The radius of a circular plate is increasing at the rate of 0.01 cm/s when the radius is 12 cm. Then, the rate at which the area increases, is

• 0.24 $\mathrm{\pi } \mathrm{cm}/\mathrm{s}$

• $60 \mathrm{\pi } \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $24\mathrm{\pi } \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $1.2\mathrm{\pi } \mathrm{sq} \mathrm{cm}/\mathrm{s}$

Observe the following statements

A: f(x) = 2x³ - 9x² + 12x - 3 is increasing outside the interval (1, 2)

R : f'(x) < 0 for x $\in$ (1, 2).

Then, which of the following is true ?

• Both A and R are true, and R is not the correct reason for A

• Both A and R are true, and R is the correct reason for A

• A is true but R is false

• A is false but R is true

If $\mathrm{\theta }$ is the angle between the curves xy = 2 and x² + 4y = 0 and x² + 4y = 0, then tan$\mathrm{\theta }$ is equal to :

• 1

• - 1

• 2

• 3

In the interval(- 3, 3) the function f(x) = $\frac{\mathrm{x}}{3} + \frac{3}{\mathrm{x}}, \mathrm{x} \ne 0$ is

• increasing

• decreasing

• neither increasing nor decreasing

• partly increasing and partly decreasing

The perimeter of a sector is a constant. If its area is to be maximum, the sectorial angle is :

• $\frac{{\mathrm{\pi }}^{\mathrm{c}}}{6}$

• $\frac{{\mathrm{\pi }}^{\mathrm{c}}}{4}$

• $4^{\mathrm{c}}$

• $2^{\mathrm{c}}$

The lengths of tangent, subtangent, normal and subnormal for the curve y = x² + x - 1 at (1, 1) are A, B, C and D respectively, then their increasing order is

• B, D, A, C

• B, A, C, D

• A, B, C, D

• B, A, D, C

The condition f(x) = x³ + px² + qx + r(x ∈ R) to have no extreme value, is

• ${\mathrm{p}}^2 < 3\mathrm{q}$

• $2{\mathrm{p}}^2 < \mathrm{q}$

• ${\mathrm{p}}^2 < \frac{1}{4}\mathrm{q}$

• ${\mathrm{p}}^2 > 3\mathrm{q}$

The circumference of a circle is measured as 56cm with an error 0.02 cm. The percentage error in its area is

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$7$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$28$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$14$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$56$}\right.$

Observe the statements given below :

Assertion (A) : f(x) = xe^{- x} has the maximum at x =1

Reason (R) : f'(1) = 0 and f'(1) < 0

Which of the following is correct ?

• Both (A) and (R) are true and (R) is the correct reason for (A)

• Both (A) and (R) are true, but (R) is not the correct reason for (A)

• (A) is true, (R) is false

• (A) is false, (R) is true

The equation of the normal to the curve y⁴ = ax³ at (a, a) is

• x + 2y = 3a

• 3x - 4y + a = 0

• 4x + 3y = 7a

• 4x - 3y = 0