=0a + b\sqrt{5} = 0, then a=0=ba = 0 = b. If a4=28a_4 = 28, then p+2q=p + 2q =
Not attempted
A
2121
B
1414
C
77
D
1212
3
If f(x)=∣cos⁡(2x)cos⁡(2x)sin⁡(2x)−cos⁡xcos⁡x−sin⁡xsin⁡xsin⁡xcos⁡x∣f(x) = \begin{vmatrix} \cos(2x) & \cos(2x) & \sin(2x) \\ -\cos x & \cos x & -\sin x \\ \sin x & \sin x & \cos x \end{vmatrix}, then
Not attempted
A
f′(x)=0f'(x) = 0 at exactly three points in (−π,π)(-\pi, \pi)
B
f′(x)=0f'(x) = 0 at more than three points in (−π,π)(-\pi, \pi)
C
f(x)f(x) attains its maximum at x=0x = 0
D
f(x)f(x) attains its minimum at x=0x = 0
4
Let α\alpha and β\beta be nonzero real numbers such that 2(cos⁡β−cos⁡α)+cos⁡αcos⁡β=12(\cos\beta - \cos\alpha) + \cos\alpha\cos\beta = 1. Then which of the following is/are true?
Not attempted
A
tan⁡(α2)+3tan⁡(β2)=0\tan\left(\frac{\alpha}{2}\right) + \sqrt{3}\tan\left(\frac{\beta}{2}\right) = 0
B
3tan⁡(α2)+tan⁡(β2)=0\sqrt{3}\tan\left(\frac{\alpha}{2}\right) + \tan\left(\frac{\beta}{2}\right) = 0
C
tan⁡(α2)−3tan⁡(β2)=0\tan\left(\frac{\alpha}{2}\right) - \sqrt{3}\tan\left(\frac{\beta}{2}\right) = 0
D
3tan⁡(α2)−tan⁡(β2)=0\sqrt{3}\tan\left(\frac{\alpha}{2}\right) - \tan\left(\frac{\beta}{2}\right) = 0
5
If g(x)=∫sin⁡xsin⁡(2x)sin⁡−1(t) dtg(x) = \int_{\sin x}^{\sin(2x)} \sin^{-1}(t)\, dt, then
Not attempted
A
g′(π2)=−2πg'\left(\frac{\pi}{2}\right) = -2\pi
B
g′(−π2)=2πg'\left(-\frac{\pi}{2}\right) = 2\pi
C
g′(π2)=2πg'\left(\frac{\pi}{2}\right) = 2\pi
D
g′(−π2)=−2πg'\left(-\frac{\pi}{2}\right) = -2\pi
6
If the line x=αx = \alpha divides the area of the region R={(x,y)∈R2:x3≤y≤x, 0≤x≤1}R = \{(x,y) \in \mathbb{R}^2 : x^3 \le y \le x,\ 0 \le x \le 1\} into two equal parts, then
Not attempted
A
0<α≤120 < \alpha \le \frac{1}{2}
B
12<α<1\frac{1}{2} < \alpha < 1
C
2α4−4α2+1=02\alpha^4 - 4\alpha^2 + 1 = 0
D
α4+4α2−1=0\alpha^4 + 4\alpha^2 - 1 = 0
7
If I=∑k=198∫kk+1k+1x(x+1) dxI = \displaystyle\sum_{k=1}^{98} \int_{k}^{k+1} \dfrac{k+1}{x(x+1)}\, dx, then
Not attempted
A
I>log⁡e99I > \log_e 99
B
I<log⁡e99I < \log_e 99
C
I<4950I < \frac{49}{50}
D
I>4950I > \frac{49}{50}
8
Three randomly chosen nonnegative integers x,yx, y and zz are found to satisfy the equation x+y+z=10x + y + z = 10. Then the probability that zz is even, is
Not attempted
A
3655\frac{36}{55}
B
611\frac{6}{11}
C
12\frac{1}{2}
D
511\frac{5}{11}
9
How many 3×33\times 3 matrices MM with entries from {0,1,2}\{0, 1, 2\} are there, for which the sum of the diagonal entries of MTMM^T M is 55?
