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JEE Advanced · Mathematics · 2019

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1
Let the circles C1:x2+y2=9C_1: x^2+y^2=9 and C2:(x−3)2+(y−4)2=16C_2: (x-3)^2+(y-4)^2=16 intersect at the points XX and YY. Suppose that another circle C3:(x−h)2+(y−k)2=r2C_3:(x-h)^2+(y-k)^2=r^2 satisfies the following conditions: (i) the centre of C3C_3 is collinear with the centres of C1C_1 and C2C_2; (ii) C1C_1 and C2C_2 both lie inside C3C_3; and (iii) C3C_3 touches C1C_1 at MM and C2C_2 at NN. Let the line through XX and YY intersect C3C_3 at ZZ and WW, and let a common tangent of C1C_1 and C3C_3 be a tangent to the parabola x2=8αyx^2=8\alpha y. There are some expressions given in List-I whose values are given in List-II below: List-I: (I) 2h+k2h+k (II) Length of ZWLength of XY\dfrac{\text{Length of }ZW}{\text{Length of }XY} (III) Area of △MZNArea of △ZMW\dfrac{\text{Area of }\triangle MZN}{\text{Area of }\triangle ZMW} (IV) α\alpha List-II: (P) 66 (Q) 6\sqrt6 (R) 54\dfrac54 (S) 215\dfrac{21}5 (T) 262\sqrt6 (U) 103\dfrac{10}3 Which of the following is the only INCORRECT combination?
Not attempted
A
(IV), (S)
B
(IV), (U)
C
(III), (R)
D
(I), (P)
2
Let the circles C1:x2+y2=9C_1: x^2+y^2=9 and C2:(x−3)2+(y−4)2=16C_2: (x-3)^2+(y-4)^2=16 intersect at the points XX and YY. Suppose that another circle C3:(x−h)2+(y−k)2=r2C_3:(x-h)^2+(y-k)^2=r^2 satisfies the following conditions: (i) the centre of C3C_3 is collinear with the centres of C1C_1 and C2C_2; (ii) C1C_1 and C2C_2 both lie inside C3C_3; and (iii) C3C_3 touches C1C_1 at MM and C2C_2 at NN. Let the line through XX and YY intersect C3C_3 at ZZ and WW, and let a common tangent of C1C_1 and C3C_3 be a tangent to the parabola x2=8αyx^2=8\alpha y. There are some expressions given in List-I whose values are given in List-II below: List-I: (I) 2h+k2h+k (II) Length of ZWLength of XY\dfrac{\text{Length of }ZW}{\text{Length of }XY} (III) Area of △MZNArea of △ZMW\dfrac{\text{Area of }\triangle MZN}{\text{Area of }\triangle ZMW} (IV) α\alpha List-II: (P) 66 (Q) 6\sqrt6 (R) 54\dfrac54 (S) 215\dfrac{21}5 (T) 262\sqrt6 (U) 103\dfrac{10}3 Which of the following is the only CORRECT combination?
Not attempted
A
(II), (T)
B
(I), (S)
C
(I), (U)
D
(II), (Q)
3
Let f(x)=sin⁡(πcos⁡x)f(x)=\sin(\pi\cos x) and g(x)=cos⁡(2πsin⁡x)g(x)=\cos(2\pi\sin x) be two functions defined for x>0x>0. Define the following sets, whose elements are written in increasing order: X={x:f(x)=0}X=\{x: f(x)=0\}, Y={x:f′(x)=0}Y=\{x: f'(x)=0\}, Z={x:g(x)=0}Z=\{x: g(x)=0\}, W={x:g′(x)=0}W=\{x: g'(x)=0\}. List-I contains the sets X,Y,Z,WX,Y,Z,W and List-II contains some information regarding these sets. List-I: (I) XX (II) YY (III) ZZ (IV) WW List-II: (P) ⊇{π2,3π2,4π,7π}\supseteq\left\{\dfrac{\pi}{2},\dfrac{3\pi}{2},4\pi,7\pi\right\} (Q) an arithmetic progression (R) NOT an arithmetic progression (S) ⊇{π6,7π6,13π6}\supseteq\left\{\dfrac{\pi}{6},\dfrac{7\pi}{6},\dfrac{13\pi}{6}\right\} (T) ⊇{π3,2π3,π}\supseteq\left\{\dfrac{\pi}{3},\dfrac{2\pi}{3},\pi\right\} (U) ⊇{π6,3π4}\supseteq\left\{\dfrac{\pi}{6},\dfrac{3\pi}{4}\right\} Which of the following is the only CORRECT combination?
