A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar

Chapter 6

Determinants - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $\alpha_{1}, \alpha_{2}$ and $\beta_{1}, \beta_{2}$ be the roots of $a x^{2}+b x+c=0$ and $p x^{2}+q x+r=0$ respectively. If the system of equations $\alpha_{1} y+\alpha_{2} z=0$ and $\beta_{1} y+\beta_{2} z=0$ has a non-trivial solution, then
(A) $\frac{b^{2}}{q^{2}}=\frac{a c}{p r}$
(B) $\frac{c^{2}}{r^{2}}=\frac{a b}{p q}$
(C) $\frac{a^{2}}{p^{2}}=\frac{b c}{q r}$
(D) None of these

Ahmad Reda

Ahmad Reda

Numerade Educator

Problem 2

$a, b, c$ are in G.P. with common ratio $r_{1}$ and $\alpha, \beta, \gamma$ are in G.P. with common ratio $r_{2}$. If the equations $a x+\infty y$. $+z=0, b x+\beta y+z=0, c x+\gamma y+z=0$ have only trivial solution, then
(A) $a, \alpha=0$
(B) $r_{1}, r_{2}=1$
(C) $r_{1}, r_{2} \neq 1$
(D) $r_{1}=r_{2}$

Ahmad Reda

Numerade Educator

Problem 3

If the value of a third order determinant is 11 , then the value of the determinant formed by its cofactors will be
(A) 11
(B) 121
(C) 1331
(D) 14641

Ahmad Reda

Numerade Educator

Problem 4

If $\frac{1}{a}, \frac{1}{b}$ and $\frac{1}{c}$ are respectively the $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ terms of an A.P., then the value of the determinant
$\left|\begin{array}{ccc}{ }^{8} C_{3} & { }^{9} C_{5} & { }^{10} C_{7} \\ { }^{8} C_{4} & { }^{9} C_{6} & { }^{10} C_{8} \\ { }^{9} C_{n} & { }^{10} C_{n+2} & { }^{11} C_{n+4}\end{array}\right|$ is
(A) $a b c$
(B) $p q r$
(C) 0
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 5

The value of the determinant
$\left|\begin{array}{ccc}\sqrt{x}+\sqrt{y} & 2 \sqrt{2} & \sqrt{z} \\ \sqrt{y z}+\sqrt{2 x} & z & \sqrt{2 z} \\ y+\sqrt{x z} & \sqrt{y z} & z\end{array}\right|$
where $x, y, z$ are positive real numbers, is
(A) $z(\sqrt{2} y-z \sqrt{y)}$
(B) $y(\sqrt{2} z-y \sqrt{z)}$
(C) $x(\sqrt{2} y-z \sqrt{y})$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 6

Let $D_{k}=\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ 2.3^{k} & 16.9^{k} & 26.27^{k} \\ \left(3^{10}-1\right) & 2\left(9^{10}-1\right) & \left(27^{10}-1\right)\end{array}\right|$ then the
value of $\sum_{k=1}^{10} D_{k}$ is
(A) $2(\alpha+\beta+\gamma)$
(B) $\alpha \beta+\alpha \gamma+\beta \gamma$
(C) $\alpha \beta \gamma$
(D) 0

Ahmad Reda

Numerade Educator

Problem 7

If $M$ is a $3 \times 3$ matrix, where $M^{\prime} M=I$ and $\operatorname{det}(M)=1$, then $\operatorname{det}(M-I)=$
(A) 1
(B) 0
(C) $-1$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 8

If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the determinant $\left|\begin{array}{ccc}{[e]} & {[\pi]} & {\left[\pi^{2}-6\right]} \\ {[\pi]} & {\left[\pi^{2}-6\right]} & {[e]} \\ {\left[\pi^{2}-6\right]} & {[e]} & {[\pi]}\end{array}\right|$, then
(A) $-8$
(B) 8
(C) 0
(D) None of these

Ahmad Reda

Numerade Educator

Problem 9

If $a_{i}, b_{i}, c_{i} \in R(i=1,2,3)$ and $x \in R$ and $\left|\begin{array}{lll}a_{1}+b_{1} x & a_{1} x+b_{1} & c_{1} \\ a_{2}+b_{2} x & a_{2} x+b_{2} & c_{2} \\ a_{3}+b_{3} x & a_{3} x+b_{3} & c_{3}\end{array}\right|=0$, then
(A) $\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|=4$
(B) $x=\pm 1$
(C) $x=2$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 10

The value of the determinant $\left|\begin{array}{ccc}\sin \theta & \cos \theta & \sin 2 \theta \\ \sin \left(\theta+\frac{2 \pi}{3}\right) & \cos \left(\theta+\frac{2 \pi}{3}\right) & \sin \left(2 \theta+\frac{4 \pi}{3}\right) \\ \sin \left(\theta-\frac{2 \pi}{3}\right) & \cos \left(\theta-\frac{2 \pi}{3}\right) & \sin \left(2 \theta-\frac{4 \pi}{3}\right)\end{array}\right|$
(A) 0
(B) $\sin \theta$
(C) $\cos \theta$
(D) independent of $\theta$

Ahmad Reda

Numerade Educator

Problem 11

If $D_{k}=\left|\begin{array}{ccc}1 & n & n \\ 2 k & n^{2}+n+2 & n^{2}+n \\ 2 k-1 & n^{2} & n^{2}+n+2\end{array}\right|$ and
$\sum_{k=1}^{n} D_{k}=48$, then $n$ equals
(A) 4
(B) 6
(C) 8
(D) None of these

Ahmad Reda

Numerade Educator

Problem 12

If $A, B, C$ are the angles of a triangle and $\left|\begin{array}{ccc}1 & 1 & 1 \\ 1+\sin A & 1+\sin B & 1+\sin C \\ \sin A+\sin ^{2} A & \sin B+\sin ^{2} B & \sin C+\sin ^{2} C\end{array}\right|=0$
then the triangle is a/an
(A) equilaterral
(B) isosceles
(C) right-angled triangle
(D) any triangle

Ahmad Reda

Numerade Educator

Problem 13

If $a_{0}, a_{1} a_{2}, a_{3}, a_{4}$ are in A.P with the common difference $d$, the value of $\left|\begin{array}{lll}a_{1} a_{2} & a_{1} & a_{0} \\ a_{2} a_{3} & a_{2} & a_{1} \\ a_{3} a_{4} & a_{3} & a_{2}\end{array}\right|$ is
(A) $2 d^{4}$
(B) $2 d^{3}$
(C) $2 d^{2}$
(D) $2 d$

Ahmad Reda

Numerade Educator

Problem 14

If $\alpha, \beta, \gamma$ are different from and are the roots of $a x^{3}+$ $b x^{2}+c x+d=0$ and
$(\beta-\gamma)(\gamma-\alpha)(\alpha-\beta)=\frac{25}{2}$, then the determinant
$\Delta=\left|\begin{array}{ccc}\frac{\alpha}{1-\alpha} & \frac{\beta}{1-\beta} & \frac{\gamma}{1-\gamma} \\ \alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2}\end{array}\right|$ equals
(A) $\frac{25 d}{2 a}$
(B) $\frac{25 d}{a}$
(C) $\frac{-25 d}{a+b+c+d}$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 15

