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

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 slope of the tangent at (x, y) to a curve passing through $\left(1, \frac{\mathrm{\pi }}{4}\right)$ is given by $\frac{\mathrm{y}}{\mathrm{x}} - {\cos }^2\left(\frac{{\displaystyle \mathrm{y}}}{{\displaystyle \mathrm{x}}}\right)$, then the equation of the curve is

• $\mathrm{y} = {\tan }^{-1}\left[\log \left(\frac{\mathrm{e}}{\mathrm{x}}\right)\right]$

• $\mathrm{y} = {\mathrm{xtan}}^{-1}\left[\log \left(\frac{\mathrm{x}}{\mathrm{e}}\right)\right]$

• $\mathrm{y} = {\mathrm{xtan}}^{-1}\left[\log \left(\frac{\mathrm{e}}{\mathrm{x}}\right)\right]$

• None of the above

If  the function f(x) = 2x³ - 9ax² + 12a²x + 1 attains its maximum and minimum at p and q respectively such that p²  = q, then a equals 

• 0

• 1

• 2

• None of these

On the interval [0, 1] the function x²⁵(1 - x)⁷⁵ takes its maximum value at the point

• 0

• 1/4

• 1/2

• 1/3

The function f(x) = $\log \left(1 + \mathrm{x}\right) - \frac{2\mathrm{x}}{2 +\hspace{0.17em}\mathrm{x}}$ is increasing on

• $\left(0, \infty \right)$

• $\left(- \infty , 0\right)$

• $\left(- \infty , \infty \right)$

• None of the above

598.

If f(x) = $\mathrm{kx} - \sin \left(\mathrm{x}\right)$, then sin(x)is monotonically increasing, then

• k > 1

• k > - 1

• k < 1

• k < - 1

The maximum area of the rectangle that can be inscribed in a circle of radius r, is

• ${\mathrm{\pi r}}^2$

• r²

• $\raisebox{1ex}{${\mathrm{\pi r}}^2$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• 2r²

The velocity of a particle at time t is given by the relation v = 6t - $\frac{{\mathrm{t}}^2}{6}$. The distance traveled in 3 s is, if s = 0 at t = 0

• $\frac{39}{2}$

• $\frac{57}{2}$

• $\frac{51}{2}$

• $\frac{33}{2}$