Not attempted
A
(II), (R), (S)
B
(I), (P), (R)
C
(II), (Q), (T)
D
(I), (Q), (U)
4
Let f(x)=sin⁡(πcos⁡x)f(x)=\sin(\pi\cos x) and g(x)=cos⁡(2πsin⁡x)g(x)=\cos(2\pi\sin x) be two functions defined for x>0x>0. Define the following sets, whose elements are written in increasing order: X={x:f(x)=0}X=\{x: f(x)=0\}, Y={x:f′(x)=0}Y=\{x: f'(x)=0\}, Z={x:g(x)=0}Z=\{x: g(x)=0\}, W={x:g′(x)=0}W=\{x: g'(x)=0\}. List-I contains the sets X,Y,Z,WX,Y,Z,W and List-II contains some information regarding these sets. List-I: (I) XX (II) YY (III) ZZ (IV) WW List-II: (P) ⊇{π2,3π2,4π,7π}\supseteq\left\{\dfrac{\pi}{2},\dfrac{3\pi}{2},4\pi,7\pi\right\} (Q) an arithmetic progression (R) NOT an arithmetic progression (S) ⊇{π6,7π6,13π6}\supseteq\left\{\dfrac{\pi}{6},\dfrac{7\pi}{6},\dfrac{13\pi}{6}\right\} (T) ⊇{π3,2π3,π}\supseteq\left\{\dfrac{\pi}{3},\dfrac{2\pi}{3},\pi\right\} (U) ⊇{π6,3π4}\supseteq\left\{\dfrac{\pi}{6},\dfrac{3\pi}{4}\right\} Which of the following is the only CORRECT combination?
Not attempted
A
(IV), (Q), (T)
B
(IV), (P), (R), (S)
C
(III), (R), (U)
D
(III), (P), (Q), (U)
5
Let a⃗=2i^+j^−k^\vec a=2\hat i+\hat j-\hat k and b⃗=i^+2j^+k^\vec b=\hat i+2\hat j+\hat k be two vectors. Consider a vector c⃗=αa⃗+βb⃗\vec c=\alpha\vec a+\beta\vec b, α,β∈R\alpha,\beta\in\mathbb{R}. If the projection of c⃗\vec c on the vector (a⃗+b⃗)(\vec a+\vec b) is 323\sqrt2, then the minimum value of (c⃗−(a⃗×b⃗))⋅c⃗(\vec c-(\vec a\times\vec b))\cdot\vec c equals ____
Not attempted
Your answer: —
Correct answer: 18
6
The value of the integral ∫0π/23cos⁡θ(cos⁡θ+sin⁡θ)5 dθ\displaystyle\int_0^{\pi/2}\dfrac{3\sqrt{\cos\theta}}{\left(\sqrt{\cos\theta}+\sqrt{\sin\theta}\right)^5}\,d\theta equals ____
Not attempted
Your answer: —
Correct answer: 0.5
7
Suppose det⁡(∑k=0nk∑k=0nnCk k2∑k=0nnCk k∑k=0nnCk 3k)=0\det\begin{pmatrix}\displaystyle\sum_{k=0}^n k & \displaystyle\sum_{k=0}^n {}^nC_k\,k^2\\[6pt] \displaystyle\sum_{k=0}^n {}^nC_k\,k & \displaystyle\sum_{k=0}^n {}^nC_k\,3^k\end{pmatrix}=0 holds for some positive integer nn. Then ∑k=0nnCkk+1\displaystyle\sum_{k=0}^n \dfrac{{}^nC_k}{k+1} equals ____
Not attempted
Your answer: —
Correct answer: 6.2
8
Five persons A,B,C,DA,B,C,D and EE are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is ____
Not attempted
Your answer: —
Correct answer: 30
9
Let ∣X∣|X| denote the number of elements in a set XX. Let S={1,2,3,4,5,6}S=\{1,2,3,4,5,6\} be a sample space, where each element is equally likely to occur. If AA and BB are independent events associated with SS, then the number of ordered pairs (A,B)(A,B) such that 1≤∣B∣<∣A∣1\le|B|<|A|, equals ____
Not attempted
Your answer: —
Correct answer: 422
10
The value of sec⁡−1(14∑k=010sec⁡ ⁣(7π12+kπ2)sec⁡ ⁣(7π12+(k+1)π2))\sec^{-1}\left(\dfrac14\displaystyle\sum_{k=0}^{10}\sec\!\left(\dfrac{7\pi}{12}+\dfrac{k\pi}{2}\right)\sec\!\left(\dfrac{7\pi}{12}+\dfrac{(k+1)\pi}{2}\right)\right) in the interval [−π4,3π4]\left[-\dfrac{\pi}{4},\dfrac{3\pi}{4}\right] equals ____
Not attempted
Your answer: —
Correct answer: 0
11
Let f(x)=sin⁡πxx2f(x)=\dfrac{\sin\pi x}{x^2}, x>0x>0. Let x1<x2<x3<⋯<xn<⋯x_1<x_2<x_3<\cdots<x_n<\cdots be all the points of local maximum of ff, and y1<y2<y3<⋯<yn<⋯y_1<y_2<y_3<\cdots<y_n<\cdots be all the points of local minimum of ff. Then which of the following options is/are correct?