Let $\left\{\Delta_{1}, \Delta_{2}, \Delta_{3}, \ldots, \Delta_{k}\right\}$ be the set of third order determinants that can be made with the distinct nonzero real numbers $a_{1}, a_{2}, a_{3}, \ldots, a_{9}$. Then
(A) $k=9 !$
(B) $\sum_{i=1}^{k} \Delta_{i}=0$
(C) at least one $\Delta_{i}=0$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 16

If $f(x)=\left|\begin{array}{ccc}(1+x)^{a} & (1+2 x)^{b} & 1 \\ 1 & (1+x)^{a} & (1+2 x)^{b} \\ (1+2 x)^{b} & 1 & (1+x)^{a}\end{array}\right|, a, b$ being positive integers, then
(A) constant term of $f(x)$ is 4
(B) coefficieent of $x$ in $f(x)$ is 0
(C) constant term in $f(x)$ is $a-b$
(D) constant term in $f(x)$ is $a+b$

Ahmad Reda

Numerade Educator

Problem 17

If $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=P A P^{\prime}$, then
$p^{\prime} Q^{2005} P$ is
(A) $\left[\begin{array}{cc}1 & 1 \\ 2005 & 1\end{array}\right]$
(B) $\left[\begin{array}{cc}1 & 2005 \\ 0 & 1\end{array}\right]$
(C) $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
(D) $\left[\begin{array}{cc}1 & 2005 \\ 2005 & 1\end{array}\right]$

Ahmad Reda

Numerade Educator

Problem 18

If $\left|\begin{array}{ccc}x^{n} & x^{n+2} & x^{n+3} \\ y^{n} & y^{n+2} & y^{n+3} \\ z^{n} & z^{n+2} & z^{n+3}\end{array}\right|$
$=(x-y)(y-z)(z-x),\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$ then $n=$
(A) $-2$
(B) $-1$
(C) 0
(D) 1

Ahmad Reda

Numerade Educator

Problem 19

If $\alpha, \beta, \gamma$ are the roots of the equation $a x^{3}+b x^{2}+c$ $=0$, then the value of the determinant $\left|\begin{array}{ccc}\alpha \beta & \beta \gamma & \gamma \alpha \\ \beta \gamma & \gamma \alpha & \alpha \beta \\ \gamma \alpha & \alpha \beta & \beta \gamma\end{array}\right|$
(A) $a$
(B) $b$
(C) 0
(D) $c$

Ahmad Reda

Numerade Educator

Problem 20

If $p+q+r=0=a+b+c$, then the value of the deter$\operatorname{minant}\left|\begin{array}{ccc}p a & q b & n c \\ q c & n a & p b \\ r b & p c & q a\end{array}\right|$ is
(A) 0
(B) $p q+q b+r c$
(C) 1
(D) None of these

Ahmad Reda

Numerade Educator

Problem 21

A determinant of second order is made with the elements 0 and $1 .$ The number of determinants with non-negative values is
(A) 3
(B) 10
(C) 11
(D) 13

Ahmad Reda

Numerade Educator

Problem 22

If $f_{j}=\sum_{i=0}^{2} a_{i j} x^{i}, j=1,2,3$ and if $f_{j}^{\prime}, f_{j}^{\prime \prime}$ denote $\frac{d f_{j}}{d x}, \frac{d^{2} f_{j}}{d x^{2}}$
respectively, then $g(x)=\left|\begin{array}{lll}f_{1} & f_{2} & f_{3} \\ f_{1}^{\prime} & f_{2}^{\prime} & f_{3}^{\prime} \\ f_{1}^{\prime \prime} & f_{2}^{\prime \prime} & f_{3}^{\prime \prime}\end{array}\right|$ is (A) a cubic in $x$
(B) a quadratic in $x$
(C) linear in $x$
(D) a constant

Ahmad Reda

Numerade Educator

Problem 23

The value of the determinant
$\Delta=\left|\begin{array}{ccc}2 a_{1} b_{1} & a_{1} b_{2}+a_{2} b_{1} & a_{1} b_{3}+a_{3} b_{1} \\ a_{1} b_{2}+a_{2} b_{1} & 2 a_{2} b_{2} & a_{2} b_{3}+a_{3} b_{2} \\ a_{1} b_{3}+a_{3} b_{1} & a_{3} b_{2}+a_{2} b_{3} & 2 a_{3} b_{3}\end{array}\right|$ is
(A) 1
(B) $-1$
(C) 0
(D) $a_{1} a_{2} a_{3} b_{1} b_{2} b_{3}$

Ahmad Reda

Numerade Educator

Problem 24

If $A+B+C=\pi, e^{i \theta}=\cos \theta+i \sin \theta$ and
$z=\left|\begin{array}{lll}e^{2 i A} & e^{-i C} & e^{-i B} \\ e^{-i C} & e^{2 i B} & e^{-i A} \\ e^{-i B} & e^{-i A} & e^{2 i C}\end{array}\right|$ then
(A) $\operatorname{Re}(z)=4$
(B) $\operatorname{Im}(z)=0$
(C) $\operatorname{Re}(z)=-4$
(D) $\operatorname{Im}(z)=-1$

Ahmad Reda

Numerade Educator

Problem 25

If $x_{i}=a_{i} b_{i} c_{i}, i=1,2,3$, are theree-digit positive integers such that each $x_{i}$ is a multiple of 19, then for some integer $n, \Delta=\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3}\end{array}\right|$ is given by
(A) $19 n+1$
(B) $19 n+2$
(C) $19 n$
(D) $19 n+3$

Ahmad Reda

Numerade Educator

Problem 26

If the system of equations $a x+b y+c=0, b x+c y+a$ $=0, c x+a y+b=0$ has a solution then the system of equations $(b+c) x+(c+a) y+(a+b) z=0$
$(c+a) x+(a+b) y+(b+c) z=0$
$(a+b) x+(b+c) y+(c+a) z=0$ has
(A) only one solution
(B) no solution
(C) infinite number of solutions
(D) None of these

Ahmad Reda

Numerade Educator

Problem 27

$(b+c)(y+z)-a x=b-c$,
$(c+a)(z+x)-b y=c-a$
$(a+b)(x+y)-c z=a-b$,
where $a+b+c \neq 0$, then $x=$
(A) $\frac{c-b}{a+b+c}$
(B) $\frac{a-c}{a+b+c}$
(C) $\frac{b-a}{a+b+c}$
(D) $\frac{1}{a+b+c}$

