Chapter 19

Let \(\theta=\frac{\pi}{5}\) and \(\mathrm{A}=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right] .\) If \(\mathrm{B}=\mathrm{A}+\mathrm{A}^{4}\), then det (B): \(\quad\) [Sep.06, 2020 (II)] (a) is one (b) lies in \((2,3)\) (c) is zero (d) lies in \((1,2)\)

If \(\Delta=\left|\begin{array}{ccc}x-2 & 2 x-3 & 3 x-4 \\ 2 x-3 & 3 x-4 & 4 x-5 \\ 3 x-5 & 5 x-8 & 10 x-17\end{array}\right|=A x^{3}+B x^{2}+C x+D\) then \(B+C\) is equal to: [Sep. 03, 2020 (I)] (a) \(-1\) (b) 1 (c) \(-3\) (d) 9

Let \(a-2 b+c=1\). If \(f(x)=\left|\begin{array}{ccc}x+a & x+2 & x+1 \\ x+b & x+3 & x+2 \\ x+c & x+4 & x+3\end{array}\right|\), then: [Jan. \(\left.9, \mathbf{2 0 2 0}(\mathbf{I I})\right]\) (a) \(f(-50)=501\) (b) \(f(-50)=-1\) (c) \(f(50)=-501\) (d) \(f(50)=1\)

If \(\Delta_{1}=\left|\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right|\) and \(\Delta_{2}=\left|\begin{array}{ccc}x & \sin 2 \theta & \cos 2 \theta \\\ -\sin 2 \theta & -x & 1 \\ \cos 2 \theta & 1 & x\end{array}\right|, x \neq 0\) then for all \(\theta \in\left(0, \frac{\pi}{2}\right)\) : (a) \(\Delta_{1}-\Delta_{2}=-2 x^{3}\) (b) \(\Delta_{1}-\Delta_{2}=x(\cos 2 \theta-\cos 4 \theta)\) (c) \(\Delta_{1} \times \Delta_{2}=-2\left(x^{3}+x-1\right)\) (d) \(\Delta_{1}+\Delta_{2}=-2 x^{3}\)

The sum of the real roots of the equation \(\left|\begin{array}{ccc}x & -6 & -1 \\ 2 & -3 x & x-3 \\ -3 & 2 x & x+2\end{array}\right|=0\), is equal to: \(\quad\) [April 10, 2019 (II)] (a) 6 (b) 0 (c) 1 (d) \(-4\)

Let \(\mathrm{A}=\left[\begin{array}{ccc}2 & \mathrm{~b} & 1 \\ \mathrm{~b} & \mathrm{~b}^{2}+1 & \mathrm{~b} \\ 1 & \mathrm{~b} & 2\end{array}\right]\) where \(\mathrm{b}>0\). Then the minimum value of \(\frac{\operatorname{det}(\mathrm{A})}{\mathrm{b}}\) is: \(\quad\) [Jan. 10,2019 (II)] (a) \(2 \sqrt{3}\) (n) \(-2 \sqrt{3}\) (c) \(-\sqrt{3}\) (d) \(\sqrt{3}\)

If \(\left|\begin{array}{lll}x-4 & 2 x & 2 x \\ 2 x & x-4 & 2 x \\ 2 x & 2 x & x-4\end{array}\right|=(A+B x)(x-A)^{2}\), then the ordered pair \((\mathrm{A}, \mathrm{B})\) is equal to: (a) \((-4,3)\) (b) \((-4,5)\) (c) \((4,5)\) (d) \((-4,-5)\)

If \(\mathrm{S}=\left\\{\mathrm{x} \in[0,2 \pi]:\left|\begin{array}{ccc}0 & \cos x & -\sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0\end{array}\right|=0\right\\}\), then \(\sum_{\mathrm{x} \in \mathrm{S}} \tan \left(\frac{\pi}{3}+\mathrm{x}\right)\) is equal to \(\quad\) [OnlineApril 8, 2017] (a) \(4+2 \sqrt{3}\) (b) \(-2+\sqrt{3}\) (c) \(-2-\sqrt{3}\) (d) \(-4-2 \sqrt{3}\)

if \(\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{array}\right|=a x-12\), then ' \(a\) ' is equal to: [Online April 11, 2015] (a) 24 (b) \(-12\) (c) \(-24\) (d) 12

