The characteristic roots of the matrix $\left[\begin{array}{ccc}1& 0& 0\\ 2& 3& 0\\ 4& 5& 6\end{array}\right]$ are

• 1, 3, 6

• 1, 2, 4

• 4, 5, 6

• 2, 4, 6

If A(adj A) = 5I where I is the identity matnx of order 3, then $\left|\mathrm{adj} \mathrm{A}\right|$ is equal to

• 125

• 25

• 5

• 10

If A = $\left|\begin{array}{ccc}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|$ and B = $\left|\begin{array}{ccc}{\mathrm{c}}_1& {\mathrm{c}}_2& {\mathrm{c}}_3\\ {\mathrm{a}}_1& {\mathrm{a}}_2& {\mathrm{a}}_3\\ {\mathrm{b}}_1& {\mathrm{b}}_2& {\mathrm{b}}_3\end{array}\right|$, then

• A = - B

• A = B

• B = 0

• B = A²

If A = $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, then A^-1 is equal to

• $- \frac{1}{2}\left[\begin{array}{cc}4& - 2\\ - 3& 1\end{array}\right]$

• $\frac{1}{2}\left[\begin{array}{cc}4& - 2\\ - 3& 1\end{array}\right]$

• $\left[\begin{array}{cc}- 2& 4\\ 1& 3\end{array}\right]$

• $\left[\begin{array}{cc}2& 4\\ 1& 3\end{array}\right]$

$\sin \left(2{\sin }^{-1}\left(\sqrt{\frac{63}{65}}\right)\right)$ is equal to

• $\frac{2\sqrt{126}}{65}$

• $\frac{4\sqrt{65}}{65}$

• $\frac{8\sqrt{63}}{65}$

• $\frac{\sqrt{63}}{65}$

6.

If f(x⁵) = 5x³, then f'(x) is equal to

• $\frac{3}{\sqrt[5]{{\mathrm{x}}^2}}$

• $\frac{3}{\sqrt[5]{\mathrm{x}}}$

• $\frac{3}{\mathrm{x}}$

• $\sqrt[5]{\mathrm{x}}$

f(x) = $$\begin{gathered} \left\{2\mathrm{a} - \mathrm{x} \mathrm{in} - \mathrm{a} < \mathrm{x} < \mathrm{a} \\ 3\mathrm{x} - 2\mathrm{a} \mathrm{in} \mathrm{a} \le \mathrm{x}\right\} \end{gathered}$$

Then, which of the following is true ?

• f(x) is discontinuous at x = a

• f(x) is not differentiable at x = a

• f(x) is differentiable at x  $\ge$ a

• f(x) is continuous at all x <a

A stone is thrown vertically upwards and the height x ft reached by the stone in t seconds is given by x = 80 t - 16t² The stone reaches the maximum height in

• 2 s

• 2.5 s

• 3 s

• 1.5 s

The maximum value of $\frac{\log \left(\mathrm{x}\right)}{\mathrm{x}} \mathrm{in} \left(2, \infty \right)$ is

• 1

• $\frac{2}{\mathrm{e}}$

• e

• $\frac{1}{\mathrm{e}}$

If f(x) = be^ax + ae^bx, then f''(0) is equal to

• 0

• 2ab

• ab(a + b)

• ab