Not attempted
A
∣xn−yn∣>1|x_n-y_n|>1 for every nn
B
x1<y1x_1<y_1
C
xn∈(2n, 2n+12)x_n\in\left(2n,\,2n+\tfrac12\right) for every nn
D
xn+1−xn>2x_{n+1}-x_n>2 for every nn
12
Let P1=I=(100010001)P_1=I=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}, P2=(100001010)P_2=\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}, P3=(010100001)P_3=\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix}, P4=(010001100)P_4=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}, P5=(001100010)P_5=\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}, P6=(001010100)P_6=\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}, and X=∑k=16Pk(213102321)PkTX=\displaystyle\sum_{k=1}^{6}P_k\begin{pmatrix}2&1&3\\1&0&2\\3&2&1\end{pmatrix}P_k^{T}, where PkTP_k^{T} denotes the transpose of PkP_k. Then which of the following options is/are correct?
Not attempted
A
X−30IX-30I is an invertible matrix
B
The sum of diagonal entries (trace) of XX is 1818
C
If X(111)=α(111)X\begin{pmatrix}1\\1\\1\end{pmatrix}=\alpha\begin{pmatrix}1\\1\\1\end{pmatrix}, then α=30\alpha=30
D
XX is a symmetric matrix
13
Let x∈Rx\in\mathbb{R} and let P=(111022003)P=\begin{pmatrix}1&1&1\\0&2&2\\0&0&3\end{pmatrix}, Q=(2xx040xx6)Q=\begin{pmatrix}2&x&x\\0&4&0\\x&x&6\end{pmatrix} and R=PQP−1R=PQP^{-1}. Then which of the following options is/are correct?
Not attempted
A
For x=1x=1, there exists a unit vector αi^+βj^+γk^\alpha\hat i+\beta\hat j+\gamma\hat k for which R(αβγ)=(000)R\begin{pmatrix}\alpha\\\beta\\\gamma\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix}
B
There exists a real number xx such that PQ=QPPQ=QP
C
det⁡R=det⁡(2xx040xx5)+8\det R = \det\begin{pmatrix}2&x&x\\0&4&0\\x&x&5\end{pmatrix}+8, for all x∈Rx\in\mathbb{R}
D
For x=0x=0, if R(1ab)=6(1ab)R\begin{pmatrix}1\\a\\b\end{pmatrix}=6\begin{pmatrix}1\\a\\b\end{pmatrix}, then a+b=5a+b=5
14
For non-negative integers nn, let f(n)=∑k=0nsin⁡ ⁣(k+1n+2π)sin⁡ ⁣(k+2n+2π)∑k=0nsin⁡2 ⁣(k+1n+2π).f(n)=\dfrac{\displaystyle\sum_{k=0}^{n}\sin\!\left(\dfrac{k+1}{n+2}\pi\right)\sin\!\left(\dfrac{k+2}{n+2}\pi\right)}{\displaystyle\sum_{k=0}^{n}\sin^2\!\left(\dfrac{k+1}{n+2}\pi\right)}. Assuming cos⁡−1x\cos^{-1}x takes values in [0,π][0,\pi], which of the following options is/are correct?
