The function f(x) = $\log \left(1 + \mathrm{x}\right) - \frac{2\mathrm{x}}{2 + \mathrm{x}}$ is increasing on

• $\left(- 1, \infty \right)$

• $\left(- \infty , 0\right)$

• $\left(- \infty , \infty \right)$

• None of these

The minimum value of ${\mathrm{x}}^2 + \frac{1}{1 + {\mathrm{x}}^2}$ is at

• x = 0

• x = 1

• x = 4

• x = 3

The maximum value of $3 \cos \left(\mathrm{\theta }\right) + 4 \sin \left(\mathrm{\theta }\right)$ is

• 3

• 4

• 5

• None of these

The function x⁵ - 5x⁴ + 5x³ - 1 is

• neither maximum nor minimum at x = 0

• maximum at x = 0

• maximum at x = 1 and minimum at x = 3

• minimum at x = 0

215.

If the radius of a circle be increasing at a uniform rate of 2 cm/s. The area of increasing of area of circle, at the instant when the radius is 20 cm, is

• $70 \mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $70 {\mathrm{cm}}^2/\mathrm{s}$

• $80 \mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $80 {\mathrm{cm}}^2/\mathrm{s}$

The equation of normal at the point (0, 3) of the ellipse 9x² + 5y² = 45 is

• x-axis

• y-axis

• y + 3 = 0

• y - 3 = 0

The maximum value of x^1/x is :

• 1/e^e

• e

• e^1/e

• 1/e

The function f defined by f(x) = 4x⁴ - 2x + 1 is increasing for :

• x < 1

• x > 0

• x < 1/2

• x > 1/2

A particle moves in a straight line so that s = $\sqrt{\mathrm{t}}$, then its acceleration is proportional to :

• (velocity)³

• velocity

• (velocity)²

• (velocity)^3/2

If the line ax + by + c = 0 is a normal to the curve y =1, then :

• a > 0, b > 0

• a > 0, b < 0

• a < 0, b < 0

• Data is insufficient