21.

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

The angle between the curves y = a^x and y = b^x is equal to

• ${\tan }^{-1}\left(\left|\frac{\mathrm{a} - \mathrm{b}}{1 + \mathrm{ab}}\right|\right)$

• ${\tan }^{-1}\left(\left|\frac{\mathrm{a} + \mathrm{b}}{1 - \mathrm{ab}}\right|\right)$

• ${\tan }^{-1}\left(\left|\frac{\log \left(\mathrm{b}\right) - \log \left(\mathrm{a}\right)}{1 + \log \left(\mathrm{a}\right)\log \left(\mathrm{b}\right)}\right|\right)$

• ${\tan }^{-1}\left(\left|\frac{\log \left(\mathrm{a}\right) - \log \left(\mathrm{b}\right)}{1 + \log \left(\mathrm{a}\right)\log \left(\mathrm{b}\right)}\right|\right)$

Let f(x) = (x - 7)²(x - 2)⁷, x $\in$ [2, 7].  The value of $\mathrm{\theta } \in \left(2, 7\right)$ such that $\mathrm{f}\text{'}\left(\mathrm{\theta }\right)$ = 0 is equal to

• $\frac{49}{4}$

• $\frac{53}{9}$

• $\frac{53}{7}$

• $\frac{49}{9}$

The domain of the function f(x) = log₂(log₃(log₄(x))) is

• $\left(- \infty , 4\right)$

• $\left(4, \infty \right)$

• (0, 4)

• $\left(1, \infty \right)$

If f : R $\to$ R and g : R $\to$ R are defined by f (x) = x - 3 and g(x) = x² + 1, then the values of x for which g{f(x)} = 10 are

• 0, - 6

• 2, - 2

• 1, - 1

• 0, 6

${\log }_{\mathrm{e}}\left(\frac{1 + 3\mathrm{x}}{1 - 2\mathrm{x}}\right)$ is equal to

• $- 5\mathrm{x} - \frac{5{\mathrm{x}}^2}{2} - \frac{35{\mathrm{x}}^3}{3} - ...$

• $- 5\mathrm{x} + \frac{5{\mathrm{x}}^2}{2} - \frac{35{\mathrm{x}}^3}{3} + ...$

• $5\mathrm{x} - \frac{5{\mathrm{x}}^2}{2} + \frac{35{\mathrm{x}}^3}{3} - ...$

• $5\mathrm{x} + \frac{5{\mathrm{x}}^2}{2} + \frac{35{\mathrm{x}}^3}{3} + ...$