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JEE Advanced · Mathematics · Matrices and Determinants

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1
Let i=−1i=\sqrt{-1}. Consider S=(0−110)S=\begin{pmatrix}0&-1\\1&0\end{pmatrix} and T=(1101)T=\begin{pmatrix}1&1\\0&1\end{pmatrix}. Let a,b,c,da,b,c,d be real numbers with ST=(abcd)ST=\begin{pmatrix}a&b\\c&d\end{pmatrix}. Let H={x+iy:x,y∈R, y>0}H=\{x+iy: x,y\in\mathbb R,\ y>0\}. Then which of the following statements is (are) TRUE?
Not attempted
A
b+iad+ic=i\dfrac{b+ia}{d+ic}=i
B
If ω=−1+i32\omega=\dfrac{-1+i\sqrt3}2, then aω+bcω+d=ω\dfrac{a\omega+b}{c\omega+d}=\omega
C
If m>2m>2 is an integer such that (ST)2=(ST)m(ST)^2=(ST)^m, then mm is an integer multiple of 88
D
If z∈Hz\in H, then az+bcz+d∈H\dfrac{az+b}{cz+d}\in H
2
Which one of the following matrices can be obtained by performing elementary row transformations on the 3×33\times3 identity matrix? (A) (111111111)\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix} (B) (111234121)\begin{pmatrix}1&1&1\\2&3&4\\1&2&1\end{pmatrix} (C) (111234258)\begin{pmatrix}1&1&1\\2&3&4\\2&5&8\end{pmatrix} (D) (111−112023)\begin{pmatrix}1&1&1\\-1&1&2\\0&2&3\end{pmatrix}
Not attempted
A
(111111111)\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}
B
(111234121)\begin{pmatrix}1&1&1\\2&3&4\\1&2&1\end{pmatrix}
C
(111234258)\begin{pmatrix}1&1&1\\2&3&4\\2&5&8\end{pmatrix}
D
(111−112023)\begin{pmatrix}1&1&1\\-1&1&2\\0&2&3\end{pmatrix}
3
Consider the matrix M=(2−110)M=\begin{pmatrix}2&-1\\1&0\end{pmatrix}. Let p,q,r,s,a,b,c,dp,q,r,s,a,b,c,d be integers such that M26=(pqrs)M^{26}=\begin{pmatrix}p&q\\r&s\end{pmatrix} and ∑k=126Mk=(abcd)\displaystyle\sum_{k=1}^{26}M^k=\begin{pmatrix}a&b\\c&d\end{pmatrix}. Then which of the following statements is (are) TRUE?
Not attempted
A
There exists a 2×22\times2 invertible matrix NN with real entries such that MN=N(1101)MN=N\begin{pmatrix}1&1\\0&1\end{pmatrix}
B
The value of aa is 378378
C
For any two given integers m,nm,n, there exist unique integers x,yx,y such that px+qy=mpx+qy=m and rx+sy=nrx+sy=n
D
For each positive real number tt, the system (a+t)x+by=1, cx+(d+t)y=−1(a+t)x+by=1,\ cx+(d+t)y=-1 has a unique solution
4
Let I=(1001)I=\begin{pmatrix}1&0\\0&1\end{pmatrix} and P=(2003)P=\begin{pmatrix}2&0\\0&3\end{pmatrix}. Let Q=(xyz4)Q=\begin{pmatrix}x&y\\z&4\end{pmatrix} for some non-zero real numbers x,y,zx,y,z, for which there is a 2×22\times2 matrix RR with all entries being non-zero real numbers, such that QR=RPQR=RP. Then which of the following statements is (are) TRUE?
Not attempted
A
The determinant of Q−2IQ-2I is zero
B
The determinant of Q−6IQ-6I is 1212
C
The determinant of Q−3IQ-3I is 1515
D
yz=2yz=2
5
Consider the matrix P=(200020003)P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}. Let the transpose of a matrix XX be denoted by XTX^T. Then the number of 3×33\times 3 invertible matrices QQ with integer entries, such that Q−1=QTandPQ=QPQ^{-1} = Q^T \quad \text{and} \quad PQ = QP, is
Not attempted
A
3232
B
88
C
1616
D
2424
6
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.