Not attempted
A
126126
B
198198
C
162162
D
135135
10
Let S={1,2,3,…,9}S = \{1, 2, 3, \ldots, 9\}. For k=1,2,…,5k = 1, 2, \ldots, 5, let NkN_k be the number of subsets of SS, each containing five elements, out of which exactly kk are odd. Then N1+N2+N3+N4+N5=N_1 + N_2 + N_3 + N_4 + N_5 =
Not attempted
A
210210
B
252252
C
125125
D
126126
11
If f:R→Rf: \mathbb{R} \to \mathbb{R} is a twice differentiable function such that f′′(x)>0f''(x) > 0 for all x∈Rx \in \mathbb{R}, and f(12)=12f\left(\frac{1}{2}\right) = \frac{1}{2}, f(1)=1f(1) = 1, then
Not attempted
A
f′(1)≤0f'(1) \le 0
B
0<f′(1)≤120 < f'(1) \le \frac{1}{2}
C
12<f′(1)≤1\frac{1}{2} < f'(1) \le 1
D
f′(1)>1f'(1) > 1
12
Let f(x)=1−x(1+∣1−x∣)∣1−x∣cos⁡(11−x)f(x) = \dfrac{1-x(1+|1-x|)}{|1-x|}\cos\left(\dfrac{1}{1-x}\right) for x≠1x \ne 1. Then
Not attempted
A
lim⁡x→1−f(x)=0\lim_{x\to 1^-} f(x) = 0
B
lim⁡x→1−f(x)\lim_{x\to 1^-} f(x) does not exist
C
lim⁡x→1+f(x)=0\lim_{x\to 1^+} f(x) = 0
D
lim⁡x→1+f(x)\lim_{x\to 1^+} f(x) does not exist
13
If y=y(x)y = y(x) satisfies the differential equation 8x(9+x)dy=(4+9+x)−1dx8\sqrt{x}\left(\sqrt{9+\sqrt{x}}\right) dy = \left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)^{-1} dx, x>0x > 0, and y(0)=7y(0) = \sqrt{7}, then y(256)=y(256) =
Not attempted
A
33
B
99
C
1616
D
8080
14
If f:R→Rf: \mathbb{R} \to \mathbb{R} is a differentiable function such that f′(x)>2f(x)f'(x) > 2f(x) for all x∈Rx \in \mathbb{R}, and f(0)=1f(0) = 1, then
Not attempted
A
f(x)f(x) is increasing in (0,∞)(0, \infty)
B
f(x)f(x) is decreasing in (0,∞)(0, \infty)
C
f(x)>e2xf(x) > e^{2x} in (0,∞)(0, \infty)
D
f′(x)<e2xf'(x) < e^{2x} in (0,∞)(0, \infty)
15
Let OO be the origin and let PQRPQR be an arbitrary triangle. The point SS is such that OP⃗⋅OQ⃗+OR⃗⋅OS⃗=OR⃗⋅OP⃗+OQ⃗⋅OS⃗=OQ⃗⋅OR⃗+OP⃗⋅OS⃗\vec{OP} \cdot \vec{OQ} + \vec{OR} \cdot \vec{OS} = \vec{OR} \cdot \vec{OP} + \vec{OQ} \cdot \vec{OS} = \vec{OQ} \cdot \vec{OR} + \vec{OP} \cdot \vec{OS} Then the triangle PQRPQR has SS as its
Not attempted
A
centroid
B
circumcentre
C
incentre
D
orthocenter
16
Paragraph\textbf{Paragraph}: Let OO be the origin, and OX⃗\vec{OX}, OY⃗\vec{OY}, OZ⃗\vec{OZ} be three unit vectors in the directions of the sides QR⃗\vec{QR}, RP⃗\vec{RP}, PQ⃗\vec{PQ}, respectively, of a triangle PQRPQR. ∣OX⃗×OY⃗∣=|\vec{OX} \times \vec{OY}| =
Not attempted
A
sin⁡(P+Q)\sin(P+Q)
B
sin⁡2R\sin 2R
C
sin⁡(P+R)\sin(P+R)
D
sin⁡(Q+R)\sin(Q+R)
17
Paragraph\textbf{Paragraph}: Let OO be the origin, and OX⃗\vec{OX}, OY⃗\vec{OY}, OZ⃗\vec{OZ} be three unit vectors in the directions of the sides QR⃗\vec{QR}, RP⃗\vec{RP}, PQ⃗\vec{PQ}, respectively, of a triangle PQRPQR. If the triangle PQRPQR varies, then the minimum value of cos⁡(P+Q)+cos⁡(Q+R)+cos⁡(R+P)\cos(P+Q) + \cos(Q+R) + \cos(R+P) is
Not attempted
A
−53-\frac{5}{3}
B
−32-\frac{3}{2}
C
32\frac{3}{2}
D
53\frac{5}{3}
18
