The number of distinct real roots of $\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is:
(1) 0
(2) 2
(3) 1
(4) 3

The mean and variance for seven observations are 8 and 16 respectively. If 5 of the observations are $2, 4, 10, 12, 14$, then the product of the remaining two observations is
(1) 48
(2) 45
(3) 49
(4) 40

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:
(1) $y(b-a)$
(2) $y(a-b)$
(3) $0$
(4) $y(a-c)$

Let $A$ be a $3 \times 3$ matrix with $\operatorname { det } ( A ) = 4$. Let $R _ { i }$ denote the $i ^ { \text {th} }$ row of $A$. If a matrix $B$ is obtained by performing the operation $R _ { 2 } \rightarrow 2 R _ { 2 } + 5 R _ { 3 }$ on $2 A$, then $\operatorname { det } ( B )$ is equal to:
(1) 64
(2) 16
(3) 128
(4) 80

If $1 , \log _ { 10 } \left( 4 ^ { x } - 2 \right)$ and $\log _ { 10 } \left( 4 ^ { x } + \frac { 18 } { 5 } \right)$ are in arithmetic progression for a real number $x$ then the value of the determinant $\left| \begin{array} { c c c } 2 \left( x - \frac { 1 } { 2 } \right) & x - 1 & x ^ { 2 } \\ 1 & 0 & x \\ x & 1 & 0 \end{array} \right|$ is equal to:

$f ( x ) = \left| \begin{array} { c c c } 2 \cos ^ { 4 } x & 2 \sin ^ { 4 } x & 3 + \sin ^ { 2 } 2 x \\ 3 + 2 \cos ^ { 4 } x & 2 \sin ^ { 4 } x & \sin ^ { 2 } 2 x \\ 2 \cos ^ { 4 } x & 3 + 2 \sin ^ { 4 } x & \sin ^ { 2 } 2 x \end{array} \right|$ then $\frac { 1 } { 5 } f ^ { \prime } ( 0 )$ is equal to:
(1) 0
(2) 1
(3) 2
(4) 6

For $\alpha , \beta \in \mathbb { R }$ and a natural number $n$, let $A _ { r } = \left| \begin{array} { c c c } r & 1 & \frac { n ^ { 2 } } { 2 } + \alpha \\ 2 r & 2 & n ^ { 2 } - \beta \\ 3 r - 2 & 3 & \frac { n ( 3 n - 1 ) } { 2 } \end{array} \right|$. Then $\sum_{r=1}^{n} A_r$ is
(1) 0
(2) $4 \alpha + 2 \beta$
(3) $2 \alpha + 4 \beta$
(4) $2 n$

$$\left|\begin{array}{rrr} 2 & -3 & 2 \\ 1 & 2 & 0 \\ 2 & 3 & 0 \end{array}\right|$$
What is the value of this determinant?
A) $-1$
B) $-2$
C) $-3$
D) $-4$
E) $-6$