7
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
8
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)
9
Let R={(a3bc2d050):a,b,c,d∈{0,3,5,7,11,13,17,19}}R=\left\{\begin{pmatrix}a&3&b\\c&2&d\\0&5&0\end{pmatrix}: a,b,c,d\in\{0,3,5,7,11,13,17,19\}\right\}. Then the number of invertible matrices in RR is
Not attempted
Your answer: —
Correct answer: 3780
10
Let M=(aij)M=(a_{ij}), i,j∈{1,2,3}i,j\in\{1,2,3\}, be the 3×33\times3 matrix such that aij=1a_{ij}=1 if j+1j+1 is divisible by ii, and aij=0a_{ij}=0 otherwise. Then which of the following statements is(are) true?
Not attempted
A
MM is invertible
B
There exists a nonzero column matrix (a1a2a3)\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix} such that M(a1a2a3)=(−a1−a2−a3)M\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix}=\begin{pmatrix}-a_1\\-a_2\\-a_3\end{pmatrix}
C
The set {X∈R3:MX=0⃗}≠{0⃗}\left\{X\in\mathbb{R}^3:MX=\vec0\right\}\ne\{\vec0\}, where 0⃗=(000)\vec0=\begin{pmatrix}0\\0\\0\end{pmatrix}
D
The matrix (M−2I)(M-2I) is invertible, where II is the 3×33\times3 identity matrix
11
Let α,β\alpha,\beta and γ\gamma be real numbers. Consider the following system of linear equations x+2y+z=7x+2y+z=7 x+αz=11x+\alpha z=11 2x−3y+βz=γ2x-3y+\beta z=\gamma Match each entry in List-I to the correct entries in List-II. List-I (P) If β=12(7α−3)\beta=\dfrac12(7\alpha-3) and γ=28\gamma=28, then the system has (Q) If β=12(7α−3)\beta=\dfrac12(7\alpha-3) and γ≠28\gamma\ne28, then the system has (R) If β≠12(7α−3)\beta\ne\dfrac12(7\alpha-3) where α=1\alpha=1 and γ≠28\gamma\ne28, then the system has (S) If β≠12(7α−3)\beta\ne\dfrac12(7\alpha-3) where α=1\alpha=1 and γ=28\gamma=28, then the system has List-II (1) a unique solution (2) no solution (3) infinitely many solutions (4) x=11x=11, y=−2y=-2 and z=0z=0 as a solution (5) x=−15x=-15, y=4y=4 and z=0z=0 as a solution The correct option is:
Not attempted
A
(P)→(3)(Q)→(2)(R)→(1)(S)→(4)(P)\to(3)\quad(Q)\to(2)\quad(R)\to(1)\quad(S)\to(4)
B
(P)→(3)(Q)→(2)(R)→(5)(S)→(4)(P)\to(3)\quad(Q)\to(2)\quad(R)\to(5)\quad(S)\to(4)
C
(P)→(2)(Q)→(1)(R)→(4)(S)→(5)(P)\to(2)\quad(Q)\to(1)\quad(R)\to(4)\quad(S)\to(5)
D
(P)→(2)(Q)→(1)(R)→(1)(S)→(3)(P)\to(2)\quad(Q)\to(1)\quad(R)\to(1)\quad(S)\to(3)
12
If M=(5232−32−12)M=\begin{pmatrix}\dfrac52 & \dfrac32\\[4pt] -\dfrac32 & -\dfrac12\end{pmatrix}, then which of the following matrices is equal to M2022M^{2022}?
Not attempted
A
(30343033−3033−3032)\begin{pmatrix}3034 & 3033\\ -3033 & -3032\end{pmatrix}
B
(3034−30333033−3032)\begin{pmatrix}3034 & -3033\\ 3033 & -3032\end{pmatrix}
C
(30333032−3032−3031)\begin{pmatrix}3033 & 3032\\ -3032 & -3031\end{pmatrix}
D
(30323031−3031−3030)\begin{pmatrix}3032 & 3031\\ -3031 & -3030\end{pmatrix}
13
Let β\beta be a real number. Consider the matrix A=(β0121−231−2)A=\begin{pmatrix}\beta & 0 & 1\\ 2 & 1 & -2\\ 3 & 1 & -2\end{pmatrix}. If A7−(β−1)A6−βA5A^7-(\beta-1)A^6-\beta A^5 is a singular matrix, then the value of 9β9\beta is _______.