Not attempted
A
sin⁡(7cos⁡−1f(5))=0\sin(7\cos^{-1}f(5))=0
B
f(4)=32f(4)=\dfrac{\sqrt3}{2}
C
lim⁡n→∞f(n)=12\lim_{n\to\infty} f(n) = \dfrac12
D
If α=tan⁡(cos⁡−1f(6))\alpha=\tan(\cos^{-1}f(6)), then α2+2α−1=0\alpha^2+2\alpha-1=0
15
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a function. We say that ff has PROPERTY 1 if lim⁡h→0f(h)−f(0)∣h∣\lim_{h\to0}\dfrac{f(h)-f(0)}{\sqrt{|h|}} exists and is finite, and PROPERTY 2 if lim⁡h→0f(h)−f(0)h2\lim_{h\to0}\dfrac{f(h)-f(0)}{h^2} exists and is finite. Then which of the following options is/are correct?
Not attempted
A
f(x)=x∣x∣f(x)=x|x| has PROPERTY 2
B
f(x)=x2/3f(x)=x^{2/3} has PROPERTY 1
C
f(x)=sin⁡xf(x)=\sin x has PROPERTY 2
D
f(x)=∣x∣f(x)=|x| has PROPERTY 1
16
Three lines L1:r⃗=λi^, λ∈RL_1: \vec r = \lambda\hat i,\ \lambda\in\mathbb{R}, L2:r⃗=k^+μj^, μ∈RL_2: \vec r = \hat k+\mu\hat j,\ \mu\in\mathbb{R}, and L3:r⃗=i^+j^+νk^, ν∈RL_3: \vec r = \hat i+\hat j+\nu\hat k,\ \nu\in\mathbb{R} are given. For which point(s) QQ on L2L_2 can we find a point PP on L1L_1 and a point RR on L3L_3 so that PP, QQ and RR are collinear?
Not attempted
A
k^+j^\hat k+\hat j
B
k^\hat k
C
k^+12j^\hat k+\tfrac12\hat j
D
k^−12j^\hat k-\tfrac12\hat j
17
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be given by f(x)=(x−1)(x−2)(x−5)f(x) = (x-1)(x-2)(x-5). Define F(x)=∫0xf(t) dtF(x) = \int_0^x f(t)\,dt for x>0x>0. Then which of the following options is/are correct?
Not attempted
A
FF has a local minimum at x=1x=1
B
FF has a local maximum at x=2x=2
C
F(x)≠0F(x)\neq 0 for all x∈(0,5)x\in(0,5)
D
FF has two local maxima and one local minimum in (0,∞)(0,\infty)
18
For a∈Ra\in\mathbb{R}, ∣a∣>1|a|>1, let lim⁡n→∞(1+23+⋯+n3n7/3(1(an+1)2+1(an+2)2+⋯+1(an+n)2))=54.\lim_{n\to\infty}\left(\dfrac{1+\sqrt[3]{2}+\cdots+\sqrt[3]{n}}{n^{7/3}\left(\dfrac{1}{(an+1)^2}+\dfrac{1}{(an+2)^2}+\cdots+\dfrac{1}{(an+n)^2}\right)}\right) = 54. Then the possible value(s) of aa is/are:
Not attempted
A
88
B
−9-9
C
−6-6
D
77
19
Let L1L_1 and L2L_2 denote the lines r⃗=i^+λ(−i^+2j^+2k^), λ∈R\vec r=\hat i+\lambda(-\hat i+2\hat j+2\hat k),\ \lambda\in\mathbb{R} and r⃗=μ(2i^−j^+2k^), μ∈R\vec r=\mu(2\hat i-\hat j+2\hat k),\ \mu\in\mathbb{R} respectively. If L3L_3 is a line which is perpendicular to both L1L_1 and L2L_2 and cuts both of them, then which of the following options describe(s) L3L_3?