Ahmad Reda

Numerade Educator

Problem 28

The equations $x+y+z=6, x+2 y+3 z=10, x+2 y+$
$m z=n$ give infinite number of values of the triplet $(x,$, $y, z$ ) if (A) $m=3, n \in R$
(B) $m=3, n \neq 10$
(C) $m=3, n=10$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 29

If $x \neq 0, y \neq 0, z \neq 0$ and $\left|\begin{array}{ccc}1+x & 1 & 1 \\ 1+y & 1+2 y & 1 \\ 1+z & 1+z & 1+3 z\end{array}\right|=0$,
then $x^{-1}+y^{-1}+z^{-1}$ is equal to
(A) $-1$
(B) $-2$
(C) $-3$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 30

If $\Delta(x)=\left|\begin{array}{ccc}x & 1+x^{2} & x^{3} \\ \log \left(1+x^{2}\right) & e^{x} & \sin x \\ \cos x & \tan x & \sin ^{2} x\end{array}\right|$ then
(A) $\Delta(x)$ is divisible by $x$
(B) $\Delta(x)=0$
(C) $\Delta^{\prime}(x)=0$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 31

The number of values of $k$ for which the linear
equations
$$
\begin{array}{r}
4 x+k y+2 z=0 \\
k x+4 y+z=0 \\
2 x+2 y+z=0
\end{array}
$$
possess a non-zero solution is
(A) 0
(B) 3
(C) 2
(D) 1

Ahmad Reda

Numerade Educator

Problem 32

Let $P$ and $Q$ be $3 \times 3$ matrices $P \neq Q$. If $P^{3}=Q^{3}$ and $P^{2} Q=Q^{2} P$, then determinant of $\left(P^{2}+Q^{2}\right)$ is equal to:
(A) $-2$
(B) 1
(C) 0
(D) $-1$

Ahmad Reda

Numerade Educator

Problem 33

The value of the determinant $\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{26} & 5 & 10 \\ 3+\sqrt{65} & \sqrt{15} & 5\end{array}\right|$
is equal to:
(A) $5 \sqrt{3}(\sqrt{6}-5)$
(B) $5 \sqrt{3}(\sqrt{6}-\sqrt{5})$
(C) $5(\sqrt{6}-5)$
(D) $\sqrt{3}(\sqrt{6}-\sqrt{5})$

Ahmad Reda

Numerade Educator

Problem 34

Let $a, b, c$ be any real numbers. Suppose that there are real numbers $x, y, z$ not all zero such that $x=c y+b z, y$ $=a z+c x$ and $z=b x+a y$. Then $a^{2}+b^{2}+c^{2}+2 a b c$ is equal to
(A) 2
(B) $-1$
(C) 0
(D) 1

Ahmad Reda

Numerade Educator

Problem 35

Let $a, b, c$ be such that $b(a+c) \neq 0$. If $\left|\begin{array}{ccc}a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1\end{array}\right|$
$+\left|\begin{array}{rrr}a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2} a & (-1)^{n+1} b & (-1)^{n} c\end{array}\right|=0$, then the value
of ' $n$ 'is
(A) zero
(B) any even integer
(C) any odd integer
(D) any integer

Ahmad Reda

Numerade Educator

Problem 36

Let $A$ be a $2 \times 2$ matrix Statement-1: $\operatorname{adj}(\operatorname{adj} A)=A$
Statement-2: $|\operatorname{adj} A|=|A|$
(A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
(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 false
(D) Statement-1 is false, Statement- 2 is true

Ahmad Reda

Numerade Educator

Problem 37

If $a, b, c, d>0 ; x \in R$ and
$\left(a^{2}+b^{2}+c^{2}\right) x^{2}-2(a b+b c+c d) x+b^{2}+c^{2}+d^{2} \leq 0$,
then $\left|\begin{array}{lll}33 & 14 & \log a \\ 65 & 27 & \log b \\ 97 & 40 & \log c\end{array}\right|=$
(A) 1
(B) $-1$
(C) 0
(D) None of these

Ahmad Reda

Numerade Educator

Problem 38

The value of the determinant
43.
$\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3} & \sqrt[2]{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{26} & 5 & \sqrt{10} \\ 3+\sqrt{65} & \sqrt{15} & 5\end{array}\right|$ is
(A) $-5 \sqrt{3}(5-\sqrt{6})$
(B) $-5 \sqrt{3}(5+\sqrt{6})$
(C) $-5 \sqrt{3}(\sqrt{6}-5)$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 39

If $A_{1} B_{1} C_{1}, A_{2} B_{2} C_{2}$ and $A_{3} B_{3} C_{3}$ are three three-digit numbers, each of which is divisible by $k$, then
$\Delta=\left|\begin{array}{lll}A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|$ is
(A) divisible by $k$
(B) divisible by $k^{2}$
(C) divisible by $2 k$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 40

If the three-digit numbers $A 28,3 B 9$ and $62 \mathrm{C}$, where $A$, $B$ and $C$ are integers between 0 and 9 , are divisible by a fixed integer $k$, then the determinant $\left|\begin{array}{ccc}A & 3 & 6 \\ 8 & 9 & C \\ 2 & B & 2\end{array}\right|$ is
(A) divisible by $k$
(B) divisible by $k^{2}$
(C) divisible by $2 k$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 41

The value of the determinant of $n$th order, being given by $\left|\begin{array}{cccc}x & 1 & 1 & \ldots \\ 1 & x & 1 & \ldots \\ 1 & 1 & x & \ldots \\ \ldots & \ldots & \ldots & \ldots\end{array}\right|$, is
(A) $(x-1)^{n-1}(x+n-1)$
(B) $(x-1)^{n}(x+n-1)$
(C) $(1-x)^{n-1}(x+n-1)$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 42

The value of the determinant
$\left|\begin{array}{ccc}\sqrt{x}+\sqrt{y} & 2 \sqrt{z} & \sqrt{z} \\ \sqrt{y z}+\sqrt{2 x} & z & \sqrt{2 z} \\ y+\sqrt{x z} & \sqrt{y z} & z\end{array}\right|$
where $x, y, z$ are positive real numbers, is
(A) $z(\sqrt{2} y-z \sqrt{y)}$
(B) $y(\sqrt{2} z-y \sqrt{z)}$
(C) $x(\sqrt{2} y-z \sqrt{y})$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 43

If $f_{j}=\sum_{i=0}^{2} a_{i j} x^{i}, j=1,2,3$ and if $f_{j}^{\prime} f_{j}^{\prime \prime}$ denote $\frac{d f_{j}}{d x}, \frac{d^{2} f_{j}}{d x^{2}}$
respectively, then $g(x)=\left|\begin{array}{lll}f_{1} & f_{2} & f_{3} \\ f_{1}^{\prime} & f_{2}^{\prime} & f_{3}^{\prime} \\ f_{1}^{\prime \prime} & f_{2}^{\prime \prime} & f_{3}^{\prime \prime}\end{array}\right|$ is
(A) a cubic in $x$
(B) a quadratic in $x$
(C) linear in $x$
(D) a constant

