If there is an error of h % in measuring the edge of a cube, then the per cent error in estimating its volume is

• k

• 3k

• $\frac{\mathrm{k}}{3}$

• None of these

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

• e

• $\frac{1}{\mathrm{e}}$

• e²

• e³

The rate of charge of the surface area of a sphere of radius r, when the radius is inacreasing at the rate of 2cm/s is proportional to

• $\frac{1}{\mathrm{r}}$

• $\frac{1}{{\mathrm{r}}^2}$

• r

• r²

The function f(x) = x³ + ax + bx + c, a² $\le$ 3b has

• one maximum value

• one minimum value

• no extreme value

• one maximum and one minimum value

A spherical balloon is expanding. If the radius is increasing at the rate of 2 cm/min, the rate at which the volume increases (in cubic centimetres per minute) when the radius is 5 cm, is

• $10\mathrm{\pi }$

• 100$\mathrm{\pi }$

• 200$\mathrm{\pi }$

• 50$\mathrm{\pi }$

496.

Let y be the number of people in a village at time t. Assume that the rate of change of the population is proportional to the number of people in the village at any time and further assume that the population never increases in time. Then, the population of the village at any fixed time t is given by

• y = e^kt + c, for some constants $\mathrm{c} \le 0 \mathrm{and} \mathrm{k} \ge 0$

• y = ce^kt, for some constants $\mathrm{c} \ge 0 \mathrm{and} \mathrm{k} \le 0$

• y = e^ct + k,  for some constants  $\mathrm{c} \le 0 \mathrm{and} \mathrm{k} \ge 0$

• y = ke^ct , for some constants $\mathrm{c} \ge 0 \mathrm{and} \mathrm{k} \le 0$

The function f (x) = x² e^{- 2x}, x > 0. Then the maximum value of f (x) is :

• $\frac{1}{\mathrm{e}}$

• $\frac{1}{2\mathrm{e}}$

• $\frac{1}{{\mathrm{e}}^2}$

• $\frac{4}{{\mathrm{e}}^4}$

The angle between the tangents at those points on the curve x = t² + 1 and y = t² - t - 6 where it meets x-axis is :

• $\pm {\tan }^{-1}\left(\frac{4}{29}\right)$

• $\pm {\tan }^{-1}\left(\frac{5}{49}\right)$

• $\pm {\tan }^{-1}\left(\frac{10}{49}\right)$

• $\pm {\tan }^{-1}\left(\frac{8}{29}\right)$

If $\mathrm{\theta }$ is semi vertical angle of a cone of maximum volume and given slant height, then tan($\mathrm{\theta }$) is equal to

• 2

• 1

• $\sqrt{2}$

• $\sqrt{3}$

A man of 2 m height walks at a uniform speed of 6 km/h away from a lamp post of 6 m height. The rate at which the length of his shadow increase, is

• 2 km/h

• 1 km/ h

• 3 km/h

• 6 km/h