The equation of the plane passing through the point (1,1,1)(1, 1, 1) and perpendicular to the planes 2x+y−2z=52x + y - 2z = 5 and 3x−6y−2z=73x - 6y - 2z = 7, is
Not attempted
A
14x+2y−15z=114x + 2y - 15z = 1
B
14x−2y+15z=2714x - 2y + 15z = 27
C
14x+2y+15z=3114x + 2y + 15z = 31
D
−14x+2y+15z=3-14x + 2y + 15z = 3
19
Let f(x)=x+ln⁡x−xln⁡x, x∈(0,∞)f(x)=x+\ln x-x\ln x,\ x\in(0,\infty). Column 1 concerns zeros of f,f′,f′′f,f',f''; Column 2 concerns the limiting behaviour of f,f′,f′′f,f',f'' at infinity; Column 3 concerns the increasing/decreasing nature of ff and f′f'. (I) f(x)=0 for some x∈(1,e2)(i) lim⁡x→∞f(x)=0(P) f is increasing in (0,1)(II) f′(x)=0 for some x∈(1,e)(ii) lim⁡x→∞f(x)=−∞(Q) f is decreasing in (e,e2)(III) f′(x)=0 for some x∈(0,1)(iii) lim⁡x→∞f′(x)=−∞(R) f′ is increasing in (0,1)(IV) f′′(x)=0 for some x∈(1,e)(iv) lim⁡x→∞f′′(x)=0(S) f′ is decreasing in (e,e2)\begin{array}{lll} \text{(I) } f(x)=0 \text{ for some } x\in(1,e^2) & \text{(i) } \lim\limits_{x\to\infty}f(x)=0 & \text{(P) } f \text{ is increasing in } (0,1)\\ \text{(II) } f'(x)=0 \text{ for some } x\in(1,e) & \text{(ii) } \lim\limits_{x\to\infty}f(x)=-\infty & \text{(Q) } f \text{ is decreasing in } (e,e^2)\\ \text{(III) } f'(x)=0 \text{ for some } x\in(0,1) & \text{(iii) } \lim\limits_{x\to\infty}f'(x)=-\infty & \text{(R) } f' \text{ is increasing in } (0,1)\\ \text{(IV) } f''(x)=0 \text{ for some } x\in(1,e) & \text{(iv) } \lim\limits_{x\to\infty}f''(x)=0 & \text{(S) } f' \text{ is decreasing in } (e,e^2) \end{array} Which of the following options is the only CORRECT combination?
Not attempted
A
(I) (i) (P)
B
(II) (ii) (Q)
C
(III) (iii) (R)
D
(IV) (iv) (S)
20
Let f(x)=x+ln⁡x−xln⁡x, x∈(0,∞)f(x)=x+\ln x-x\ln x,\ x\in(0,\infty). Column 1 concerns zeros of f,f′,f′′f,f',f''; Column 2 concerns the limiting behaviour of f,f′,f′′f,f',f'' at infinity; Column 3 concerns the increasing/decreasing nature of ff and f′f'. (I) f(x)=0 for some x∈(1,e2)(i) lim⁡x→∞f(x)=0(P) f is increasing in (0,1)(II) f′(x)=0 for some x∈(1,e)(ii) lim⁡x→∞f(x)=−∞(Q) f is decreasing in (e,e2)(III) f′(x)=0 for some x∈(0,1)(iii) lim⁡x→∞f′(x)=−∞(R) f′ is increasing in (0,1)(IV) f′′(x)=0 for some x∈(1,e)(iv) lim⁡x→∞f′′(x)=0(S) f′ is decreasing in (e,e2)\begin{array}{lll} \text{(I) } f(x)=0 \text{ for some } x\in(1,e^2) & \text{(i) } \lim\limits_{x\to\infty}f(x)=0 & \text{(P) } f \text{ is increasing in } (0,1)\\ \text{(II) } f'(x)=0 \text{ for some } x\in(1,e) & \text{(ii) } \lim\limits_{x\to\infty}f(x)=-\infty & \text{(Q) } f \text{ is decreasing in } (e,e^2)\\ \text{(III) } f'(x)=0 \text{ for some } x\in(0,1) & \text{(iii) } \lim\limits_{x\to\infty}f'(x)=-\infty & \text{(R) } f' \text{ is increasing in } (0,1)\\ \text{(IV) } f''(x)=0 \text{ for some } x\in(1,e) & \text{(iv) } \lim\limits_{x\to\infty}f''(x)=0 & \text{(S) } f' \text{ is decreasing in } (e,e^2) \end{array} Which of the following options is the only CORRECT combination?
