If the line lx + my + n = 0 is tangent to the parabola y² = 4ax, then

• mn = al²

• lm = an²

• ln = am²

• None of the above

All the points on the curve y² = 4a$\left|\mathrm{x} + \mathrm{asin}\left(\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{a}$}\right.\right)\right|$, where the tangent is parallel to the axis of x are lies on

• circle

• parabola

• straight line

• None of these

The length of normal at any point to the curve $\mathrm{y} = \mathrm{c} \cosh \left(\frac{\mathrm{x}}{\mathrm{c}}\right)$ is

• fixed

• $\frac{{\mathrm{y}}^2}{{\mathrm{c}}^2}$

• $\frac{{\mathrm{y}}^2}{\mathrm{c}}$

• $\frac{\mathrm{y}}{{\mathrm{c}}^2}$

The height of right circular cylinder of maximum volume inscribed in a sphere of diameter 2a is

• $2\sqrt{3}\mathrm{a}$

• $\sqrt{3}\mathrm{a}$

• $\frac{2\mathrm{a}}{\sqrt{3}}$

• $\frac{\mathrm{a}}{\sqrt{3}}$

The approximate value of f(x) = x³ + 5x² - 7x + 9 at x = 1.1 is

• 8.6

• 8.5

• 8.4

• 8.3

The point on the curve 6y = x³ + 2 at which y-coordinate is changing 8 times as fast as x-coordinate is

• (4, 11)

• (4, - 11)

• (- 4, 11)

• (- 4, - 11)

The maximum value of $\mathrm{f}\left(\mathrm{x}\right) = \frac{\log \left(\mathrm{x}\right)}{\mathrm{x}}\left(\mathrm{x} \ne 0, \mathrm{x} \ne 1\right)$ is

• e

• $\frac{1}{\mathrm{e}}$

• e²

• $\frac{1}{{\mathrm{e}}^2}$

If the volume of spherical ball is increasing at the rate of 4$\mathrm{\pi }$ cm³/s, then the rate of change of its surface area when the volume is 288 $\mathrm{\pi }$ cm³, is

• $\frac{4}{3}\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $\frac{2}{3}\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $4\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $2\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

The equation of displacement of a particle is x(t) = 5t² - 7t + 3. The acceleration at the moment when its velocity becomes 5 m/sec is

• 3 m/sec²

• 7 m/sec²

• 10 m/sec²

• 8 m/sec²

The mean value of the function $\mathrm{f}\left(\mathrm{x}\right) = \frac{2}{{\mathrm{e}}^{\mathrm{x}} + 1}$ on the interval [0, 2] is

• $2 - {\log }_{\mathrm{e}}\left(\frac{2}{{\mathrm{e}}^2 + 1}\right)$

• $2 + {\log }_{\mathrm{e}}\left(\frac{2}{{\mathrm{e}}^2 + 1}\right)$

• $2 + {\log }_{\mathrm{e}}\left(\frac{2}{{\mathrm{e}}^2 - 1}\right)$

• $- 2 + {\log }_{\mathrm{e}}\left(\frac{2}{{\mathrm{e}}^2 - 1}\right)$