Not attempted
Your answer: —
Correct answer: 3
14
Let p,q,rp,q,r be nonzero real numbers that are, respectively, the 10th10^{th}, 100th100^{th} and 1000th1000^{th} terms of a harmonic progression. Consider the system of linear equations x+y+z=1x+y+z=1 10x+100y+1000z=010x+100y+1000z=0 qr x+pr y+pq z=0qr\,x+pr\,y+pq\,z=0 List-I (I) If qr=10\dfrac qr=10, then the system of linear equations has (II) If pr≠100\dfrac pr\ne100, then the system of linear equations has (III) If pq≠10\dfrac pq\ne10, then the system of linear equations has (IV) If pq=10\dfrac pq=10, then the system of linear equations has List-II (P) x=0, y=109, z=−19x=0,\ y=\dfrac{10}9,\ z=-\dfrac19 as a solution (Q) x=109, y=−19, z=0x=\dfrac{10}9,\ y=-\dfrac19,\ z=0 as a solution (R) infinitely many solutions (S) no solution (T) at least one solution The correct option is:
Not attempted
A
(I)→(T); (II)→(R); (III)→(S); (IV)→(T)(I)\to(T);\ (II)\to(R);\ (III)\to(S);\ (IV)\to(T)
B
(I)→(Q); (II)→(S); (III)→(S); (IV)→(R)(I)\to(Q);\ (II)\to(S);\ (III)\to(S);\ (IV)\to(R)
C
(I)→(Q); (II)→(R); (III)→(P); (IV)→(R)(I)\to(Q);\ (II)\to(R);\ (III)\to(P);\ (IV)\to(R)
D
(I)→(T); (II)→(S); (III)→(P); (IV)→(T)(I)\to(T);\ (II)\to(S);\ (III)\to(P);\ (IV)\to(T)
15
Let ∣M∣|M| denote the determinant of a square matrix MM. Let g:[0,π2]→Rg:\left[0,\dfrac{\pi}{2}\right]\to\mathbb{R} be the function defined by g(θ)=f(θ)−1+f(π2−θ)−1g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\dfrac{\pi}{2}-\theta\right)-1} where f(θ)=12∣1sin⁡θ1−sin⁡θ1sin⁡θ−1−sin⁡θ1∣+∣sin⁡πcos⁡(θ+π4)tan⁡(θ−π4)sin⁡(θ−π4)−cos⁡π2log⁡e(4π)cot⁡(θ+π4)log⁡e(π4)tan⁡π∣f(\theta)=\dfrac12\begin{vmatrix}1 & \sin\theta & 1\\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1\end{vmatrix} + \begin{vmatrix}\sin\pi & \cos\left(\theta+\dfrac{\pi}{4}\right) & \tan\left(\theta-\dfrac{\pi}{4}\right)\\ \sin\left(\theta-\dfrac{\pi}{4}\right) & -\cos\dfrac{\pi}{2} & \log_e\left(\dfrac{4}{\pi}\right)\\ \cot\left(\theta+\dfrac{\pi}{4}\right) & \log_e\left(\dfrac{\pi}{4}\right) & \tan\pi\end{vmatrix}. Let p(x)p(x) be a quadratic polynomial whose roots are the maximum and minimum values of the function g(θ)g(\theta), and p(2)=2−2p(2)=2-\sqrt2. Then, which of the following is/are TRUE?
Not attempted
A
p(3+24)<0p\left(\dfrac{3+\sqrt2}{4}\right)<0
B
p(1+324)>0p\left(\dfrac{1+3\sqrt2}{4}\right)>0
C
p(52−14)>0p\left(\dfrac{5\sqrt2-1}{4}\right)>0
D
p(5−24)<0p\left(\dfrac{5-\sqrt2}{4}\right)<0
16
For any 3×33\times3 matrix MM, let ∣M∣|M| denote the determinant of MM. Let II be the 3×33\times3 identity matrix. Let EE and FF be two 3×33\times3 matrices such that (I−EF)(I-EF) is invertible. If G=(I−EF)−1G=(I-EF)^{-1}, then which of the following statements is (are) TRUE?
Not attempted
A
∣FE∣=∣I−FE∣∣FGE∣|FE|=|I-FE||FGE|
B
(I−FE)(I+FGE)=I(I-FE)(I+FGE)=I
C
EFG=GEFEFG=GEF
D
(I−FE)(I−FGE)=I(I-FE)(I-FGE)=I
17
For any 3×33\times3 matrix MM, let ∣M∣|M| denote the determinant of MM. Let E=(12323481318)E=\begin{pmatrix}1&2&3\\2&3&4\\8&13&18\end{pmatrix}, P=(100001010)P=\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix} and F=(13281813243)F=\begin{pmatrix}1&3&2\\8&18&13\\2&4&3\end{pmatrix} If QQ is a nonsingular matrix of order 3×33\times3, then which of the following statements is (are) TRUE?
