The function $\mathrm{y} = \sqrt{2\mathrm{x} - {\mathrm{x}}^2}$

• increases in (0, 1) but decreases in (1, 2)

• decreases in (0, 2)

• increases m (1, 2) but decreases in (0, 1)

• increases in (0, 2)

The interval in which the function $\mathrm{y} = \mathrm{x} - 2\sin \left(\mathrm{x}\right)$; $0 \le \mathrm{x} \le 2\mathrm{\pi }$ increases throughout is

• $\left(\frac{5\mathrm{\pi }}{3}, 2\mathrm{\pi }\right)$

• $\left(0, \frac{\mathrm{\pi }}{3}\right)$

• $\left(\frac{\mathrm{\pi }}{3}, \frac{5\mathrm{\pi }}{3}\right)$

• $\left(0, \frac{\mathrm{\pi }}{4}\right)$

623.

The points of the curve y = x³ + x - 2 at which its tangent are parallel to the straight line y = 4x - 1 are

• (2, 7), (- 2, - 11)

• (0, 2), (2^1/3, 2^1/3)

• (- 2^1/3, - 2^1/3), (0, - 4)

• (1, 0), (- 1, - 4)

The equation of the normal to the curve y = - $\sqrt{\mathrm{x}}$ + 2 at the point of its intersection with the bisector of the first quadrant is

• 4x - y + 16 = 0

• 4x - y = 16

• 2x - y - 1 = 0

• 2x - y + 1 = 0

The angle at which the curve y = x² and the curve x = $\frac{5}{3}\cos \left(\mathrm{t}\right)$, y = $\frac{5}{4}\sin \left(\mathrm{t}\right)$ intersect is

• ${\tan }^{-1}\left(\frac{2}{41}\right)$

• ${\tan }^{-1}\left(\frac{41}{2}\right)$

• $- {\tan }^{-1}\left(\frac{2}{41}\right)$

• $2{\tan }^{-1}\left(\frac{41}{2}\right)$

The maximum value of the function $\mathrm{y} = 2\tan \left(\mathrm{x}\right) - {\tan }^2\left(\mathrm{x}\right)$ over $\left[0, \frac{\mathrm{\pi }}{2}\right]$ is

• $\infty$

• 1

• 3

• 2

The values of a and b for which the function y = alog_e(x ) + bx² + x, has extremum at the points x₁ = 1 and x₂ = 2 are

• $\mathrm{a} = \frac{2}{3}, \mathrm{b} = - \frac{1}{6}$

• $\mathrm{a} = - \frac{2}{3}, \mathrm{b} = - \frac{1}{6}$

• $\mathrm{a} = - \frac{2}{3}, \mathrm{b} = \frac{1}{6}$

• $\mathrm{a} = - \frac{1}{3}, \mathrm{b} = - \frac{1}{6}$

The value of maxima of ${\left(\frac{1}{\mathrm{x}}\right)}^{\mathrm{x}}$ is

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

• e^e

• e

• e^1/e

A point particle moves along a straight line such that x = $\sqrt{\mathrm{t}}$, where t is time. Then, ratio of acceleration to cube of the velocity is

• - 1

• - 0.5

• - 3

• - 2

The tangents to curve y = x³ - 2x² + x - 2 which are parallel to straight line y = x, are

• $\mathrm{x} + \mathrm{y} = 2 \mathrm{and} \mathrm{x} - \mathrm{y} = \frac{86}{27}$

• $\mathrm{x} - \mathrm{y} = 2 \mathrm{and} \mathrm{x} - \mathrm{y} = \frac{86}{27}$

• $\mathrm{x} - \mathrm{y} = 2 \mathrm{and} \mathrm{x} + \mathrm{y} = \frac{86}{27}$

• $\mathrm{x} + \mathrm{y} = 2 \mathrm{and} \mathrm{x} + \mathrm{y} = \frac{86}{27}$