Not attempted
A
r⃗=13(2i^+k^)+t(2i^+2j^−k^), t∈R\vec r=\dfrac13(2\hat i+\hat k)+t(2\hat i+2\hat j-\hat k),\ t\in\mathbb{R}
B
r⃗=29(2i^−j^+2k^)+t(2i^+2j^−k^), t∈R\vec r=\dfrac29(2\hat i-\hat j+2\hat k)+t(2\hat i+2\hat j-\hat k),\ t\in\mathbb{R}
C
r⃗=t(2i^+2j^−k^), t∈R\vec r=t(2\hat i+2\hat j-\hat k),\ t\in\mathbb{R}
D
r⃗=29(4i^+j^+k^)+t(2i^+2j^−k^), t∈R\vec r=\dfrac29(4\hat i+\hat j+\hat k)+t(2\hat i+2\hat j-\hat k),\ t\in\mathbb{R}
20
Three lines are given by r⃗=λi^, λ∈R\vec r=\lambda\hat i,\ \lambda\in\mathbb{R}, r⃗=μ(i^+j^), μ∈R\vec r=\mu(\hat i+\hat j),\ \mu\in\mathbb{R} and r⃗=ν(i^+j^+k^), ν∈R\vec r=\nu(\hat i+\hat j+\hat k),\ \nu\in\mathbb{R}. Let the lines cut the plane x+y+z=1x+y+z=1 at the points AA, BB and CC respectively. If the area of the triangle ABCABC is Δ\Delta then the value of (6Δ)2(6\Delta)^2 equals _____
Not attempted
Your answer: —
Correct answer: 0.75
21
In a non-right-angled triangle △PQR\triangle PQR, let p,q,rp,q,r denote the lengths of the sides opposite to the angles at P,Q,RP,Q,R respectively. The median from RR meets the side PQPQ at SS, the perpendicular from PP meets the side QRQR at EE, and RSRS and PEPE intersect at OO. If p=3p=\sqrt3, q=1q=1, and the radius of the circumcircle of the △PQR\triangle PQR equals 11, then which of the following options is/are correct?
Not attempted
A
Area of △SOE=312\triangle SOE=\dfrac{\sqrt3}{12}
B
Radius of incircle of △PQR=32(2−3)\triangle PQR=\dfrac{\sqrt3}{2}(2-\sqrt3)
C
Length of RS=72RS=\dfrac{\sqrt7}{2}
D
Length of OE=16OE=\dfrac16
22
Let Γ\Gamma denote a curve y=y(x)y=y(x) which is in the first quadrant and let the point (1,0)(1,0) lie on it. Let the tangent to Γ\Gamma at a point PP intersect the yy-axis at YPY_P. If PYPPY_P has length 11 for each point PP on Γ\Gamma, then which of the following options is/are correct?
Not attempted
A
y=log⁡e ⁣(1+1−x2x)−1−x2y=\log_e\!\left(\dfrac{1+\sqrt{1-x^2}}{x}\right)-\sqrt{1-x^2}
B
xy′−1−x2=0xy'-\sqrt{1-x^2}=0
C
y=−log⁡e ⁣(1+1−x2x)+1−x2y=-\log_e\!\left(\dfrac{1+\sqrt{1-x^2}}{x}\right)+\sqrt{1-x^2}
D
xy′+1−x2=0xy'+\sqrt{1-x^2}=0
23
Let α\alpha and β\beta be the roots of x2−x−1=0x^2-x-1=0, with α>β\alpha>\beta. For all positive integers nn, define an=αn−βnα−βa_n=\dfrac{\alpha^n-\beta^n}{\alpha-\beta}, n≥1n\ge1, b1=1b_1=1 and bn=an−1+an+1b_n=a_{n-1}+a_{n+1}, n≥2n\ge2. Then which of the following options is/are correct?
Not attempted
A
a1+a2+a3+⋯+an=an+2−1a_1+a_2+a_3+\cdots+a_n=a_{n+2}-1 for all n≥1n\ge1
B
∑n=1∞an10n=1089\displaystyle\sum_{n=1}^{\infty}\dfrac{a_n}{10^n}=\dfrac{10}{89}
C
∑n=1∞bn10n=889\displaystyle\sum_{n=1}^{\infty}\dfrac{b_n}{10^n}=\dfrac{8}{89}
D
bn=αn+βnb_n=\alpha^n+\beta^n for all n≥1n\ge1
24
Let AP(a;d)AP(a;d) denote the set of all the terms of an infinite arithmetic progression with first term aa and common difference d>0d>0. If AP(1;3)∩AP(2;5)∩AP(3;7)=AP(a;d)AP(1;3)\cap AP(2;5)\cap AP(3;7)=AP(a;d) then a+da+d equals _____
Not attempted
Your answer: —
Correct answer: 157
25
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be given by f(x)={x5+5x4+10x3+10x2+3x+1,x<0x2−x+1,0≤x<123x3−4x2+7x−83,1≤x<3(x−2)log⁡e(x−2)−x+103,x≥3f(x)=\begin{cases}x^5+5x^4+10x^3+10x^2+3x+1, & x<0\\x^2-x+1, & 0\le x<1\\\dfrac23x^3-4x^2+7x-\dfrac83, & 1\le x<3\\(x-2)\log_e(x-2)-x+\dfrac{10}{3}, & x\ge3\end{cases}. Then which of the following options is/are correct?
