## Matrices and Determinants Questions and Answers Part-14

1. if $f\left(x\right)=\begin{bmatrix}\sec^{2}x & 1 & 1 \\\cos^{2} x & \cos^{2} x & cosec^{2} x \\1 & \cos^{2} x & \cot^{2} x\end{bmatrix}$
then
a) $\int_{-\pi/4}^{\pi/4} f\left(x\right)dx=\frac{1}{32}\left(3\pi+8\right)$
b) $f'\left(\pi/2\right)=0$
c) minimun value of $f\left(x\right)$ is 1
d) All of the above

Explanation:

2. Let $\triangle\left(x\right)=\begin{bmatrix}3 & 3x & 3x^{2}+2a^{2} \\3x & 3x^{2}+2a^{2} & 3x^{3}+6a^{2} x \\3x^{2}+2a^{2} & 3x^{3}+6a^{2} x & 3x^{4}+12a^{2}x^{2}+2a^{4} \end{bmatrix}$
then
a) $\triangle$'(x) = 0
b) $\triangle$(x) is independent of x
c) $\int_{0}^{1} \triangle \left(x\right)dx=4a^{6}$
d) All of the above

Explanation:

3. Let $\lambda$ and $\alpha$ be real. Let S denote the set of all values of $\lambda$ for which the system of linear equations
$\lambda x+\left(\sin \alpha\right)y+\left(\cos \alpha\right)z=0$
$x+\left(\cos \alpha\right)y+\left(\sin \alpha\right)z=0$
$-x+\left(\sin \alpha\right)y-\left(\cos \alpha\right)z=0$
has a non-trivial solution then S contains
a) (– 1, 1)
b) $\left[-\sqrt{2},-1\right]$
c) $\left[1,\sqrt{2}\right]$
d) All of the above

Explanation: The given system of linear equations will have a non-trivial solution if

4. Consider the system of linear equations in x, y and z:
(sin 3 $\theta$ ) x – y + z = 0
(cos 2 $\theta$ ) x + 4y + 3z = 0
2x + 7y + 7z = 0
The values of $\theta$ for which the system of equations has a non-trivial solution are
a) $\left\{{n\pi : n\epsilon I}\right\}$
b) $\left\{{m\pi +\left(-1\right)^{m}\pi/6 : m\epsilon I}\right\}$
c) $\left\{{n\pi +\left(-1\right)^{n}\pi/3 : n\epsilon I}\right\}$
d) Both a and b

Explanation: The given system of equations will have a non-trivial solution if

5. Let a, b, c, d be four real numbers and let
$\triangle=\begin{bmatrix}2 & a+b+c+d & ac+cd \\a+b+c+d & 2\left(a+b\right)\left(c+d\right) & ab\left(c+d\right)+cd\left(a+b\right) \\ab+cd & ab\left(c+d\right)+cd\left(a+b\right) & 2abcd\end{bmatrix}$
then $\triangle$ is equal to
a) 2abcd (a + b) (c + d)
b) 2abcd (a + b + c + d)
c) (a + b) (c + d)
d) 0

Explanation:

6. If $\theta\epsilon R$  maximum value of $\triangle=\begin{bmatrix}1 & 1 & 1\\1 & 1+\sin\theta & 1 \\1 & 1 & 1+\cos\theta\end{bmatrix}$
is
a) 1/2
b) $\sqrt{3}/2$
c) $\sqrt{2}$
d) $3\sqrt{2}/4$

Explanation:

7. if $\triangle=\begin{bmatrix}x+a& b & c\\a & x+b & c \\a & b & x+c\end{bmatrix}=0$
then x equals
a) a + b + c
b) – (a + b + c)
c) 0, a + b + c
d) 0, – (a + b + c)

Explanation: Use C1 $\rightarrow$ C1 + C2 + C3

8. The determinant $\triangle=\begin{bmatrix}\sec^{2} \theta& \tan^{2}\theta & 1\\\tan^{2}\theta & \sec^{2}\theta & -1 \\12 & 10 & 2\end{bmatrix}$
equals
a) $2\sin^{2} \theta$
b) $12\sec^{2} \theta-10\tan^{2} \theta$
c) $12\sec^{2} \theta-10\tan^{2} \theta+5$
d) 0

Explanation: Use C1 $\rightarrow$ C1 - C2

9.Let f and g be two function defined on R such that f(x + y) = f(x) g(y) + g(x) f(y) for each x, $y\epsilon R$ . Let
$\triangle=\begin{bmatrix}f\left(\theta+\alpha\right)& f\left(\alpha\right) & g\left(\alpha\right)\\f\left(\theta+\beta\right)& f\left(\beta\right) & g\left(\beta\right) \\f\left(\theta+\gamma\right) & f\left(\gamma\right) & g\left(\gamma\right)\end{bmatrix}$
then $\triangle$ equal
a) $f(\alpha) f(\beta) f(\gamma)$
b) $g(\alpha) g(\beta) g(\gamma)$
c) $f(\alpha) g(\alpha) + f(\beta) g(\beta) + f(\gamma) g(\gamma)$
d) 0

10. If a, $b\epsilon R$ and $\triangle=\begin{bmatrix}a & a+bi & bi \\a+bi & bi & a \\bi & a & a+bi \end{bmatrix}$
then $\triangle$ equals
a) $a^{3}+b^{3}i$
b) $2\left(a^{3}+b^{3}\right)$
c) $a^{3}-b^{3}i$
d) $-2\left(a^{3}-b^{3}i\right)$
Explanation: Use C1 $\rightarrow$ C1 + C2 + C3