The least value of the product \(x y z\) for which the determinant \(\left|\begin{array}{ccc}x & 1 & 1 \\ 1 & y & 1 \\ 1 & 1 & z\end{array}\right|\) is non-negative, is: [Online April 10, 2015] (a) \(-2 \sqrt{2}\) (b) \(-1\) (c) \(-16 \sqrt{2}\) (d) \(-8\)

If \(\mathrm{f}(\theta)=\left|\begin{array}{ccc}1 & \cos \theta & 1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1\end{array}\right|\) and \(\mathrm{A}\) and \(\mathrm{B}\) are respectively the maximum and the minimum values of \(f(\theta)\), then \((A, B)\) is equal to: \mathrm{\\{} O n l i n e ~ A p r i l ~ 1 2 , ~ 2 0 1 4 ] ~ (a) \((3,-1)\) (b) \((4,2-\sqrt{2})\) (c) \((2+\sqrt{2}, 2-\sqrt{2})\) (d) \((2+\sqrt{2},-1)\)

If \(\mathrm{B}\) is a \(3 \times 3\) matrix such that \(\mathrm{B}^{2}=0\), then det. \(\left[(\mathrm{I}+\mathrm{B})^{50}-50 \mathrm{~B}\right]\) is equal to: \(\quad\) Online April \(\left.\mathbf{9}, \mathbf{2 0 1 4}\right]\) (a) 1 (b) 2 (c) 3 (d) 50

Let \(S=\left\\{\left(\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right): a_{i j} \in\\{0,1,2\\}, a_{11}=a_{22}\right\\}\) Then the number of non-singular matrices in the set \(\mathrm{S}\) is : [Online April 25, 2013] (a) 27 (b) 24 (c) 10 (d) 20

Let \(\mathrm{A}\), other than \(\mathrm{I}\) or \(-\mathrm{I}\), be a \(2 \times 2\) real matrix such that \(\mathrm{A}^{2}=\mathrm{I}, \mathrm{I}\) being the unit matrix. Let \(\operatorname{Tr}(\mathrm{A})\) be the sum of diagonal elements of A. \(\quad\) [Online April 23, 2013] Statement-1: \(\operatorname{Tr}(\mathrm{A})=0\) Statement-2: \(\operatorname{det}(\mathrm{A})=-1\) (a) Statement- 1 is true; Statement- 2 is false. (b) Statement- 1 is true; Statement- 2 is true; Statement-2 is not a correct explanation for Statement-1. (c) Statement- 1 is true; Statement- 2 is true; Statement-2 is a correct explanation for Statement- 1 . (d) Statement- 1 is false; Statement- 2 is true.

Statement \(-1:\) Determinant of a skew-symmetric matrix of order 3 is zero. Statement - 2 : For any matrix \(\mathrm{A}, \operatorname{det}(\mathrm{A})^{\mathrm{T}}=\operatorname{det}(\mathrm{A})\) and \(\operatorname{det}(-\mathrm{A})=-\operatorname{det}(\mathrm{A})\). Where det (B) denotes the determinant of matrix B. Then: (a) Both statements are true (b) Both statements are false (c) Statement- 1 is false and statement- 2 is true (d) Statement- 1 is true and statement- 2 is false

Let \(A\) be a \(2 \times 2\) matrix with non-zero entries and let \(A^{2}=I\), where \(I\) is \(2 \times 2\) identity matrix. Define \(\operatorname{Tr}(A)=\) sum of diagonal elements of \(A\) and \(|A|=\) determinant of matrix \(A\). Statement - \(\mathbf{1 :} \operatorname{Tr}(A)=0\). Statement \(-2:|A|=1\) [2010] (a) Statement- 1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statement \(-1\). (b) Statement \(-1\) is true, Statement \(-2\) is false. (c) Statement \(-1\) is false, Statement \(-2\) is true. (d) Statement - 1 is true, Statement 2 is true; Statement \(-2\) is a correct explanation for Statement \(-1\).

