If $\mathrm{A}\left(\mathrm{\alpha }\right) = \left[\begin{array}{cc}\cos \left(\mathrm{\alpha }\right)& \sin \left(\mathrm{\alpha }\right)\\ - \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\end{array}\right]$, then the matrix ${\mathrm{A}}^2\left(\mathrm{\alpha }\right)$ is

• $\mathrm{A}\left(2\mathrm{\alpha }\right)$

• $\mathrm{A}\left(\mathrm{\alpha }\right)$

   

• $\mathrm{A}\left(3\mathrm{\alpha }\right)$

• $\mathrm{A}\left(4\mathrm{\alpha }\right)$

The value of $\left(\frac{{∆}^2}{\mathrm{E}}\right){\mathrm{x}}^2$ at the interval h = 1 is

• 0

• 1

• 2

• 4

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

• 0

• 2

• 1/e

• - 1

The value of f at x = 0 so that function $\mathrm{f}\left(\mathrm{x}\right) = \frac{2^{\mathrm{x}} - 2^{- \mathrm{x}}}{\mathrm{x}}$0, $\mathrm{x} \ne 0$ is continuous at x = 0, is

• 0

• log(2)

• 4

• log(4)

If y = a^x . b^{2x - 1}, then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is

• ${\mathrm{y}}^2\log \left({\mathrm{ab}}^2\right)$

• $\mathrm{y} . {\left(\log \left({\mathrm{ab}}^2\right)\right)}^2$

• y²

• $\mathrm{y} . {\left(\log \left({\mathrm{a}}^2\mathrm{b}\right)\right)}^2$

If a particle moves such that the displacement is proportional to the square of the velocity acquired, then its acceleration is

• proportional to s²

• proportional to $\frac{1}{{\mathrm{s}}^2}$

• proportional to $\frac{1}{\mathrm{s}}$

• a constant

7.

The function $\mathrm{f}\left(\mathrm{x}\right) = {\tan }^{-1}\left(\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right)$, x > 0 is always an increasing function on the interval

• $\left(0, \mathrm{\pi }\right)$

• $\left(0, \frac{\mathrm{\pi }}{2}\right)$

• $\left(0, \frac{\mathrm{\pi }}{4}\right)$

• $\left(0, \frac{3\mathrm{\pi }}{4}\right)$

A ladder 10 m long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of 3 cm/s. The height of the upper end while it is descending at the rate of 4 cm/s, is

• $4\sqrt{3}$ m

• $5\sqrt{3}$ m

• $5\sqrt{2}$ m

• 6 m

The value of ${\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}{\sin }^3\left(\mathrm{x}\right){\cos }^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• 1

• 2

• 3

• 0

$\frac{\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)}{\overrightarrow{\mathrm{b}} . \left(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}\right)} + \frac{\overrightarrow{\mathrm{b}} . \left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)}{\overrightarrow{\mathrm{a}} . \left(\overrightarrow{\mathrm{b}} \times \stackrel{ \to }{\mathrm{c}}\right)}$ is equal to

• 1

• 2

• 0

• $\infty$