## Matrices and Determinants Questions and Answers Part-16

1. If $f\left(x\right)=\begin{bmatrix}\sin 2x\left(1+2\cos x\right) &\sin2x & \sin3x\\3+4\sin x & 3 & 4\sin x \\1+\sin x & \sin x & 1\end{bmatrix}$
then the value of $\int_{0}^{\pi/2} f\left(x\right) d(x)$    is
a) 3
b) 2/3
c) 1/3
d) 0

Explanation:

2. If $\alpha,\beta,\gamma$   are the roots of $x^{3}+ax^{2}+b=0$     , then the determinant $\triangle$  , where
$\triangle=\begin{bmatrix}\alpha & \beta & \gamma \\\beta & \gamma & \alpha \\\gamma & \alpha & \beta\end{bmatrix}$
equals
a) $-a^{3}$
b) $a^{3}-3b$
c) $a^{2}-3b$
d) a3

Explanation:

3. The number of distinct real roots of $\triangle=\begin{bmatrix}\tan x & \cot x & \cot x \\\cot x & \tan x & \cot x \\\cot x & \cot x & \tan x\end{bmatrix}=0$
in the interval$-\pi/4\leq x \leq\pi/4$     is
a) 0
b) 2
c) 1
d) 3

Explanation:

4. If $\omega \neq 1$   is a complex cube root of unity ,and $x+iy =\begin{bmatrix}1 & i & -\omega \\-i & 1 & \omega^{2} \\\omega & -\omega^{2} & 1\end{bmatrix}$
then
a) x = – 1, y = 0
b) x = 1, y = – 1
c) x = 1, y = 1
d) x = 0, y = 0

Explanation: Simplify to obtain x + iy = - 1

5. If $e^{ix}=\cos x+i \sin x$     and $x+iy=\begin{bmatrix}1 & e^{\pi i/4} & e^{\pi i/3} \\e^{-\pi i/4} & 1 & e^{2\pi i/3} \\e^{-\pi i/3} & e^{-2\pi i/3} & e^{2\pi i}\end{bmatrix}$
then
a) $x = – 1, y =\sqrt{2}$
b) $x = 1, y =-\sqrt{2}$
c) $x = – \sqrt{2}, y =\sqrt{2}$
d) none of these

Explanation:

6. If a, b, $c\epsilon R$ , the number of real roots of the equation $\triangle=\begin{bmatrix}x & c & -b \\-c & x & a \\b & -a & x\end{bmatrix} = 0$
is
a) 0
b) 1
c) 2
d) 3

Explanation:

7. If a, b, $c\epsilon R$ and a2+b2-ab-a-b+1 $\leq$ 0 and $\alpha+\beta+\gamma=0$    , then $\triangle=\begin{bmatrix}1 & \cos\gamma & \cos\beta \\\cos\gamma & a & \cos\alpha \\\cos\alpha & \cos\beta & b\end{bmatrix}$
equals
a) ab
b) 1
c) 2
d) 3

Explanation:

8. If $\triangle=\begin{bmatrix}\sin\alpha &\cos \alpha&\sin\alpha+\cos\beta \\\sin\beta& \cos \alpha & \sin\beta+\cos\beta \\\sin\gamma & \cos \alpha & \sin\gamma+\cos\beta\end{bmatrix}$
then $\triangle$ is equal to
a) $\sin\alpha\sin\beta\sin\gamma$
b) $\cos\alpha\sec\beta\tan\gamma$
c) $\sin\alpha\sin\left(\alpha+\beta\right)+\cos\alpha\cos\left(\gamma+\beta\right)$
d) 0

Explanation:

9. Let $f\left(x\right)=\begin{bmatrix}cosec x & \sin x & cosec^2 x+\tan x \sec x \\\sin^{2}x & \sin^{2}x & \sec^{2}x \\1 & \sin^{2}x & \sin^{2}x\end{bmatrix}$
then $\int_{0}^{\pi/2} f\left(x\right) dx$    equals
a) $-\left(\frac{\pi}{4}+\frac{8}{15}\right)$
b) $\frac{\pi}{4}$
c) $\frac{\pi}{4}+\frac{1}{5}$
d) $\pi$

10. The values of $\theta$  lying between $\theta=0$  and $\theta=\pi/2$   and satisfying the equation $\begin{bmatrix}1+\sin^{2}\theta & \cos^{2}\theta & 4\sin6\theta \\\sin^{2}\theta & 1+\cos^{2}\theta &4\sin6\theta \\\sin^{2}\theta & \cos^{2}\theta &1+ 4\sin6\theta\end{bmatrix}=0$
a) $\pi/36, 5\pi/36$
b) $7\pi/36, 11\pi/36$
c) $5\pi/36, 7\pi/36$
d) $11\pi/36, \pi/36$