Ahmad Reda

Numerade Educator

Problem 44

If $x_{i}=a_{i} b_{r} c_{i}, i=1,2,3$, are theree-digit positive integers such that each $x_{i}$ is a multiple of 19 , then for some integer $n, \Delta=\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3}\end{array}\right|$ is given by
(A) $19 n+1$
(B) $19 n+2$
(C) $19 n$
(D) $19 n+3$

Ahmad Reda

Numerade Educator

Problem 45

$(b+c)(y+z)-a x=b-c$,
$(c+a)(z+x)-b y=c-a$,
$(a+b)(x+y)-c z=a-b$,
where $a+b+c \neq 0$, then $x=$
(A) $\frac{c-b}{a+b+c}$
(B) $\frac{a-c}{a+b+c}$
(C) $\frac{b-a}{a+b+c}$
(D) $\frac{1}{a+b+c}$

Ahmad Reda

Numerade Educator

Problem 46

If $x \neq 0, y \neq 0, z \neq 0$ and $\left|\begin{array}{ccc}1+x & 1 & 1 \\ 1+y & 1+2 y & 1 \\ 1+z & 1+z & 1+3 z\end{array}\right|=0$,
then $x^{-1}+y^{-1}+z^{-1}$ is equal to
(A) $-1$
(B) $-2$
(C) $-3$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 47

If $2 s=a+b+c$ and $\left|\begin{array}{ccc}a^{2} & (s-a)^{2} & (s-a)^{2} \\ (s-b)^{2} & b^{2} & (s-b)^{2} \\ (s-c)^{2} & (s-c)^{2} & c^{2}\end{array}\right|=$
$k(s-a)(s-b)(s-c)$, then $k$ is equal to
(A) 2
(B) $2 s$
(C) $2 s^{2}$
(D) $2 s^{3}$

Gaurav Kalra

Numerade Educator

Problem 48

Let $\alpha, \beta$ be the roots of the equation $a x^{2}+b x+c=0$. Let $s_{n}=\alpha^{n}+\beta^{n}$ for $n \geq 1$. Then, the value of the determinant $\left|\begin{array}{ccc}3 & 1+s_{1} & 1+s_{2} \\ 1+s_{1} & 1+s_{2} & 1+s_{3} \\ 1+s_{2} & 1+s_{3} & 1+s_{4}\end{array}\right|$ is
(A) $\frac{(a+b+c)\left(b^{2}-4 a c\right)}{a^{4}}$
(B) $\frac{(a+b+c)^{2}\left(b^{2}-4 a c\right)}{a^{4}}$
(C) $\frac{(a+b+c)^{2}\left(b^{2}-4 a c\right)}{a^{2}}$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 49

$\begin{aligned}&\text { The value of the determinant } \\&\begin{array}{llll}a & b-c & c+b \\ \text { (A) } a^{2}+b^{2}+c^{2}\end{array} & \begin{array}{ccc}a & b & c-a \\ a-b & a+b & c\end{array} \mid \text { is }\end{aligned}$
(B) $a b c(a+b+c)$
(C) $\left(a^{2}+b^{2}+c^{2}\right)(a+b+c)$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 50

If $\left|\begin{array}{ccc}\operatorname{cosec} \alpha & 1 & 0 \\ 1 & 2 \operatorname{cosec} \alpha & 1 \\ 0 & 1 & 2 \operatorname{cosec} \alpha\end{array}\right|=\frac{1}{2}\left(z^{3}+\frac{1}{z^{3}}\right)$,
then $z$ is equal to
(A) $\sin \alpha / 2$
(B) $\cos \alpha / 2$
(C) $\tan \alpha / 2$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 51

If $\left|\begin{array}{ccc}a^{2} & b^{2} & c^{2} \\ (a+1)^{2} & (b+1)^{2} & (c+1)^{2} \\ (a-1)^{2} & (b-1)^{2} & (c-1)^{2}\end{array}\right|=k(a-b)(b-c)$
$(c-a)$, then $k$ is equal to
(A) 4
(B) $-4$
(C) 2
(D) $-2$

Ahmad Reda

Numerade Educator

Problem 52

The value of the determinant
$\left|\begin{array}{lll}\left(a-a_{1}\right)^{-2} & \left(a-a_{1}\right)^{-1} & a_{1}^{-1} \\ \left(a-a_{2}\right)^{-2} & \left(a-a_{2}\right)^{-1} & a_{2}^{-1} \\ \left(a-a_{3}\right)^{-2} & \left(a-a_{3}\right)^{-1} & a_{3}^{-1}\end{array}\right|$
(A) $\frac{a^{2} \Pi\left(a_{i}-a_{j}\right)}{\pi a_{i} \Pi\left(a-a_{i}\right)^{2}}$
(B) $\frac{-a^{2} \Pi\left(a_{i}-a_{j}\right)}{\Pi a_{i} \Pi\left(a-a_{i}\right)^{2}}$
(C) $\frac{\Pi a_{i} \Pi\left(a-a_{i}\right)^{2}}{a^{2} \Pi\left(a_{i}-a_{j}\right)}$
(D) $-\frac{\Pi a_{i} \Pi\left(a-a_{i}\right)^{2}}{a^{2} \Pi\left(a_{i}-a_{j}\right)}$

Gaurav Kalra

Numerade Educator

Problem 53

If $\left|\begin{array}{ccc}\frac{1}{a+x} & \frac{1}{b+x} & \frac{1}{c+x} \\ \frac{1}{a+y} & \frac{1}{b+y} & \frac{1}{c+y} \\ \frac{1}{a+z} & \frac{1}{b+z} & \frac{1}{c+z}\end{array}\right|=\frac{P}{Q}$, where $Q$ is the
product of denominators, then $P$ is equal to
(A) $(a-b)(b-c)(c-a)$
(B) $(x-y)(y-z)(z-x)$
(C) $(a-b)(b-c)(c-a)(x-y)(y-z)(z-x)$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 54

If $a, b, c, d$ are the roots of the equation $\alpha x^{4}+\beta x^{3}+\gamma x^{2}$ $+\delta x+\xi=0$, then the value of the determinant
$\left|\begin{array}{cccc}1+a & 1 & 1 & 1 \\ 1 & 1+b & 1 & 1 \\ 1 & 1 & 1+c & 1 \\ 1 & 1 & 1 & 1+d\end{array}\right|$ is
(A) $\frac{\delta-\gamma}{\alpha}$
(B) $\frac{\xi-\delta}{\alpha}$
(C) $\frac{\alpha-\beta}{\alpha}$
(D) $\frac{\beta-\alpha}{\alpha}$

Gaurav Kalra

Numerade Educator

Problem 55

The value of the determinant $\left|\begin{array}{cccc}0 & x & y & z \\ -x & 0 & c & b \\ -y & -c & 0 & a \\ -z & -b & -a & 0\end{array}\right|$ is
(A) $(a x+b y+c z)^{2}$
(B) $(a x-b y+c z)^{2}$
(C) $(a x+b y-c z)^{2}$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 56