Let \(A\) be \(a 2 \times 2\) matrix with real entries. Let \(I\) be the \(2 \times 2\) identity matrix. Denote by \(\operatorname{tr}(A)\), the sum of diagonal entries of \(a\). Assume that \(A^{2}=I\). \(\quad\) [2008] Statement- 1 : If \(A \neq I\) and \(A \neq-I\), then \(\operatorname{det}(A)=-1\) Statement- \(2:\) If \(A \neq I\) and \(A \neq-I\), then \(\operatorname{tr}(A) \neq 0\) (a) Statement \(-1\) is false, Statement- 2 is true (b) Statement \(-1\) is true, Statement- 2 is true; Statement \(-2\) is a correct explanation for Statement- 1 (c) Statement \(-1\) is true, Statement- 2 is true; Statement \(-2\) is not a correct explanation for Statement- 1 (d) Statement \(-1\) is true, Statement- 2 is false

Let \(A=\left|\begin{array}{ccc}5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\\ 0 & 0 & 5\end{array}\right|\). If \(\left|A^{2}\right|=25\), then \(|\alpha|\) equals [2007] (a) \(1 / 5\) (b) 5 (c) \(5^{2}\) (d) 1

If \(1, \omega, \omega^{2}\) are the cube roots of unity, then \(\Delta=\left|\begin{array}{ccc}1 & \omega^{n} & \omega^{2 n} \\ \omega^{n} & \omega^{2 n} & 1 \\ \omega^{2 n} & 1 & \omega^{n}\end{array}\right|\) is equal to (a) \(\omega^{2}\) (b) 0 (c) 1 (d) \(\omega\)

If the minimum and the maximum values of the function \(f:\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}\), defined by \(f(\theta)=\left|\begin{array}{ccc}-\sin ^{2} \theta & -1-\sin ^{2} \theta & 1 \\\ -\cos ^{2} \theta & -1-\cos ^{2} \theta & 1 \\ 12 & 10 & -2\end{array}\right|\) are \(m\) and \(M\) respec- tively, then the ordered pair ( \(m, M)\) is equal to: 28 [Sep. \(\mathbf{0 5}, \mathbf{2 0 2 0}(\mathbf{I})]\) (a) \((0,2 \sqrt{2})\) (b) \((-4,0)\) (c) \((-4,4)\) (d) \((0,4)\)

If \(a+x=b+y=c+z+1\), where \(a, b, c, x, y, z\) are non-zero distinct real numbers, then \(\left|\begin{array}{lll}x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c\end{array}\right|\) is equal to: [Sep. 05, 2020 (II)] (a) \(y(b-a)\) (b) \(y(a-b)\) (c) 0 (d) \(y(a-c)\)

Let two points be \(A(1,-1)\) and \(B(0,2)\). If \(a\) point \(P\left(x^{\prime}, y^{\prime}\right)\) be such that the area of \(\Delta P A B=5 \mathrm{sq}\). units and it lies on the line, \(3 x+y-4 \lambda=0\), then a value of \(\lambda\) is: [Jan. 8, \(\mathbf{2 0 2 0}\) (I)] (a) 4 (b) 3 (c) 1 (d) \(-3\)

Let \(A=\left[a_{i j}\right]\) and \(B=\left[b_{y}\right]\) be two \(3 \times 3\) real matrices such that \(b_{i j}=(3)^{i+j-2)} a_{i j}\), where \(i, j=1,2,3\). If the determinant of \(B\) is 81 , then the determinant of \(A\) is: \(\quad\) [Jan. \(7, \mathbf{2 0 2 0}\) (II)] (a) \(1 / 3\) (b) 3 (c) \(1 / 81\) (d) \(1 / 9\)

A value of \(\theta \in(0, \pi / 3)\), for which \(\left|\begin{array}{ccc}1+\cos ^{2} \theta & \sin ^{2} \theta & 4 \cos 6 \theta \\ \cos ^{2} \theta & 1+\sin ^{2} \theta & 4 \cos 6 \theta \\ \cos ^{2} \theta & \sin ^{2} \theta & 1+4 \cos 6 \theta\end{array}\right|=0\), is [April 12, 2019 (II)] (a) \(\frac{\pi}{9}\) (b) \(\frac{\pi}{18}\) (c) \(\frac{7 \pi}{24}\) (d) \(\frac{7 \pi}{36}\)

Let \(\alpha\) and \(\beta\) be the roots of the equation \(x^{2}+x+1=0\). Then for \(y^{\prime \prime \prime} 0\) in \(R,\left|\begin{array}{ccc}y+1 & \alpha & \beta \\ \alpha & y+\beta & 1 \\ \beta & 1 & y+\alpha\end{array}\right|\) is equal to: [April09, 2019 (I)] (a) \(y\left(y^{2}-1\right)\) (b) \(y\left(y^{2}-3\right)\) (c) \(\mathrm{y}^{3}\) (d) \(\mathrm{y}^{3}-1\)