Not attempted
A
f′f' has a local maximum at x=1x=1
B
ff is onto
C
ff is increasing on (−∞,0)(-\infty,0)
D
f′f' is NOT differentiable at x=1x=1
26
There are three bags B1B_1, B2B_2 and B3B_3. The bag B1B_1 contains 55 red and 55 green balls, B2B_2 contains 33 red and 55 green balls, and B3B_3 contains 55 red and 33 green balls. Bags B1B_1, B2B_2 and B3B_3 have probabilities 310,310\dfrac{3}{10},\dfrac{3}{10} and 410\dfrac{4}{10} respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?
Not attempted
A
Probability that the selected bag is B3B_3 and the chosen ball is green equals 310\dfrac{3}{10}
B
Probability that the chosen ball is green equals 3980\dfrac{39}{80}
C
Probability that the chosen ball is green, given that the selected bag is B3B_3, equals 38\dfrac38
D
Probability that the selected bag is B3B_3, given that the chosen ball is green, equals 513\dfrac{5}{13}
27
Let SS be the sample space of all 3×33\times3 matrices with entries from the set {0,1}\{0,1\}. Let the events E1E_1 and E2E_2 be given by E1={A∈S:det⁡A=0}E_1=\{A\in S:\det A=0\} and E2={A∈S:sum of entries of A is 7}E_2=\{A\in S:\text{sum of entries of }A\text{ is }7\}. If a matrix is chosen at random from SS, then the conditional probability P(E1∣E2)P(E_1\mid E_2) equals _____
Not attempted
Your answer: —
Correct answer: 0.5
28
The area of the region {(x,y):xy≤8, 1≤y≤x2}\{(x,y):xy\le8,\ 1\le y\le x^2\} is
Not attempted
A
8log⁡e2−1438\log_e2-\dfrac{14}{3}
B
16log⁡e2−14316\log_e2-\dfrac{14}{3}
C
16log⁡e2−616\log_e2-6
D
8log⁡e2−738\log_e2-\dfrac{7}{3}
29
If I=2π∫−π/4π/4dx(1+esin⁡x)(2−cos⁡2x)I=\dfrac{2}{\pi}\displaystyle\int_{-\pi/4}^{\pi/4}\dfrac{dx}{(1+e^{\sin x})(2-\cos2x)} then 27I227I^2 equals _____
Not attempted
Your answer: —
Correct answer: 4
30
Let SS be the set of all complex numbers zz satisfying ∣z−2+i∣≥5|z-2+i|\ge\sqrt5. If the complex number z0z_0 is such that 1∣z0−1∣\dfrac{1}{|z_0-1|} is the maximum of the set {1∣z−1∣:z∈S}\left\{\dfrac{1}{|z-1|}:z\in S\right\}, then the principal argument of 4−z0−zˉ0z0−zˉ0+2i\dfrac{4-z_0-\bar z_0}{z_0-\bar z_0+2i} is
Not attempted
A
π4\dfrac{\pi}{4}
B
−π2-\dfrac{\pi}{2}
C
3π4\dfrac{3\pi}{4}
D
π2\dfrac{\pi}{2}
31
Let ω≠1\omega\ne1 be a cube root of unity. Then the minimum of the set {∣a+bω+cω2∣2:a,b,c distinct non-zero integers}\{|a+b\omega+c\omega^2|^2 : a,b,c\text{ distinct non-zero integers}\} equals _____
Not attempted
Your answer: —
Correct answer: 3
32
A line y=mx+1y=mx+1 intersects the circle (x−3)2+(y+2)2=25(x-3)^2+(y+2)^2=25 at the points PP and QQ. If the midpoint of the line segment PQPQ has xx-coordinate −35-\dfrac35, then which one of the following options is correct?
