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

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

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

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

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

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

513.

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 distance between the origin and the normal to the curve y = e^2x + x² at x = 0 is

• 2

• $\frac{2}{\sqrt{3}}$

• $\frac{2}{\sqrt{5}}$

• $\frac{1}{2}$

The point on the curve x² + y² = a², y $\ge$ 0 at which the tangent is parallel to x-axis is

• (a, 0)

• (- a, 0)

• $\left(\frac{\mathrm{a}}{2}, \frac{\sqrt{3}}{2}\mathrm{a}\right)$

• (0, a)