The value of the determinant
$\left|\begin{array}{ccc}b^{2}+c^{2} & a b & a c \\ a b & c^{2}+a^{2} & b c \\ c a & c b & a^{2}+b^{2}\end{array}\right|$ is
(A) $a^{2} b^{2} c^{2}$
(B) $2 a^{2} b^{2} c^{2}$
(C) $4 a^{2} b^{2} c^{2}$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 57

If $f(x)=\left|\begin{array}{ccc}x+c_{1} & x+a & x+a \\ x+b & x+c_{2} & x+a \\ x+b & x+b & x+c_{3}\end{array}\right|$ and $g(x)=\left(c_{1}-x\right)$
$\left(c_{2}-x\right)\left(c_{3}-x\right)$, then $f(0)$ is equal to
(A) $\frac{b g(a)-a g(b)}{(b-a)}$
(B) $\frac{b g(a)+a g(b)}{(b+a)}$
(C) $\frac{b g(a)-a g(b)}{(b+a)}$
(D) $\frac{b g(a)+a g(b)}{(b-a)}$

Gaurav Kalra

Numerade Educator

Problem 58

If $\left|\begin{array}{ccc}2 b c-a^{2} & c^{2} & b^{2} \\ c^{2} & 2 c a-b^{2} & a^{2} \\ b^{2} & a^{2} & 2 a b-c^{2}\end{array}\right|$
$=\left(a^{3}+b^{3}+c^{3}+k a b c\right)^{2}$, then $k$ is equal to
(A) 2
(B) $-2$
(C) 3
(D) $-3$

Ahmad Reda

Numerade Educator

Problem 59

The value of the determinant
is $\left|\begin{array}{ccc}\beta \gamma & \beta \gamma^{\prime}+\beta^{\prime} \gamma & \beta^{\prime} \gamma^{\prime} \\ \gamma \alpha & \gamma \alpha^{\prime}+\gamma^{\prime} \alpha & \gamma^{\prime} \alpha^{\prime} \\ \alpha \beta & \alpha \beta^{\prime}+\alpha^{\prime} \beta & \alpha^{\prime} \beta^{\prime}\end{array}\right|$
(A) $\left(\alpha \beta^{\prime}-\alpha^{\prime} \beta\right)\left(\beta \gamma^{\prime}-\beta^{\prime} \gamma\right)\left(\gamma \alpha^{\prime}-\gamma^{\prime} \alpha\right)$
(B) $\alpha \beta \gamma(\alpha+\beta+\gamma)\left(\alpha^{\prime}+\beta^{\prime}+\gamma^{\prime}\right)$
(C) $\alpha^{\prime} \beta^{\prime} \gamma^{\prime}(\alpha+\beta+\gamma)\left(\alpha^{\prime}+\beta^{\prime}+\gamma\right)$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 60

If $a \neq 0, a \neq 1$ and
$\left|\begin{array}{ccc}x+1 & x & x \\ x & x+a & x \\ x & x & x+a^{2}\end{array}\right|=a^{3}+f(x) \cdot a\left(a^{2}+a+1\right)$, then
(A) $f(x)=x$
(B) $f(x)=x^{2}$
(C) $f(x)=x^{3}$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 61

The value of the determinant $\left|\begin{array}{ccc}-b c & b^{2}+b c & c^{2}+b c \\ a^{2}+a c & -a c & c^{2}+a c \\ a^{2}+a b & b^{2}+a b & -a b\end{array}\right|$ is
(A) $\left(a^{2}+b^{2}+c^{2}\right)^{3}$
(B) $(a b+b c+c a)^{3}$
(C) $\left(a^{2}+b^{2}+c^{2}\right)(a b+b c+c a)^{2}$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 62

If $\left|\begin{array}{ccc}x+a^{2} & a b & a c \\ a b & x+b^{2} & b c \\ a c & b c & x+c^{2}\end{array}\right|=0$ and $x(\neq 0) \in R$ then
$x$ is equal to
(A) $a^{2}+b^{2}+c^{2}$
(B) $-\left(a^{2}+b^{2}+c^{2}\right)$
(C) $2\left(a^{2}+b^{2}+c^{2}\right)$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 63

The values of $m$ for which the system of equations $3 x+m y=m$ and $2 x-5 y=20$ has a solution satisfying the condition $x>0, y>0$, are
(A) $m \in\left(-\infty, \frac{-15}{2}\right) \cup(0, \infty)$
(B) $m \in\left(-\infty, \frac{-15}{2}\right) \cup(30, \infty)$
(C) $m \in\left(-\infty, \frac{-15}{2}\right) \cup(0,30)$
(D) None of these

Ahmad Reda

Numerade Educator

Problem 64

If $a=\cos \theta+i \sin \theta, b=\cos 2 \theta-i \sin 2 \theta, c=\cos 3$
$\theta+i \sin 3 \theta$ and if $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=0$ then $\theta$ is equal to
(A) $\overline{n \pi}$
(B) $2 n \pi$
(C) $(2 n+1) \frac{\pi}{2}$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 65

The value of the determinant $\left|\begin{array}{ccc}\frac{1}{a} & \frac{1}{a(a+d)} & \frac{1}{(a+d)(a+2 d)} \\ \frac{1}{a+d} & \frac{1}{(a+d)(a+2 d)} & \frac{1}{(a+2 d)(a+3 d)} \\ \frac{1}{a+2 d} & \frac{1}{(a+2 d)(a+3 d)} & \frac{1}{(a+3 d)(a+4 d)}\end{array}\right|$
where $a, d>0$, is
(A) $-\frac{4 d^{4}}{a(a+d)^{2}(a+2 d)^{3}(a+3 d)^{2}(a+4 d)}$
(B) $\frac{4 d^{4}}{a(a+d)^{2}(a+2 d)^{3}(a+3 d)^{2}(a+4 d)}$
(C) $\frac{4 d^{4}}{a(a+d)^{2}(a+2 d)^{3}(a+3 d)^{2}(a+4 d)^{2}}$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 66

The value of the determinant $\left|\begin{array}{ccc}(b+c)^{2} & c^{2} & b^{2} \\ c^{2} & (c+a)^{2} & a^{2} \\ b^{2} & a^{2} & (a+b)^{2}\end{array}\right|$ is
(A) $2(a b+b c+c a)^{3}$
(B) $(a b+b c+c a)^{3}$
(C) $4(a b+b c+c a)^{3}$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 67

If the equations $(a+1)^{3} x+(a+2)^{3} y=(a+3)^{3},(a+1) x+(a+2) y$
$=a+3, x+y=1$ are consistent then $a$ is equal to
(A) 1
(B) $-1$
(C) 2
(D) $-2$

