21.

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 s

• a constant

The maximum value of xy when x + 2y = 8 is :

• 20

• 16

• 24

• 8

The function f(x) = tan^-1(sin(x) + cos(x)), x > 0 is always an increasing function on the interval :

• $\left(0, \mathrm{\pi }\right)$

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

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

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

The radius of a cylinder is increasing at the rate of 3 m/s and its altitude is decreasing at the rate of 4 m/s. The rate of change of volume when radius is 4 m and altitude is 6 m, is :

• $80 \mathrm{\pi } \mathrm{cu} \mathrm{m}/\mathrm{s}$

• $144 \mathrm{\pi } \mathrm{cu} \mathrm{m}/\mathrm{s}$

• $- 80 \mathrm{\pi } \mathrm{cu} \mathrm{m}/\mathrm{s}$

• $64 \mathrm{\pi } \mathrm{cu} \mathrm{m}/\mathrm{s}$

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 emfs. The height of the upper end while it is descending at the rate of 4 emfs, is :

• 4$\sqrt{3}$

• 5$\sqrt{3}$

• 5$\sqrt{2}$

• 6 m

If N_a = {an : n $\in$ N}, then N₅ n N₇ is equal to :

• N₇

• N

• N₃₅

• N₅

If $\mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{\alpha x}}{\mathrm{x} + 1}, \mathrm{x} \ne - 1$, for what value of $\mathrm{\alpha }$ is f$\left[\mathrm{f}\left(\mathrm{x}\right)\right]$ = x ?

• $\sqrt{2}$

• $- \sqrt{2}$

• 1

• - 1

If A_i = $\left[\begin{array}{cc}{\mathrm{a}}^{\mathrm{i}}& {\mathrm{b}}^{\mathrm{i}}\\ {\mathrm{b}}^{\mathrm{i}}& {\mathrm{a}}^{\mathrm{i}}\end{array}\right]$ and, if $\left|\mathrm{a}\right| < 1, \left|\mathrm{b}\right| < 1$, then $\sum _{\mathrm{i} = 1}^{\infty }\det \left({\mathrm{A}}_{\mathrm{i}}\right)$ is equa to :

• $\frac{{\mathrm{a}}^2}{\left(1 - {\mathrm{a}}^2\right)} - \frac{{\mathrm{b}}^2}{\left(1 - {\mathrm{b}}^2\right)}$

• $\frac{{\mathrm{a}}^2 - {\mathrm{b}}^2}{\left(1 - {\mathrm{a}}^2\right)\left(1 - {\mathrm{b}}^2\right)}$

• $\frac{{\mathrm{a}}^2}{\left(1 - {\mathrm{a}}^2\right)} + \frac{{\mathrm{b}}^2}{\left(1 - {\mathrm{b}}^2\right)}$

• $\frac{{\mathrm{a}}^2}{\left(1 + {\mathrm{a}}^2\right)} - \frac{{\mathrm{b}}^2}{\left(1 + {\mathrm{b}}^2\right)}$

If w is a complex cube-root of unity, then $\left|\begin{array}{ccc}1& \mathrm{w}& {\mathrm{w}}^2\\ \mathrm{w}& {\mathrm{w}}^2& 1\\ {\mathrm{w}}^2& 1& \mathrm{w}\end{array}\right|$ is equal to

• - 1

• 1

• 0

• w

The soluton of ${\sin }^{-1}\left(\mathrm{x}\right) - {\sin }^{-1}\left(2\mathrm{x}\right) = \pm \frac{\mathrm{\pi }}{3}$ is

• $\pm \frac{1}{3}$

• $\pm \frac{1}{4}$

• $\pm \frac{\sqrt{3}}{2}$

• $\pm \frac{1}{2}$