Not attempted
A
6≤m<86 \le m < 8
B
2≤m<42 \le m < 4
C
4≤m<64 \le m < 6
D
−3≤m<−1-3 \le m < -1
33
Define the collections {E1,E2,E3,…}\{E_1,E_2,E_3,\ldots\} of ellipses and {R1,R2,R3,…}\{R_1,R_2,R_3,\ldots\} of rectangles as follows: E1:x29+y24=1E_1:\dfrac{x^2}{9}+\dfrac{y^2}{4}=1; R1R_1: rectangle of largest area, with sides parallel to the axes, inscribed in E1E_1; EnE_n: ellipse x2an2+y2bn2=1\dfrac{x^2}{a_n^2}+\dfrac{y^2}{b_n^2}=1 of largest area inscribed in Rn−1R_{n-1}, n>1n>1; RnR_n: rectangle of largest area, with sides parallel to the axes, inscribed in EnE_n, n>1n>1. Then which of the following options is/are correct?
Not attempted
A
The eccentricities of E18E_{18} and E19E_{19} are NOT equal
B
The distance of a focus from the centre in E9E_9 is 532\dfrac{\sqrt5}{32}
C
The length of latus rectum of E9E_9 is 16\dfrac16
D
∑n=1N(area of Rn)<24\displaystyle\sum_{n=1}^{N}(\text{area of }R_n)<24, for each positive integer NN
34
Let the point BB be the reflection of the point A(2,3)A(2,3) with respect to the line 8x−6y−23=08x-6y-23=0. Let ΓA\Gamma_A and ΓB\Gamma_B be circles of radii 22 and 11 with centres AA and BB respectively. Let TT be a common tangent to the circles ΓA\Gamma_A and ΓB\Gamma_B such that both circles are on the same side of TT. If CC is the point of intersection of TT and the line passing through AA and BB, then the length of the line segment ACAC is _____
Not attempted
Your answer: —
Correct answer: 10
35
Let M=(sin⁡4θ−1−sin⁡2θ1+cos⁡2θcos⁡4θ)=αI+βM−1M = \begin{pmatrix}\sin^4\theta & -1-\sin^2\theta\\1+\cos^2\theta & \cos^4\theta\end{pmatrix} = \alpha I + \beta M^{-1}, where α=α(θ)\alpha = \alpha(\theta) and β=β(θ)\beta = \beta(\theta) are real numbers, and II is the 2×22\times 2 identity matrix. If α∗\alpha^* is the minimum of the set {α(θ):θ∈[0,2π)}\{\alpha(\theta) : \theta \in [0, 2\pi)\} and β∗\beta^* is the minimum of the set {β(θ):θ∈[0,2π)}\{\beta(\theta) : \theta \in [0, 2\pi)\}, then the value of α∗+β∗\alpha^*+\beta^* is
Not attempted
A
−3716-\dfrac{37}{16}
B
−2916-\dfrac{29}{16}
C
−3116-\dfrac{31}{16}
D
−1716-\dfrac{17}{16}
36
Let M=(01a1233b1)M=\begin{pmatrix}0 & 1 & a\\1 & 2 & 3\\3 & b & 1\end{pmatrix} and adj M=(−11−18−62−53−1)\text{adj}\,M=\begin{pmatrix}-1 & 1 & -1\\8 & -6 & 2\\-5 & 3 & -1\end{pmatrix} where aa and bb are real numbers. Which of the following options is/are correct?
Not attempted
A
a+b=3a+b=3
B
det⁡(adj M2)=81\det(\text{adj}\,M^2)=81
C
(adj M)−1+adj(M−1)=−M(\text{adj}\,M)^{-1}+\text{adj}(M^{-1})=-M
D
If M(αβγ)=(123)M\begin{pmatrix}\alpha\\\beta\\\gamma\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix}, then α−β+γ=3\alpha-\beta+\gamma=3
Hint:\textbf{Hint:} Read off a,ba,b from single cofactor entries of adj M\text{adj}\,M (e.g. (adj M)11(\text{adj}\,M)_{11} and (adj M)22(\text{adj}\,M)_{22}), then use the standard identities det⁡(adj A)=(det⁡A)n−1\det(\text{adj}\,A)=(\det A)^{n-1}, adj A=(det⁡A)A−1\text{adj}\,A=(\det A)A^{-1}, and adj(A−1)=A/det⁡A\text{adj}(A^{-1})=A/\det A.
