A spherical iron ball ofradius 10 cm, coated with a layer of ice of uniform thickness, melts at a rate of 100 $\mathrm{\pi }$ cm/min. The rate at which the thickness of decreases when the thickness of ice is 5 cm, is

• 1 cm/min

• 2 cm/min

• $\frac{1}{376} \mathrm{cm}/\min$

• 5 cm/min

If ax² + bx + 4 attains its minimum value - 1 at x = 1, then the values of a and bare respectively

• 5, - 10

• 5, - 5

• 5, 5

• 10, - 5

The function f(x) = (9 - x²)² increases in

• $\left(- 3, 0\right) \cup \left(3, \infty \right)$

• $\left(- \infty , - 3\right) \cup \left(3, \infty \right)$

• $\left(- \infty , - 3\right) \cup \left(0, 3\right)$

• (- 3, 3)

Let $$\begin{gathered} \mathrm{g}\left(\mathrm{x}\right) = \left\{2\mathrm{e}, \mathrm{if} \mathrm{x} \le 1 \\ \log \left(\mathrm{x} - 1\right), \mathrm{if} \mathrm{x} > 1\right\} \end{gathered}$$. The equation  of the normal to y = g(x) at the point (3, log(2)), is

• y - 2x = 6 + log(2)

• y + 2x = 6 + log(2)

• y + 2x = 6 - log(2)

• y + 2x = - 6 + log(2)

If a and b are positive numbers such that a > b, then the minimum value of $\mathrm{asec}\left(\mathrm{\theta }\right) - \mathrm{btan}\left(0 < \mathrm{\theta } < \frac{\mathrm{\pi }}{2}\right)$ is

• $\frac{1}{\sqrt{{\mathrm{a}}^2 - {\mathrm{b}}^2}}$

• $\frac{1}{\sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2}}$

• $\sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2}$

• $\sqrt{{\mathrm{a}}^2 - {\mathrm{b}}^2}$

76.

If the curves $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{12} = 1 \mathrm{and} {\mathrm{y}}^3 = 8\mathrm{x}$  intersect at right angle, then the value of a is equal to

• 16

• 12

• 8

• 4

If the function f(x) = x³ -12ax² + 36a²x - 4(a > 0) attains its maximum and minimum at x = p and x = q respectively and if 3p = q², then a is equal to

• $\frac{1}{6}$

• $\frac{1}{36}$

• $\frac{1}{3}$

• 18

The equation of the tangent to the curve $\mathrm{y} = 4{\mathrm{e}}^{\frac{\mathrm{x}}{4}}$ at the point where the curve crosses y-axis is equal to

• 3x + 4y = 16

• 4x + y = 4

• x + y = 4

• 4x - 3y = - 12

The diagonal of a square is changing at the rate of 0.5 cms^-1. Then, the rate of change of area, when the area is 400 cm² is equal to

• $20\sqrt{2} {\mathrm{cm}}^2/\mathrm{s}$

• $10\sqrt{2} {\mathrm{cm}}^2/\mathrm{s}$

• $\frac{1}{10\sqrt{2}} {\mathrm{cm}}^2/\mathrm{s}$

• $\frac{10}{\sqrt{2}} {\mathrm{cm}}^2/\mathrm{s}$

The equation of the tangent to the curve x² - 2.xy + y² + 2x + y - 6 = 0 at (2, 2) is

• 2x + y - 6 = 0

• 2y + x - 6 = 0

• x + 3y - 8 = 0

• 3x + y - 8 = 0