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.$

765.

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

The angle between the curves y = 4x + 4 and y² = 36(9 - x) is

• 30°

• 45°

• 60°

• 90°

If m and M respectively denote the minimum and maximum of f(x) = (x - 1)² + 3 for x $\in$ [- 3, 1], then the ordered pair (m, M) is equal to

• (- 3, 19)

• (3, 19)

• (- 19, 3)

• (- 19, - 3)

The length of the subtangent at (2, 2) to thecurve x⁵ = 2y⁴ is

• $\frac{5}{2}$

• $\frac{8}{5}$

• $\frac{2}{5}$

• $\frac{5}{8}$

There is an error of $\pm 0.04$cm in the measurement of the diameter of a sphere. When radius is 10cm, the percentage error in the volume of the sphere is

• $\pm 1.2$

• $\pm 1.0$

• $\pm 0.8$

• $\pm 0.6$