Let the numbers \(2, b, c\) be in an A.P. and \(\mathrm{A}=\left[\begin{array}{ccc}1 & 1 & 1 \\ 2 & b & c \\ 4 & b^{2} & c^{2}\end{array}\right] .\) If \(\operatorname{det}(\mathrm{A}) ?[2,16]\), then \(c\) lies in the interval : \(\quad\) [April 08, 2019 (II)] (a) \([2,3)\) (b) \(\left(2+2^{34}, 4\right)\) (c) \([4,6]\) (d) \(\left[3,2+2^{34}\right]\)

If \(A=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right]\); then for all \(\theta \in\left(\frac{3 \pi}{4}, \frac{5 \pi}{4}\right)\) det \((\) A) lies in the interval : [Jan. 12, 2019 (II)] (a) \(\left(1, \frac{5}{2}\right]\) (b) \(\left[\frac{5}{2}, 4\right)\) (c) \(\left(0, \frac{3}{2}\right]\) (d) \(\left(\frac{3}{2}, 3\right]\)

If \(\left|\begin{array}{ccc}a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b\end{array}\right|\) \(=(a+b+c)(x+a+b+c)^{2}, x \neq 0\) and \(a+b+c \neq 0\), then \(x\) is equal to: [Jan. 11, 2019 (II)] (a) \(a b c\) (b) \(-(a+b+c)\) (c) \(2(a+b+c)\) (d) \(-2(a+b+c)\)

Let \(\mathrm{d} \in \mathbf{R}\), and \(A=\left[\begin{array}{ccc}-2 & 4+\mathrm{d} & (\sin \theta)^{-2} \\ 1 & (\sin \theta)+2 & \mathrm{~d} \\ 5 & (2 \sin \theta)-\mathrm{d} & (-\sin \theta)+2+2 \mathrm{~d}\end{array}\right]\) \(\theta \in[0,2 \pi]\). If the minimum value of det (A) is 8 , then a value of d is: \(\quad[\) Jan 10,2019 (I)] (a) \(-5\) (b) \(-7\) (c) \(2(\sqrt{2}+1)\) (d) \(2(\sqrt{2}+2)\)

Let \(a_{1}, a_{2}, a_{3}, \ldots, a_{10}\) be in G.P. with \(a_{i}>0\) for \(i=1,2, \ldots, 10\) and \(S\) be the set of pairs \((r, k), r, k \in N\) (the set of natural numbers) for which $$ \left|\begin{array}{lll} \log _{e} a_{1}^{r} a_{2}^{k} & \log _{e} a_{2}^{r} a_{3}^{k} & \log _{e} a_{3}^{r} & a_{4}^{k} \\ \log _{e} a_{4}^{r} a_{5}^{k} & \log _{e} a_{5}^{r} a_{6}^{k} & \log _{e} a_{6}^{r} & a_{7}^{k} \\ \log _{e} a_{7}^{r} a_{8}^{k} & \log _{e} a_{8}^{r} a_{9}^{k} & \log _{e} a_{9}^{r} a_{10}^{k} \end{array}\right|=0 $$ Then the number of elements in \(\mathrm{S}\), is : [Jan. \(\mathbf{1 0}, \mathbf{2 0 1 9}\) (II)] (a) 4 (b) infinitely many (c) 2 (d) 10

If \(A=\left[\begin{array}{ccc}e^{t} & e^{-t} \cos t & e^{-t} \sin t \\ e^{t} & -e^{-t} \cos t-e^{-t} \sin t & -e^{-t} \sin t+e^{-t} \cos t \\ e^{t} & 2 e^{-t} \sin t & -2 e^{-t} \cos t\end{array}\right]\) then A is: \(\quad\) [Jan. 09, 2019 (II)] (a) invertible for all \(\mathrm{t} \in \mathbf{R}\). (b) invertible only if \(\mathrm{t}=\pi\). (c) not invertible for any \(t \in \mathbf{R}\). (d) invertible only if \(t=\frac{\pi}{2}\).