Not attempted
A
F=PEPF=PEP and P2=(100010001)P^2=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}
B
∣EQ+PFQ−1∣=∣EQ∣+∣PFQ−1∣|EQ+PFQ^{-1}|=|EQ|+|PFQ^{-1}|
C
∣(EF)3∣>∣EF∣2|(EF)^3|>|EF|^2
D
Sum of the diagonal entries of P−1EP+FP^{-1}EP+F is equal to the sum of diagonal entries of E+P−1FPE+P^{-1}FP
18
Let α,β\alpha, \beta and γ\gamma be real numbers such that the system of linear equations x+2y+3z=αx+2y+3z=\alpha 4x+5y+6z=β4x+5y+6z=\beta 7x+8y+9z=γ−17x+8y+9z=\gamma-1 is consistent. Let ∣M∣|M| represent the determinant of the matrix M=(α2γβ10−101)M=\begin{pmatrix}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1\end{pmatrix} Let PP be the plane containing all those (α,β,γ)(\alpha,\beta,\gamma) for which the above system of linear equations is consistent, and DD be the square of the distance of the point (0,1,0)(0,1,0) from the plane PP. The value of ∣M∣|M| is ____.
Not attempted
Your answer: —
Correct answer: 1
19
The trace of a square matrix is defined to be the sum of its diagonal entries. If AA is a 2×22\times2 matrix such that the trace of AA is 33 and the trace of A3A^3 is −18-18, then the value of the determinant of AA is ____
Not attempted
Your answer: —
Correct answer: 5
20
Let MM be a 3×33 \times 3 invertible matrix with real entries and let II denote the 3×33 \times 3 identity matrix. If M−1=adj(adj M)M^{-1} = \text{adj}(\text{adj}\, M), then which of the following statements is/are ALWAYS TRUE?
Not attempted
A
M=IM = I
B
det⁡M=1\det M = 1
C
M2=IM^2 = I
D
(adj M)2=I(\text{adj}\, M)^2 = I
21
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
22
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
23
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}
24
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
25
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
26
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
27
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
28
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
29
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}
30
Let a,λ,μ∈Ra,\lambda,\mu\in\mathbb{R}. Consider the system of linear equations ax+2y=λax+2y=\lambda, 3x−2y=μ3x-2y=\mu. Which of the following statement(s) is(are) correct?
Not attempted
A
If a=−3a=-3, then the system has infinitely many solutions for all values of λ\lambda and μ\mu
B
If a≠−3a\ne-3, then the system has a unique solution for all values of λ\lambda and μ\mu
C
If λ+μ=0\lambda+\mu=0, then the system has infinitely many solutions for a=−3a=-3
D
If λ+μ≠0\lambda+\mu\ne0, then the system has no solution for a=−3a=-3
31
Let P=(1004101641)P=\begin{pmatrix}1&0&0\\4&1&0\\16&4&1\end{pmatrix} and II be the identity matrix of order 33. If Q=[qij]Q=[q_{ij}] is a matrix such that P50−Q=IP^{50}-Q=I, then q31+q32q21\dfrac{q_{31}+q_{32}}{q_{21}} equals
Not attempted
A
5252
B
103103
C
201201
D
205205
32
The total number of distinct x∈Rx\in\mathbb{R} for which ∣xx21+x32x4x21+8x33x9x21+27x3∣=10\begin{vmatrix}x&x^2&1+x^3\\2x&4x^2&1+8x^3\\3x&9x^2&1+27x^3\end{vmatrix}=10 is
Not attempted
Your answer: —
Correct answer: 2
33
Let P=(3−1−220α3−50)P=\begin{pmatrix}3&-1&-2\\2&0&\alpha\\3&-5&0\end{pmatrix}, where α∈R\alpha\in\mathbb{R}. Suppose Q=[qij]Q=[q_{ij}] is a matrix such that PQ=kIPQ=kI, where k∈Rk\in\mathbb{R}, k≠0k\ne0 and II is the identity matrix of order 3. If q23=−k8q_{23}=-\dfrac{k}{8} and det⁡(Q)=k22\det(Q)=\dfrac{k^2}{2}, then
Not attempted
A
α=0, k=8\alpha=0,\ k=8
B
4α−k+8=04\alpha-k+8=0
C
det⁡(Padj⁡Q)=29\det(P\operatorname{adj}Q)=2^9
D
det⁡(Qadj⁡P)=213\det(Q\operatorname{adj}P)=2^{13}
34
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}
35
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
36
Let MM and NN be two 3×33\times 3 matrices such that MN=NMMN = NM. Further, if M≠N2M \neq N^2 and M2=N4M^2 = N^4, then which of the following statement(s) is(are) correct?
