The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is :

• $\frac{2}{3}\sqrt{3}$

• $\sqrt{3}$

• $2\sqrt{3}$

• $\sqrt{6}$

Given that the slope of the tangent to a curve y = y(x) at any point (x, y) is $\frac{2\mathrm{y}}{{\mathrm{x}}^2}$. If the curve passes through the centre of the circle x² + y² - 2x - 2y = 0, then its equation is :

• ${\mathrm{xlog}}_{\mathrm{e}}\left|\mathrm{y}\right| = 2\left(\mathrm{x} - 1\right)$

• ${\mathrm{xlog}}_{\mathrm{e}}\left|\mathrm{y}\right| = - 2\left(\mathrm{x} - 1\right)$

• ${\mathrm{x}}^2{\log }_{\mathrm{e}}\left|\mathrm{y}\right| = - 2\left(\mathrm{x} - 1\right)$

• ${\mathrm{xlog}}_{\mathrm{e}}\left|\mathrm{y}\right| = \left(\mathrm{x} - 1\right)$

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is ${\tan }^{-1}\left(\frac{1}{2}\right)$. Water is poured into it at a constant rate of 5 cubic meter per minute. Then the rate (in m/min), at which the level of water is rising at the instant when the depth of water in the tank is 10m; is :

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10\mathrm{\pi }$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$15\mathrm{\pi }$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5\mathrm{\pi }$}\right.$

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

If the tangent to the curve, y = x³ + ax - b at the point (- 1, - 5) is perpendicular to the line, - x + y + 4 = 0, then which one of the following points lies on the curve ?

• (2, - 1)

• (- 2, 2)

• (2, - 2)

• (- 2, 1)

585.

Let f(x) = e^x - x and g(x) = x² - x, $\forall \mathrm{x} \in \mathrm{R}$. Then the set of all x $\in$ R, where the function h(x) = (fog)(x) is increasing, is :

• $[- \frac{1}{2}, 0] \cup [1, \infty )$

• $[- 1, - \frac{1}{2}] \cup [\frac{1}{2}, \infty )$

• $[0, \infty )$

• $[0, \frac{1}{2}] \cup [1, \infty )$

The tangent and normal to the ellipse 3x² + 5y² = 32 at the point P(2, 2) meet the x-axis at Q and R, respectively .Then the area (in sq. units) of the triangle PQR is :

• $\frac{68}{15}$

• $\frac{16}{3}$

• $\frac{14}{3}$

• $\frac{34}{15}$

A spherical iron ball of radius 10cm is coated with a layer of ice of uniform thickness that melts a rate of 50cm³/min When the thickness of the ice 5cm, then the rate at which the thickness (in cm/min) of the ice decreases, is :

• $\frac{5}{6\mathrm{\pi }}$

• $\frac{1}{18\mathrm{\pi }}$

• $\frac{1}{9\mathrm{\pi }}$

• $\frac{1}{36\mathrm{\pi }}$

   

If x + y = 8, then maximum value of x²y is

• $\frac{2048}{9}$

• $\frac{2048}{81}$

• $\frac{2048}{3}$

• $\frac{2048}{27}$

If f(x) = x³ - 9x + 1 and x = 3, then using Newton Raphson method, first iteration is

• $\frac{53}{18}$

• $\frac{53}{9}$

• $\frac{35}{18}$

• $\frac{35}{9}$

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

• 0

• 2

• 1/e

• - 1