Gaurav Kalra

Numerade Educator

Problem 68

If the system of equations $x \sin \alpha+y \sin \beta+z \sin \gamma=0, x \cos \alpha+y \cos \beta+z \cos \gamma$
$=0, x+y+z=0$, where $\alpha, \beta, \gamma$ are angles of a triangle, have a non-trivial solution, then the triangle must be
(A) isosceles
(B) equilateral
(C) right angled
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 69

If $x_{1} \neq 0, x_{2} \neq 0, x_{3} \neq 0$, then the determinant
$\left|\begin{array}{ccc}x_{1}+a_{1} b_{1} & a_{1} b_{2} & a_{1} b_{3} \\ a_{2} b_{1} & x_{2}+a_{2} b_{2} & a_{2} b_{3} \\ a_{3} b_{1} & a_{3} b_{2} & x_{3}+a_{3} b_{3}\end{array}\right|$ is equal to
(A) $x_{1} x_{2} x_{3}\left(1+\frac{a_{1} b_{1}}{x_{1}}+\frac{a_{2} b_{2}}{x_{2}}+\frac{a_{3} b_{3}}{x_{3}}\right)$
(B) $-x_{1} x_{2} x_{3}\left(1+\frac{a_{1} b_{1}}{x_{1}}+\frac{a_{2} b_{2}}{x_{2}}+\frac{a_{3} b_{3}}{x_{3}}\right)$
(C) $x_{1} x_{2} x_{3}\left(1-\frac{a_{1} b_{1}}{x_{1}}-\frac{a_{2} b_{2}}{x_{2}}-\frac{a_{3} b_{3}}{x_{3}}\right)$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 70

If $\left|\begin{array}{ccc}a & a+d & a+2 d \\ a^{2} & (a+d)^{2} & (a+2 d)^{2} \\ 2 a+3 d & 2(a+d) & 2 a+d\end{array}\right|=0$, then
(A) $a+d=0$
(B) $d=0$
(C) $d=0$ or $a+d=0$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 71

Let $\left|\begin{array}{ccc}x+3 & x+2 & (x+2)^{3} \\ x+2 & x+3 & (x+2)^{3} \\ (x+2)^{3} & x+2 & x+3\end{array}\right|$
$=a x^{7}+b x^{6}+c x^{5}+d x^{4}+e x^{3}+f x^{2}+g x+h$ be an iden-
tity in $x$, where $a, b, c, d, e, f, g, h$ are independent of $x$, then the value of $g$ is
(A) $-213$
(B) 213
(C) 0
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 72

If $\left|\begin{array}{ccc}x^{n} & y^{n} & z^{n} \\ x^{n+2} & y^{n+2} & z^{n+2} \\ x^{n+3} & y^{n+3} & z^{n+3}\end{array}\right|$
$=(x-y)(y-z)(z-x)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$ then
(A) $n=1$
(B) $n=-1$
(C) $n=2$
(D) $n=-2$

Gaurav Kalra

Numerade Educator

Problem 73

The value of the determinant
$\left|\begin{array}{ccc}\sin \alpha \cos \beta & \cos \alpha \cos \beta & -\sin \alpha \sin \beta \\ \sin \alpha \sin \beta & \cos \alpha \sin \beta & \sin \alpha \cos \beta \\ \cos \alpha & -\sin \alpha & 0\end{array}\right|$ is
(A) is independent of $\alpha$
(B) independent of $\beta$
(C) independent of $\alpha$ and $\beta$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 74

If $\alpha, \beta, \gamma$ are the roots of the equation $x^{3}+p x+q=0$, then the value of the determinant
$\left|\begin{array}{ccc}1+\alpha & 1 & 1 \\ 1 & 1+\beta & 1 \\ 1 & 1 & 1+\gamma\end{array}\right|$ is
(A) $p^{2}-2 q$
(B) $3 p q$
(C) $p-q$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 75

The value of a determinant of third order whose all elements are 1 or $-1$ is
(A) an even number
(B) an odd number
(C) a prime number
(D) cannot be determined

Gaurav Kalra

Numerade Educator

Problem 76

If square matrices $A$ and $B$ are such that $A A^{\theta}=A^{\theta} A$, $B B^{\theta}=B^{\theta} B$ and $A B^{\theta}=B^{\theta} A$, then $(A B)(A B)^{\theta}$ is equal to
(A) $B^{\theta} A^{\theta} A B$
(B) $B A^{\theta} A B$
(C) $B A^{\theta} A B^{\theta}$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 77

Let $\Delta(x)=\left|\begin{array}{ccc}x & 2 & x \\ x^{2} & x & 6 \\ x & x & 6\end{array}\right|=A x^{4}+B x^{3}+C x^{2}+D x+E$

Gaurav Kalra

Numerade Educator

Problem 78

If $\Delta_{1}=$
$\left|\begin{array}{ccc}y^{5} z^{6}\left(z^{3}-y^{3}\right) & x^{4} z^{6}\left(x^{3}-z^{3}\right) & x^{4} z^{5}\left(y^{3}-x^{3}\right) \\ y^{2} z^{3}\left(y^{6}-z^{6}\right) & x z^{3}\left(z^{6}-x^{6}\right) & x y^{2}\left(x^{6}-y^{6}\right) \\ y^{2} z^{3}\left(z^{3}-y^{3}\right) & x z^{3}\left(x^{3}-z^{3}\right) & x y^{2}\left(y^{3}-x^{3}\right)\end{array}\right|$
and, $\Delta_{2}=\left|\begin{array}{ccc}x & y^{2} & z^{3} \\ x^{4} & y^{5} & z^{6} \\ x^{7} & y^{8} & z^{9}\end{array}\right|$, then $\Delta_{1} \Delta_{2}=$
(A) $\Delta_{22}^{2}$
(C) $\Delta_{2}^{4}$
(B) $\Delta_{2}^{3}$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 79

If $a b c=\gamma, A=\left[\begin{array}{lll}a & b & c \\ c & a & b \\ b & c & a\end{array}\right]$ and $A A^{\prime}=I$, then $a, b, c$ are
the roots of the equation.
(A) $x^{3} \pm x^{2}+\gamma=0$
(B) $x^{3} \pm 2 x^{2}+\gamma=0$
(C) $x^{3} \pm x^{2}-\gamma=0$
(D) $x^{3} \pm 2 x^{2}-\gamma=0$

Gaurav Kalra

Numerade Educator

Problem 80

If $a, b, c$ are the sides of a triangle $A B C$ such that $\left|\begin{array}{ccc}a^{2} & b^{2} & c^{2} \\ (a+1)^{2} & (b+1)^{2} & (c+1)^{2} \\ (a-1)^{2} & (b-1)^{2} & (c-1)^{2}\end{array}\right|=0$, then $\Delta A B C$ is
(A) a right angled triangle
(B) an isosceles triangle
(C) an equilateral triangle
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 81