Let \(\mathrm{k}\) be an integer such that triangle with vertices \((\mathrm{k},-3 \mathrm{k}),(5, \mathrm{k})\) and \((-\mathrm{k}, 2)\) has area \(28 \mathrm{sq}\). units. Then the orthocentre of this triangle is at the point: \(\quad\) [2017] (a) \(\left(2, \frac{1}{2}\right)\) (b) \(\left(2,-\frac{1}{2}\right)\) (c) \(\left(1, \frac{3}{4}\right)\) (d) \(\left(1,-\frac{3}{4}\right)\)

The number of distinct real roots of the equaiton, \(\left|\begin{array}{lll}\cos x & \sin x & \sin x \\ \sin x & \cos x & \sin x \\\ \sin x & \sin x & \cos x\end{array}\right|=0\) in the interval \(\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]\) is : [OnlineApril 9, 2016] (a) 1 (b) 4 (c) 2 (d) 3

If \(\alpha, \beta \neq 0\), and \(f(n)=\alpha^{n}+\beta^{n}\) and \(\left|\begin{array}{ccc}3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\\ 1+f(2) & 1+f(3) & 1+f(4)\end{array}\right|=K(1-\alpha)^{2}(1-\beta)^{2}(\alpha-\beta)^{2}\), then \(K\) is equal to: [2014] (a) 1 (b) \(-1\) (c) \(\alpha \beta\) (d) \(\frac{1}{\alpha \beta}\)

If \(\Delta_{\mathrm{r}}=\left|\begin{array}{ccc}\mathrm{r} & 2 \mathrm{r}-1 & 3 \mathrm{r}-2 \\ \frac{\mathrm{n}}{2} & \mathrm{n}-1 & \mathrm{a} \\\ \frac{1}{2} \mathrm{n}(\mathrm{n}-1) & (\mathrm{n}-1)^{2} & \frac{1}{2}(\mathrm{n}-1)(3 \mathrm{n}-4)\end{array}\right|\) then the value of \(\sum_{\mathrm{r}=1}^{\mathrm{n}-1} \Delta_{\mathrm{r}}\) (a) depends only on a (b) depends only on \(\mathrm{n}\) (c) depends both on a and \(\mathrm{n}\) (d) is independent of both a and \(\mathrm{n}\)

If \(\left|\begin{array}{ccc}a^{2} & b^{2} & c^{2} \\ (a+\lambda)^{2} & (b+\lambda)^{2} & (c+\lambda)^{2} \\ (a-\lambda)^{2} & (b-\lambda)^{2} & (c-\lambda)^{2}\end{array}\right|=k \lambda\left|\begin{array}{ccc}a^{2} & b^{2} & c^{2} \\ a & b & c \\ 1 & 1 & 1\end{array}\right|, \lambda \neq 0\) then \(\mathrm{k}\) is equal to: \(\quad\) [Online April 12, 2014] (a) \(4 \lambda\) abc (b) \(-4 \lambda a b c\) (c) \(4 \lambda^{2}\) (d) \(-4 \lambda^{2}\)

If \(a, b, c\) are sides of a scalene triangle, then the value of \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is : [OnlineApril 9, 2013] (a) non-negative (b) negative (c) positive (d) non-positive

If \(a, b, c\), are non zero complex numbers satisfying \(a^{2}+b^{2}+c^{2}=0\) and \(\left|\begin{array}{ccc}b^{2}+c^{2} & a b & a c \\ a b & c^{2}+a^{2} & b c \\\ a c & b c & a^{2}+b^{2}\end{array}\right|=k a^{2} b^{2} c^{2}\), then \(k\) is equal to [Online May 19, 2012] (a) 1 (b) 3 (c) 4 (d) 2

If \(\left|\begin{array}{ccc}-2 a & a+b & a+c \\ b+a & -2 b & b+c \\ c+a & b+c & -2 c\end{array}\right|=\alpha(a+b)(b+c)(c+a) \neq 0\) then \(\alpha\) is equal to \(\quad\) Online May 12, 2012] (a) \(a+b+c\) (b) \(a b c\) (c) 4 (d) 1

The area of the triangle whose vertices are complex numbers \(z, i z, z+i z\) in the Argand diagram is \([\) Online May 12, 2012] (a) \(2|z|^{2}\) (b) \(1 / 2|z|^{2}\) (c) \(4|z|^{2}\) (d) \(|z|^{2}\)

The area of triangle formed by the lines joining the vertex of the parabola, \(x^{2}=8 y\), to the extremities of its latus rectum is \mathrm{\\{} D n l i n e ~ M a y ~ 1 2 , ~ 2 0 1 2 ] ~ (a) 2 (b) 8 (c) 1 (d) 4

Let \(a, b, c\) be such that \(b(a+c) \neq 0\) if [2009] \(\left|\begin{array}{ccc}a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1\end{array}\right|+\left|\begin{array}{ccc}a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2} a & (-1)^{n+1} b & (-1)^{n} c\end{array}\right|=0\) then the value of \(n\) is : (a) any even integer (b) any odd integer (c) any integer (d) zero