Not attempted
A
(I) (ii) (R)
B
(II) (iii) (S)
C
(III) (iv) (P)
D
(IV) (i) (S)
21
Let f(x)=x+ln⁡x−xln⁡x, x∈(0,∞)f(x)=x+\ln x-x\ln x,\ x\in(0,\infty). Column 1 concerns zeros of f,f′,f′′f,f',f''; Column 2 concerns the limiting behaviour of f,f′,f′′f,f',f'' at infinity; Column 3 concerns the increasing/decreasing nature of ff and f′f'. (I) f(x)=0 for some x∈(1,e2)(i) lim⁡x→∞f(x)=0(P) f is increasing in (0,1)(II) f′(x)=0 for some x∈(1,e)(ii) lim⁡x→∞f(x)=−∞(Q) f is decreasing in (e,e2)(III) f′(x)=0 for some x∈(0,1)(iii) lim⁡x→∞f′(x)=−∞(R) f′ is increasing in (0,1)(IV) f′′(x)=0 for some x∈(1,e)(iv) lim⁡x→∞f′′(x)=0(S) f′ is decreasing in (e,e2)\begin{array}{lll} \text{(I) } f(x)=0 \text{ for some } x\in(1,e^2) & \text{(i) } \lim\limits_{x\to\infty}f(x)=0 & \text{(P) } f \text{ is increasing in } (0,1)\\ \text{(II) } f'(x)=0 \text{ for some } x\in(1,e) & \text{(ii) } \lim\limits_{x\to\infty}f(x)=-\infty & \text{(Q) } f \text{ is decreasing in } (e,e^2)\\ \text{(III) } f'(x)=0 \text{ for some } x\in(0,1) & \text{(iii) } \lim\limits_{x\to\infty}f'(x)=-\infty & \text{(R) } f' \text{ is increasing in } (0,1)\\ \text{(IV) } f''(x)=0 \text{ for some } x\in(1,e) & \text{(iv) } \lim\limits_{x\to\infty}f''(x)=0 & \text{(S) } f' \text{ is decreasing in } (e,e^2) \end{array} Which of the following options is the only INCORRECT combination?
Not attempted
A
(I) (iii) (P)
B
(II) (iv) (Q)
C
(III) (i) (R)
D
(II) (iii) (P)
22
Columns 1, 2 and 3 contain conics, equations of tangents to the conics, and points of contact, respectively. (I) x2+y2=a2(i) my=m2x+a(P) (am2,2am)(II) x2+a2y2=a2(ii) y=mx+am2+1(Q) (−mam2+1,am2+1)(III) y2=4ax(iii) y=mx+a2m2−1(R) (−a2ma2m2+1,1a2m2+1)(IV) x2−a2y2=a2(iv) y=mx+a2m2+1(S) (−a2ma2m2−1,−1a2m2−1)\begin{array}{lll} \text{(I) } x^2+y^2=a^2 & \text{(i) } my=m^2x+a & \text{(P) } \left(\dfrac{a}{m^2},\dfrac{2a}{m}\right)\\ \text{(II) } x^2+a^2y^2=a^2 & \text{(ii) } y=mx+a\sqrt{m^2+1} & \text{(Q) } \left(\dfrac{-ma}{\sqrt{m^2+1}},\dfrac{a}{\sqrt{m^2+1}}\right)\\ \text{(III) } y^2=4ax & \text{(iii) } y=mx+\sqrt{a^2m^2-1} & \text{(R) } \left(\dfrac{-a^2m}{\sqrt{a^2m^2+1}},\dfrac{1}{\sqrt{a^2m^2+1}}\right)\\ \text{(IV) } x^2-a^2y^2=a^2 & \text{(iv) } y=mx+\sqrt{a^2m^2+1} & \text{(S) } \left(\dfrac{-a^2m}{\sqrt{a^2m^2-1}},\dfrac{-1}{\sqrt{a^2m^2-1}}\right) \end{array} For a=2a=\sqrt2, if a tangent is drawn to a suitable conic (Column 1) at the point of contact (−1,1)(-1,1), then which of the following options is the only CORRECT combination for obtaining its equation?