The set of equations : $\lambda x-y+(\cos \theta) z=0 ; 3 x+y+2 z$
$=0 ;(\cos \theta) x+y+2 z=0,0 \leq \theta<2 \pi$, has non-trivial
solutions.
(A) for no values of $\lambda$ and $\theta$
(B) for all values of $\lambda$ and $\theta$
(C) for all values of $\lambda$ and only two values of $\theta$
(D) for only one value of $\lambda$ and all values of $\theta$

Gaurav Kalra

Numerade Educator

Problem 82

The value of $\lambda$ for which the equations $x+y-3=0$, $(1+\lambda) x+(2+\lambda) y-8=0, x-(1+\lambda) y+(2+\lambda)=0$
are consistent is
(A) 1
(B) $5 / 3$
(C) $-5 / 3$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 83

Let $\left\{\Delta_{1}, \Delta_{2}, \Delta_{3}, \ldots, \Delta_{k}\right\}$ be the set of third order determinants that can be made with the distinct non-zero real numbers $a_{1}, a_{2}, a_{3}, \ldots, a_{9}$. Then,
(A) $k=9 !$
(B) $\sum_{i=1}^{k} \Delta_{i}=0$
(C) at least one $\Delta_{i}=0$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 84

If $A+B+C=\pi, e^{i \theta}=\cos \theta+i \sin \theta$ and
$z=\left|\begin{array}{lll}e^{2 i A} & e^{-i C} & e^{-i B} \\ e^{-i C} & e^{2 i B} & e^{-i A} \\ e^{-i B} & e^{-i A} & e^{2 i C}\end{array}\right|$ then
(A) $\operatorname{Re}(z)=4$
(B) $\operatorname{Im}(z)=0$
(C) $\operatorname{Re}(z)=-4$
(D) $\operatorname{Im}(z)=-1$

Gaurav Kalra

Numerade Educator

Problem 85

If $\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{array}\right|=A x+B$, then
(A) $A=\left|\begin{array}{lll}4 & 0 & 0 \\ 2 & 3 & 3 \\ 4 & 0 & 2\end{array}\right|$
(B) $B=\left|\begin{array}{ccc}4 & 0 & 0 \\ 2 & 3 & 3 \\ 4 & 0 & -1\end{array}\right|$
(C) $A=\left|\begin{array}{ccc}4 & 0 & 0 \\ 2 & 3 & -3 \\ 4 & 0 & 2\end{array}\right|$
(D) $B=\left|\begin{array}{ccc}4 & 0 & 0 \\ 2 & 3 & -3 \\ 4 & 0 & -1\end{array}\right|$

Gaurav Kalra

Numerade Educator

Problem 86

5. If $\left|\begin{array}{ccc}0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0\end{array}\right|=0, a \neq b \neq c$, then
(A) $x=0$ if $b(a+c) \leq a c$
(B) $x=$ ? $\sqrt{b(a+c)-a c}$ if $b(a+c) \geq a c$
(C) $x=0, \pm \sqrt{b(a+c)-a c}$ if $b(a \neq c)>a c$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 87

If $\left|\begin{array}{ccc}b c-a^{2} & c a-b^{2} & a b-c^{2} \\ c a-b^{2} & a b-c^{2} & b c-a^{2} \\ a b-c^{2} & b c-a^{2} & c a-b^{2}\end{array}\right|=\left|\begin{array}{ccc}\alpha^{2} & \beta^{2} & \beta^{2} \\ \beta^{2} & \alpha^{2} & \beta^{2} \\ \beta^{2} & \beta^{2} & \alpha^{2}\end{array}\right|$
then
(A) $\alpha^{2}=a^{2}+b^{2}+c^{2}$
(B) $\beta^{2}=a b+b c+c a$
(C) $\alpha^{2}=a b+b c+c a$
(D) $\beta^{2}=a^{2}+b^{2}+c^{2}$

Gaurav Kalra

Numerade Educator

Problem 88

The determinant $\left|\begin{array}{ccc}\sin x & \sin y & \sin z \\ \cos x & \cos y & \cos z \\ \cos ^{3} x & \cos ^{3} y & \cos ^{3} z\end{array}\right| ; 0<x, y$,
$z<\frac{\pi}{2}$, is equal to zero if
(A) $x=y$
(B) $y=z$
(C) $z=x$
(D) $x+y+z=\frac{\pi}{2}$

Gaurav Kalra

Numerade Educator

Problem 89

The value of the determinant
$\left|\begin{array}{ccc}\cos (\theta+\alpha) & -\sin (\theta+\alpha) & \cos 2 \alpha \\ \sin \theta & \cos \theta & \sin \alpha \\ -\cos \theta & \sin \theta & \lambda \cos \alpha\end{array}\right|$ is
(A) independent of $\theta$ for all $\lambda \in \mathrm{R}$
(B) independent of $\theta$ and $\alpha$ when $\lambda=1$
(C) independent of $\theta$ and $\alpha$ when $\lambda=-1$
(D) None of these

Gaurav Kalra

Numerade Educator

Problem 90

The value of $\theta$ lying between $\theta=0$ and $\theta=\frac{\pi}{2}$ and satisfying the equation $\left|\begin{array}{ccc}1+\sin ^{2} \theta & \cos ^{2} \theta & 4 \sin 4 \theta \\ \sin ^{2} \theta & 1+\cos ^{2} \theta & 4 \sin 4 \theta \\ \sin ^{2} \theta & \cos ^{2} \theta & 1+4 \sin 4 \theta\end{array}\right|=0$ is
(A) $\frac{7 \pi}{24}$
(B) $\frac{5 \pi}{24}$
(C) $\frac{11 \pi}{24}$
(D) $\frac{\pi}{24}$

Gaurav Kalra

Numerade Educator

Problem 91

If $a_{n}=\int_{0}^{\pi / 2} \frac{1-\cos 2 n x}{1-\cos 2 x} d x$, then
(A) $a_{n+1}$ is A.M. between $a_{n}$ and $a_{n+2}$
(B) $a_{n+1}$ is G.M between $a_{n}$ and $a_{n+2}$
(C) $a_{n+1}$ is H.M. between $a_{n}$ and $a_{n+2}$
(D) $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|=0$

Gaurav Kalra

Numerade Educator

Problem 92

If $\alpha, \beta, \gamma$ are non-zero real numbers such that
$\left|\begin{array}{ccc}\beta \gamma & \gamma \alpha & \alpha \beta \\ \gamma \alpha & \alpha \beta & \beta \gamma \\ \alpha \beta & \beta \gamma & \gamma \alpha\end{array}\right|=0$, then
(A) $\frac{1}{\gamma}+\frac{1}{\alpha \omega}+\frac{1}{\beta \omega^{2}}=0$
(B) $\frac{1}{\beta}+\frac{1}{\alpha \omega}+\frac{1}{\gamma \omega^{2}}=0$
(C) $\frac{1}{\beta}+\frac{1}{\gamma \omega}+\frac{1}{\alpha \omega^{2}}=0$
(D) $(\alpha \beta)^{3}+(\beta \gamma)^{3}+(\gamma \alpha)^{3}=3 \alpha^{2} \beta^{2} \gamma^{2}$