Not attempted
A
Determinant of (M2+MN2)(M^2 + MN^2) is 00.
B
There is a 3×33\times 3 non-zero matrix UU such that (M2+MN2)U(M^2+MN^2)U is the zero matrix.
C
Determinant of (M2+MN2)≥1(M^2+MN^2) \geq 1.
D
For a 3×33\times 3 matrix UU, if (M2+MN2)U(M^2+MN^2)U equals the zero matrix then UU is the zero matrix.
37
Let MM be a 2×22\times 2 symmetric matrix with integer entries. Then MM is invertible if:
Not attempted
A
the first column of MM is the transpose of the second row of MM
B
the second row of MM is the transpose of the first column of MM
C
MM is a diagonal matrix with nonzero entries in the main diagonal
D
the product of entries in the main diagonal of MM is not the square of an integer
38
Let ω\omega be a complex cube root of unity with ω≠1\omega\neq1 and P=[pij]P=[p_{ij}] be an n×nn\times n matrix with pij=ωi+jp_{ij}=\omega^{i+j}. Then P2≠0P^2\neq0, when n=n=
Not attempted
A
5757
B
5555
C
5858
D
5656
39
For 3×33\times3 matrices MM and NN, which of the following statement(s) is (are) NOT correct?
Not attempted
A
NTMNN^TMN is symmetric or skew symmetric, according as MM is symmetric or skew symmetric
B
MN−NMMN-NM is skew symmetric for all symmetric matrices MM and NN
C
MNMN is symmetric for all symmetric matrices MM and NN
D
(adj M)(adj N)=adj(MN)(\text{adj}\,M)(\text{adj}\,N)=\text{adj}(MN) for all invertible matrices MM and NN
Check each statement using the transpose properties (AB)T=BTAT(AB)^T=B^TA^T; a statement is only true if it holds for ALL symmetric/invertible M,NM,N, not just special cases where they happen to commute.
Deconstruction\textbf{Deconstruction}: Four claims about matrix symmetry/skew-symmetry and adjugates for general 3×33\times3 matrices; identify which are NOT universally true. Strategy\textbf{Strategy}: Use (ABC)T=CTBTAT(ABC)^T=C^TB^TA^T and known adjugate identities, testing each claim for a fully general (not necessarily commuting) pair M,NM,N. Execution\textbf{Execution}: (A) (NTMN)T=NTMTN(N^TMN)^T=N^TM^TN. If MT=MM^T=M, this equals NTMNN^TMN (symmetric); if MT=−MM^T=-M, this equals −NTMN-N^TMN (skew-symmetric). So (A) is correct\textbf{correct} (always true). - (B) For symmetric M,NM,N: (MN−NM)T=NTMT−MTNT=NM−MN=−(MN−NM)(MN-NM)^T=N^TM^T-M^TN^T=NM-MN=-(MN-NM), so MN−NMMN-NM is skew-symmetric always. (B) is correct\textbf{correct}. - (C) For symmetric M,NM,N: (MN)T=NTMT=NM(MN)^T=N^TM^T=NM. This equals MNMN only if MN=NMMN=NM, which is not guaranteed for general symmetric matrices (they need not commute). So (C) is NOT correct\textbf{NOT correct} in general. - (D) The genuine identity is adj(MN)=adj(N) adj(M)\text{adj}(MN)=\text{adj}(N)\,\text{adj}(M) (reversed order), not (adj M)(adj N)=adj(MN)(\text{adj}\,M)(\text{adj}\,N)=\text{adj}(MN) as stated — these agree only when M,NM,N commute, not for all invertible M,NM,N. So (D) is NOT correct\textbf{NOT correct} in general. Verification\textbf{Verification}: A concrete counterexample for (C): take M=(100000000)M=\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}, N=(010100000)N=\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix} (both symmetric); MN=(010000000)MN=\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix} is visibly not symmetric, confirming (C) fails in general, and the reversed-order adjugate identity is a standard textbook result confirming (D). Final Answer\textbf{Final Answer}: (C) and (D)