Not attempted
A
(I) (i) (P)
B
(I) (ii) (Q)
C
(II) (ii) (Q)
D
(III) (i) (P)
23
Columns 1, 2 and 3 contain conics, equations of tangents to the conics, and points of contact, respectively. (I) x2+y2=a2(i) my=m2x+a(P) (am2,2am)(II) x2+a2y2=a2(ii) y=mx+am2+1(Q) (−mam2+1,am2+1)(III) y2=4ax(iii) y=mx+a2m2−1(R) (−a2ma2m2+1,1a2m2+1)(IV) x2−a2y2=a2(iv) y=mx+a2m2+1(S) (−a2ma2m2−1,−1a2m2−1)\begin{array}{lll} \text{(I) } x^2+y^2=a^2 & \text{(i) } my=m^2x+a & \text{(P) } \left(\dfrac{a}{m^2},\dfrac{2a}{m}\right)\\ \text{(II) } x^2+a^2y^2=a^2 & \text{(ii) } y=mx+a\sqrt{m^2+1} & \text{(Q) } \left(\dfrac{-ma}{\sqrt{m^2+1}},\dfrac{a}{\sqrt{m^2+1}}\right)\\ \text{(III) } y^2=4ax & \text{(iii) } y=mx+\sqrt{a^2m^2-1} & \text{(R) } \left(\dfrac{-a^2m}{\sqrt{a^2m^2+1}},\dfrac{1}{\sqrt{a^2m^2+1}}\right)\\ \text{(IV) } x^2-a^2y^2=a^2 & \text{(iv) } y=mx+\sqrt{a^2m^2+1} & \text{(S) } \left(\dfrac{-a^2m}{\sqrt{a^2m^2-1}},\dfrac{-1}{\sqrt{a^2m^2-1}}\right) \end{array} If a tangent to a suitable conic (Column 1) is found to be y=x+8y=x+8 and its point of contact is (8,16)(8,16), then which of the following options is the only CORRECT combination?
Not attempted
A
(I) (ii) (Q)
B
(II) (iv) (R)
C
(III) (i) (P)
D
(III) (ii) (Q)
24
Columns 1, 2 and 3 contain conics, equations of tangents to the conics, and points of contact, respectively. (I) x2+y2=a2(i) my=m2x+a(P) (am2,2am)(II) x2+a2y2=a2(ii) y=mx+am2+1(Q) (−mam2+1,am2+1)(III) y2=4ax(iii) y=mx+a2m2−1(R) (−a2ma2m2+1,1a2m2+1)(IV) x2−a2y2=a2(iv) y=mx+a2m2+1(S) (−a2ma2m2−1,−1a2m2−1)\begin{array}{lll} \text{(I) } x^2+y^2=a^2 & \text{(i) } my=m^2x+a & \text{(P) } \left(\dfrac{a}{m^2},\dfrac{2a}{m}\right)\\ \text{(II) } x^2+a^2y^2=a^2 & \text{(ii) } y=mx+a\sqrt{m^2+1} & \text{(Q) } \left(\dfrac{-ma}{\sqrt{m^2+1}},\dfrac{a}{\sqrt{m^2+1}}\right)\\ \text{(III) } y^2=4ax & \text{(iii) } y=mx+\sqrt{a^2m^2-1} & \text{(R) } \left(\dfrac{-a^2m}{\sqrt{a^2m^2+1}},\dfrac{1}{\sqrt{a^2m^2+1}}\right)\\ \text{(IV) } x^2-a^2y^2=a^2 & \text{(iv) } y=mx+\sqrt{a^2m^2+1} & \text{(S) } \left(\dfrac{-a^2m}{\sqrt{a^2m^2-1}},\dfrac{-1}{\sqrt{a^2m^2-1}}\right) \end{array} The tangent to a suitable conic (Column 1) at (3,12)\left(\sqrt3,\dfrac12\right) is found to be 3x+2y=4\sqrt3x+2y=4, then which of the following options is the only CORRECT combination?