Gaurav Kalra

Numerade Educator

Problem 93

The positive integral solutions of the equation $\left|\begin{array}{ccr}x^{3}+1 & x^{2} y & x^{2} z \\ x y^{2} & y^{3}+1 & y^{2} z \\ x z^{2} & y z^{2} & z^{3}+1\end{array}\right|=30$ are
(A) $(3,1,1)$
(B) $(1,3,1)$
(C) $(1,1,3)$
(D) $(-1,1,3)$

Gaurav Kalra

Numerade Educator

Problem 94

If $f(x)=\left|\begin{array}{ccc}e^{x} & \sin x & 1 \\ \cos x & \log \left(1+x^{2}\right) & 1 \\ x & x^{2} & 1\end{array}\right|=a+b x+c x^{2}$, then
(A) $a=0$
(B) $a=1$
(C) $b=-1$
(D) $b=-2$

Gaurav Kalra

Numerade Educator

Problem 95

If maximum and minimum values of the determinant $\left|\begin{array}{ccc}1+\sin ^{2} x & \cos ^{2} x & \sin 2 x \\ \sin ^{2} x & 1+\cos ^{2} x & \sin 2 x \\ \sin ^{2} x & \cos ^{2} x & 1+\sin 2 x\end{array}\right|$ are $\alpha$ and $\beta$, then
(A) $\alpha+\beta^{99}=4$
(B) $\alpha^{3}-\beta^{17}=26$
(C) $\left(\alpha^{2 n}-\beta^{2 n}\right)$ is always an even integer for $n \in N$
(D) a triangle can be constructed having its sides as $\alpha-\beta, \alpha+\beta$ and $\alpha+3 \beta$

Gaurav Kalra

Numerade Educator

Problem 96

Let $A=\left[a_{i j}\right.$ be an $n \times n$ matrix. The matrix $A-\lambda I$ is called the characteristics matrix of $A$, where $\lambda$ is a scalar and $I$ is the identity matrix. The determinant $|A-\lambda I|$ is a non-null polynomial of degree $n$ in $\lambda$ and is called the characteristic polynomial of $A$. The equation $|A-\lambda I|=0$ is called the characteristic equation of $A$ and its roots are called the characteristic roots or latent roots or eigen values of $A$. The set of all eigenvalues of the matrix $A$ is called the spectrum of
A. The product of the eigenvalues of a matrix $A$ is equal to the determinant $A$.
$\left.\begin{array}{l}\text { The characteristic roots of the matrix } A=\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 1 & 2 \\ 1 & 2 & 0\end{array}\right]\end{array}\right]$
(A) 1
(B) 2
(C) $-2$
(D) 3

Gaurav Kalra

Numerade Educator

Problem 97

Let $A=\left[a_{i j}\right.$ be an $n \times n$ matrix. The matrix $A-\lambda I$ is called the characteristics matrix of $A$, where $\lambda$ is a scalar and $I$ is the identity matrix. The determinant $|A-\lambda I|$ is a non-null polynomial of degree $n$ in $\lambda$ and is called the characteristic polynomial of $A$. The equation $|A-\lambda I|=0$ is called the characteristic equation of $A$ and its roots are called the characteristic roots or latent roots or eigen values of $A$. The set of all eigenvalues of the matrix $A$ is called the spectrum of
A. The product of the eigenvalues of a matrix $A$ is equal to the determinant $A$.

The given values of the matrix $A=\left[\begin{array}{lll}1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4\end{array}\right]$ are
(A) $4,-2,-2$,
(B) $-4,2,-2$
(C) $-4,2,2$
(D) $4,-4,2$

Gaurav Kalra

Numerade Educator

Problem 98

Let $A=\left[a_{i j}\right.$ be an $n \times n$ matrix. The matrix $A-\lambda I$ is called the characteristics matrix of $A$, where $\lambda$ is a scalar and $I$ is the identity matrix. The determinant $|A-\lambda I|$ is a non-null polynomial of degree $n$ in $\lambda$ and is called the characteristic polynomial of $A$. The equation $|A-\lambda I|=0$ is called the characteristic equation of $A$ and its roots are called the characteristic roots or latent roots or eigen values of $A$. The set of all eigenvalues of the matrix $A$ is called the spectrum of
A. The product of the eigenvalues of a matrix $A$ is equal to the determinant $A$.
Which of the following statements are true? If $A$ is any $n \times n$ matrix and $\lambda$ is a characteristic root of $A$, then
(A) $A$ and $A^{\prime}$ have the same characteristic roots
(B) $k \lambda$ is a characteristic root of $k A$ ( $k$ being scalar)
(C) $\lambda^{n}$ is a characteristic root of $A^{n}$ ( $n$ being positive integer)
(D) $\frac{1}{\lambda}$ is a characteristic root of $A^{-1}$

Gaurav Kalra

Numerade Educator

Problem 99

Let $A=\left[a_{i j}\right.$ be an $n \times n$ matrix. The matrix $A-\lambda I$ is called the characteristics matrix of $A$, where $\lambda$ is a scalar and $I$ is the identity matrix. The determinant $|A-\lambda I|$ is a non-null polynomial of degree $n$ in $\lambda$ and is called the characteristic polynomial of $A$. The equation $|A-\lambda I|=0$ is called the characteristic equation of $A$ and its roots are called the characteristic roots or latent roots or eigen values of $A$. The set of all eigenvalues of the matrix $A$ is called the spectrum of
A. The product of the eigenvalues of a matrix $A$ is equal to the determinant $A$.
Which of the following statements are correct?
(A) If $A, B$ are $n$ rowed square matrices and $A$ is non-singular, then $A^{-1} B$ and $B A^{-1}$ has same character-istic roots.
(B) If $A$ and $P$ are square matrices of same order and $P$ is non-singular, then $A$ and $P^{-1} A P$ have same characteristic roots.
(C) If $A$ and $B$ be two square matrices of same order, then $A B$ and $B A$ have same characteristic roots.
(D) All of these

Gaurav Kalra

Numerade Educator

Problem 100

If $\left|\begin{array}{ccc}1+x & x & x^{2} \\ x & 1+x & x^{2} \\ x^{2} & x & 1+x\end{array}\right|=p x^{5}+q x^{4}+r x^{3}+s x^{2}+t x+w$, then
$$
\begin{array}{ll}
\hline \text { Column-I } & \text { Column-II } \\
\hline \text { I. } w \text { is equal to } & \text { (A) } 3 \\
\text { II. } t \text { is equal to } & \text { (B) } 1 \\
\text { III. } p+r \text { is equal to } & \text { (C) }-1 \\
\text { IV. } q+s \text { is equal to } & \text { (D) } 0 \\
\hline
\end{array}
$$

Gaurav Kalra

Numerade Educator