If \(D=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right|\) for \(x \neq 0, y \neq 0\), then D is (a) divisible by \(x\) but not \(y\) (b) divisible by \(y\) but \(\operatorname{not} x\) (c) divisible by neither \(x\) nor \(y\) (d) divisible by both \(x\) and \(y\)

If \(a^{2}+b^{2}+c^{2}=-2\) and \(f(x)=\left|\begin{array}{ccc}1+a^{2} x & \left(1+b^{2}\right) x & \left(1+c^{2}\right) x \\ \left(1+a^{2}\right) x & 1+b^{2} x & \left(1+c^{2}\right) x \\ \left(1+a^{2}\right) x & \left(1+b^{2}\right) x & 1+c^{2} x\end{array}\right|\) then \(\mathrm{f}(\mathrm{x})\) is a polynomial of degree (a) 1 (b) 0 (c) 3 (d) 2

If \(a>0\) and discriminant of \(a x^{2}+2 b x+c\) is \(-\) ve, then \(\left|\begin{array}{ccc}a & b & a x+b \\ b & c & b x+c \\ a x+b & b x+c & 0\end{array}\right|\) is equal to (a) \(+\mathrm{ve}\) (b) \(\left(a c-b^{2}\right)\left(a x^{2}+2 b x+c\right)\) (c) \(-\mathrm{ve}\) (d) 0

\(l, m, n\) are the \(p^{t h}, q^{t h}\) and \(r^{t h}\) term of a G. P. all positive, then \(\left|\begin{array}{lll}\log l & p & 1 \\ \log m & q & 1 \\ \log n & r & 1\end{array}\right|\) equals \(\quad\) [2002] (a) \(-1\) (b) 2 (c) 1 (d) 0

Let \(A\) be a \(3 \times 3\) matrix such that adj \(A=\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1\end{array}\right]\) and \(B=\operatorname{adj}(\operatorname{adj} A)\). If \(|A|=\lambda\) and \(\left|\left(B^{-1}\right)^{T}\right|=\mu\), then the ordered pair, \((|\lambda|, \mu)\) is equal to: \(\quad\) [Sep. 03, 2020 (II)] (a) \(\left(3, \frac{1}{81}\right)\) (b) \(\left(9, \frac{1}{9}\right)\) (c) \((3,81)\) (d) \(\left(9, \frac{1}{81}\right)\)

If the matrices \(\mathrm{A}=\left[\begin{array}{ccc}1 & 1 & 2 \\ 1 & 3 & 4 \\\ 1 & -1 & 3\end{array}\right], \mathrm{B}=\operatorname{adj} \mathrm{A}\) and \(\mathrm{C}=3 \mathrm{~A}\), then \(\frac{|\operatorname{adj} \mathrm{B}|}{|\mathrm{C}|}\) is equal to: \([\) Jan. \(9, \mathbf{2 0 2 0}(\mathrm{I})]\) (a) 8 (b) 16 (c) 72 (d) 2

If \(\mathrm{B}=\left[\begin{array}{ccc}5 & 2 \alpha & 1 \\ 0 & 2 & 1 \\\ \alpha & 3 & -1\end{array}\right]\) is the inverse of a \(3 \times 3\) matrix \(\mathrm{A}\), then the sum of all values of \(\alpha\) for which det \((\mathrm{A})+1=0\), is : [April 12, 2019 (I)] (a) 0 (b) \(-1\) (c) 1 (d) 2

If \(\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right] \cdot\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right] \cdot\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right] \ldots \ldots \ldots . .\left[\begin{array}{cc}1 & \mathrm{n}-1 \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}1 & 78 \\ 0 & 1\end{array}\right]\), then the inverse of \(\left[\begin{array}{ll}1 & n \\ 0 & 1\end{array}\right]\) is: \(\quad\) [April 09, \(\left.\mathbf{2 0 1 9}(\mathbf{I I})\right]\) (a) \(\left[\begin{array}{cc}1 & 0 \\ 12 & 1\end{array}\right]\) (b) \(\left[\begin{array}{cc}1 & -13 \\ 0 & 1\end{array}\right]\) (c) \(\left[\begin{array}{cc}1 & -12 \\ 0 & 1\end{array}\right]\) (d) \(\left[\begin{array}{cc}1 & 0 \\ 13 & 1\end{array}\right]\)