Not attempted
A
(IV) (iii) (S)
B
(IV) (iv) (S)
C
(II) (iii) (R)
D
(II) (iv) (R)
25
The sides of a right angled triangle are in arithmetic progression. If the triangle has area 2424, then what is the length of its smallest side?
Not attempted
Your answer: —
Correct answer: 6
26
Words of length 1010 are formed using the letters A,B,C,D,E,F,G,H,I,JA,B,C,D,E,F,G,H,I,J. Let xx be the number of such words where no letter is repeated, and let yy be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then y9x=\dfrac{y}{9x}=
Not attempted
Your answer: —
Correct answer: 5
27
For a real number α\alpha, if the system of linear equations (1αα2α1αα2α1)(xyz)=(1−11)\begin{pmatrix}1&\alpha&\alpha^2\\ \alpha&1&\alpha\\ \alpha^2&\alpha&1\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\-1\\1\end{pmatrix} has infinitely many solutions, then 1+α+α2=1+\alpha+\alpha^2=
Not attempted
Your answer: —
Correct answer: 1
28
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a differentiable function such that f(0)=0f(0)=0, f(π2)=3f\left(\dfrac{\pi}{2}\right)=3 and f′(0)=1f'(0)=1. If g(x)=∫xπ2[f′(t)csc⁡t−cot⁡t csc⁡t f(t)]dtg(x)=\displaystyle\int_x^{\frac{\pi}{2}}\left[f'(t)\csc t-\cot t\,\csc t\,f(t)\right]dt for x∈(0,π2]x\in\left(0,\dfrac{\pi}{2}\right], then lim⁡x→0g(x)=\displaystyle\lim_{x\to0}g(x)=
Not attempted
Your answer: —
Correct answer: 2
29
For how many values of pp do the circle x2+y2+2x+4y−p=0x^2+y^2+2x+4y-p=0 and the coordinate axes have exactly three common points?
Not attempted
Your answer: —
Correct answer: 2
30
Let XX and YY be two events such that P(X)=13P(X)=\dfrac{1}{3}, P(X∣Y)=12P(X\mid Y)=\dfrac{1}{2} and P(Y∣X)=25P(Y\mid X)=\dfrac{2}{5}. Then
Not attempted
A
P(Y)=415P(Y)=\dfrac{4}{15}
B
P(X′∣Y)=12P(X'\mid Y)=\dfrac{1}{2}
C
P(X∩Y)=15P(X\cap Y)=\dfrac{1}{5}
D
P(X∪Y)=25P(X\cup Y)=\dfrac{2}{5}
31
Let a,b,xa, b, x and yy be real numbers such that a−b=1a-b=1 and y≠0y\neq0. If the complex number z=x+iyz=x+iy satisfies Im(az+bz+1)=y\text{Im}\left(\dfrac{az+b}{z+1}\right)=y, then which of the following is(are) possible value(s) of xx?
Not attempted
A
−1+1−y2-1+\sqrt{1-y^2}
B
−1−1−y2-1-\sqrt{1-y^2}
C
1+1+y21+\sqrt{1+y^2}
D
1−1+y21-\sqrt{1+y^2}
32
Which of the following is(are) NOT the square of a 3×33\times3 matrix with real entries?
