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

155.

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