let y = y(x) be the solution of the differential equation $\sin x \frac{dy}{dx} + y \cos x = 4x, x\in (0, \pi )$. If $y = \left(\frac{\pi }{2}\right) = 0, then y\left(\frac{\pi }{6}\right) is equal to:$

• $-\frac{4}{9}{\pi }^2$

• $\frac{4}{9\sqrt{3}}{\pi }^2$

• $\frac{-8}{9\sqrt{3}}{\pi }^2$

• $-\frac{8}{9}{\pi }^2$

If the line ax + by + c = 0, ab $\ne$ 0, is a tangent to the curve xy = 1- 2x, then

• a> 0, b < 0

• a>0, b> 0

• a< 0, b > 0

• a< 0, b < 0

Time period T of a simple pendulum of length l is given by T = $2\mathrm{\pi }\sqrt{\frac{1}{\mathrm{g}}}$. If the length is increased by 2%, then an approximate change in the time period is

• 2 %

• 1 %

• $\frac{1}{2}$ %

• None of these

The number of values of k, for which the equation x² - 3x + k=0 has two distinct roots lying in the interval (0, 1), are

• three

• two

• infinitely many

• no value of k satisfies the requirement

The smallest positive root of the equation tan(x) - x = 0 lies in

• $\left(0, \frac{\mathrm{\pi }}{2}\right)$

• $\left(\frac{\mathrm{\pi }}{2}, \mathrm{\pi }\right)$

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

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

$\mathrm{Let} \mathrm{y} = {\mathrm{e}}^{{\mathrm{x}}^2} \mathrm{and} \mathrm{y} = {\mathrm{e}}^{{\mathrm{x}}^2}\sin \left(\mathrm{x}\right)$be two given curves. Then, angle between the tangents to the curves at any point of their intersection is

• 0

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

Suppose that the equation f (x) = x² + bx + c = 0 has two distinct real roots $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$. The angle between the tangent to the curve y = f (x) at the point $\left(\frac{\mathrm{\alpha } + \mathrm{\beta }}{2}, \mathrm{f}\left(\frac{{\displaystyle \mathrm{\alpha } + \mathrm{\beta }}}{{\displaystyle 2}}\right)\right)$ and the positive direction of the x-axis is

• 0°

• 30°

• 60°

• 90°

The angle of intersection between the curves $\mathrm{y} = \left[\left|\sin \left(\mathrm{x}\right)\right| + \left|\cos \left(\mathrm{x}\right)\right|\right]$ and x² + y² = 10, where [x] denotes the greatest integer $\le \mathrm{x}$, is

• ${\tan }^{-1}\left(3\right)$

• ${\tan }^{-1}\left(- 3\right)$

• ${\tan }^{-1}\left(\sqrt{3}\right)$

• ${\tan }^{-1}\left(1/\sqrt{3}\right)$

For the curve x² + 4xy + 8y = 64 the tangents are parallel to the x-axis only at the points

• $\left(0, 2\sqrt{2}\right) \mathrm{and} \left(0, - 2\sqrt{2}\right)$

• (8, - 4) and (- 8, 4)

• $\left(8\sqrt{2}, - 2\sqrt{2}\right) \mathrm{and} \left(- 8\sqrt{2}, 2\sqrt{2}\right)$

• (9, 0) and (- 8, 0)

30.

Let exp (x) denote the exponential function e^x. If f (x) = $\exp \left({\mathrm{x}}^{\frac{1}{\mathrm{x}}}\right)$, x > 0, then the minimum value off in the interval [2, 5] is

• $\exp \left({\mathrm{e}}^{\frac{1}{\mathrm{e}}}\right)$

• $\exp \left(2^{\frac{1}{2}}\right)$

• $\exp \left(5^{\frac{1}{5}}\right)$

• $\exp \left(3^{\frac{1}{3}}\right)$