Not attempted
A
(100010001)\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}
B
(10001000−1)\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}
C
(1000−1000−1)\begin{pmatrix}1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}
D
(−1000−1000−1)\begin{pmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}
33
Let [x][x] be the greatest integer less than or equal to xx. Then, at which of the following point(s) is the function f(x)=xcos⁡(π(x+[x]))f(x) = x\cos(\pi(x+[x])) discontinuous?
Not attempted
A
x=−1x=-1
B
x=0x=0
C
x=1x=1
D
x=2x=2
34
Let f:R→(0,1)f:\mathbb{R}\to(0,1) be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0,1)(0,1)?
Not attempted
A
x9−f(x)x^9 - f(x)
B
x−∫0π2−xf(t)cos⁡t dtx - \displaystyle\int_0^{\frac{\pi}{2}-x} f(t)\cos t\, dt
C
ex−∫0xf(t)sin⁡t dte^x - \displaystyle\int_0^{x} f(t)\sin t\, dt
D
f(x)+∫0π2f(t)sin⁡t dtf(x) + \displaystyle\int_0^{\frac{\pi}{2}} f(t)\sin t\, dt
35
If 2x−y+1=02x - y + 1 = 0 is a tangent to the hyperbola x2a2−y216=1\dfrac{x^2}{a^2} - \dfrac{y^2}{16} = 1, then which of the following CANNOT be sides of a right angled triangle?
Not attempted
A
a,4,1a, 4, 1
B
a,4,2a, 4, 2
C
2a,8,12a, 8, 1
D
2a,4,12a, 4, 1
36
If a chord, which is not a tangent, of the parabola y2=16xy^2=16x has the equation 2x+y=p2x+y=p, and midpoint (h,k)(h,k), then which of the following is(are) possible value(s) of pp, hh and kk?
Not attempted
A
p=−2, h=2, k=−4p=-2,\ h=2,\ k=-4
B
p=−1, h=1, k=−3p=-1,\ h=1,\ k=-3
C
p=2, h=3, k=−4p=2,\ h=3,\ k=-4
D
p=5, h=4, k=−3p=5,\ h=4,\ k=-3
Hint\textbf{Hint}: For y2=4axy^2=4ax (a=4a=4 here), the chord with midpoint (h,k)(h,k) has equation ky−2a(x+h)=k2−4ahky-2a(x+h)=k^2-4ah (the T=S1T=S_1 form). Compare its coefficients with 2x+y=p2x+y=p.
Deconstruction\textbf{Deconstruction}: A chord of y2=16xy^2=16x has equation 2x+y=p2x+y=p and midpoint (h,k)(h,k). We need the consistent triple (p,h,k)(p,h,k) among the given options. Strategy\textbf{Strategy}: For y2=4axy^2=4ax, the chord with a given midpoint (h,k)(h,k) has equation (the T=S1T=S_1 form) ky−2a(x+h)=k2−4ahky-2a(x+h)=k^2-4ah. Here 4a=16⇒a=44a=16 \Rightarrow a=4. Rearranged: 8x−ky=8h−k28x-ky=8h-k^2. Compare coefficients with 2x+y=p2x+y=p scaled by 44: 8x+4y=4p8x+4y=4p. Execution\textbf{Execution}: Matching the yy-coefficients: −k=4⇒k=−4-k=4 \Rightarrow k=-4. This immediately eliminates [B] and [D], both of which state k=−3k=-3. Matching the constant terms: 8h−k2=4p⇒8h−16=4p⇒p=2h−48h-k^2=4p \Rightarrow 8h-16=4p \Rightarrow p=2h-4. Check [A]: h=2⇒ph=2 \Rightarrow p should be 2(2)−4=02(2)-4=0, but [A] states p=−2p=-2. Fails. Check [C]: h=3⇒p=2(3)−4=2h=3 \Rightarrow p=2(3)-4=2, and [C] states p=2p=2. Matches exactly. Verification\textbf{Verification}: An independent check — the midpoint must itself lie on the chord line 2x+y=p2x+y=p. For [C]: 2(3)+(−4)=2=p2(3)+(-4)=2=p ✓ Consistent. For [A]: 2(2)+(−4)=0≠−2=p2(2)+(-4)=0\neq -2=p ✗ Confirms [A] is inconsistent. Final Answer\textbf{Final Answer}: Only [C] is a possible value set.