Deconstruction:\textbf{Deconstruction:} Multiple-correct question on adjugate-matrix identities for a 3×33\times3 matrix with two unknown entries a,ba,b to be determined first from the given adj M\text{adj}\,M. Strategy:\textbf{Strategy:} Use (adj M)ij=(\text{adj}\,M)_{ij}= cofactor CjiC_{ji} of MM to pin down a,ba,b from two specific entries. Then use det⁡(adj M2)=(det⁡M)4\det(\text{adj}\,M^2)=(\det M)^4 for a 3×33\times3 matrix, (adj M)−1=M/det⁡M(\text{adj}\,M)^{-1}=M/\det M and adj(M−1)=M/det⁡M\text{adj}(M^{-1})=M/\det M for option (3), and solve MX=(1,2,3)TMX=(1,2,3)^T via X=(adj M/det⁡M)(1,2,3)TX=(\text{adj}\,M/\det M)(1,2,3)^T for option (4). Execution:\textbf{Execution:} (adj M)11=C11=2(1)−3b=2−3b=−1⇒b=1(\text{adj}\,M)_{11}=C_{11}=2(1)-3b=2-3b=-1\Rightarrow b=1. (adj M)22=C22=0(1)−a(3)=−3a=−6⇒a=2(\text{adj}\,M)_{22}=C_{22}=0(1)-a(3)=-3a=-6\Rightarrow a=2. So a+b=2+1=3a+b=2+1=3 — option (1) is TRUE. M=(012123311)M=\begin{pmatrix}0 & 1 & 2\\1 & 2 & 3\\3 & 1 & 1\end{pmatrix}. Expanding, det⁡M=0(2−3)−1(1−9)+2(1−6)=0+8−10=−2\det M=0(2-3)-1(1-9)+2(1-6)=0+8-10=-2. det⁡(adj M2)=(det⁡M2)3−1=(det⁡M)4=(−2)4=16≠81\det(\text{adj}\,M^2)=(\det M^2)^{3-1}=(\det M)^4=(-2)^4=16\ne81 — option (2) is FALSE. Since adj M=(det⁡M)M−1=−2M−1\text{adj}\,M=(\det M)M^{-1}=-2M^{-1}, we get (adj M)−1=−12M(\text{adj}\,M)^{-1}=\dfrac{-1}{2}M. Also adj(M−1)=(det⁡M−1)(M−1)−1=1det⁡MM=−12M\text{adj}(M^{-1})=(\det M^{-1})(M^{-1})^{-1}=\dfrac{1}{\det M}M=\dfrac{-1}{2}M. Adding: (adj M)−1+adj(M−1)=−M(\text{adj}\,M)^{-1}+\text{adj}(M^{-1})=-M — option (3) is TRUE. For option (4): M(αβγ)=(123)⇒(αβγ)=M−1(123)=adj Mdet⁡M(123)=−12(−11−18−62−53−1)(123)=−12(−22−2)=(1−11)M\begin{pmatrix}\alpha\\\beta\\\gamma\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix}\Rightarrow\begin{pmatrix}\alpha\\\beta\\\gamma\end{pmatrix}=M^{-1}\begin{pmatrix}1\\2\\3\end{pmatrix}=\dfrac{\text{adj}\,M}{\det M}\begin{pmatrix}1\\2\\3\end{pmatrix}=\dfrac{-1}{2}\begin{pmatrix}-1 & 1 & -1\\8 & -6 & 2\\-5 & 3 & -1\end{pmatrix}\begin{pmatrix}1\\2\\3\end{pmatrix}=\dfrac{-1}{2}\begin{pmatrix}-2\\2\\-2\end{pmatrix}=\begin{pmatrix}1\\-1\\1\end{pmatrix}. So α−β+γ=1−(−1)+1=3\alpha-\beta+\gamma=1-(-1)+1=3 — option (4) is TRUE. Verification:\textbf{Verification:} Recompute det⁡M\det M a second way (row reduction): R2→R2,R3→R3−3R1R_2\to R_2, R_3\to R_3-3R_1 (using R1=(0,1,2)R_1=(0,1,2) is awkward as pivot; instead expand along column 1 directly as above) gives the same −2-2. Also verify (adj M)11=C11=det⁡(2311)=2−3=−1(\text{adj}\,M)_{11}=C_{11}=\det\begin{pmatrix}2 & 3\\1 & 1\end{pmatrix}=2-3=-1, matching the given adjugate exactly — confirming a=2,b=1a=2,b=1 are correct. Final Answer:\textbf{Final Answer:} Options (1), (3), and (4).