Let f(x) = 2x³ - 9ax² + 12a²x + 1, where a > 0. The minimum of f is attained at a point q and the maximum is attained at a point p. If p = q, then a is equal to

• 1

• 3

• 2

• 0

The difference between the maximum and minimum value of of the function $\mathrm{f}\left(\mathrm{x}\right) = {\int }_0^{\mathrm{x}}\left({\mathrm{t}}^2 + \mathrm{t} + 1\right)\mathrm{dt}$ on [2, 3] is

• 39/6

• 49/6

• 59/6

• 69/6

If a and b are the non-zero distinct roots of x² + ax + b = 0, then the minimum value of x² + ax + b is

• 2/3

• 9/4

• - 9/4

• - 2/3

The equation of the tangent to the curve (1 + x²)y = 2 - x where it crosses the x-axis, is :

• x + 5y = 2

• x - 5y = 2

• 5x - y = 2

• 5x + y - 2 = 0

The sides of an equilateral triangle are increasing at the rate of 2 cm/s. The rate at which the area increases, when the side is 10 cm, is:

• $\sqrt{3} \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• 10 sq cm/s

• $10\sqrt{3} \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $\frac{10}{\sqrt{3}} \mathrm{sq} \mathrm{cm}/\mathrm{s}$

The function f(x) = 1 - x³ - x⁵ is decreasing for :

• $1 \le \mathrm{x} \le 5$

• $\mathrm{x} \le 1$

• $\mathrm{x} \ge 1$

• all values of x

127.

If PQ and PR are the two sides of a triangle, then the angle between them which gives maximum area of the triangle, is :

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

The function y = a(1 - cos(x)) is maximum when x is equal to

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $- \frac{\mathrm{\pi }}{2}$

• - $\frac{\mathrm{\pi }}{6}$

The maximum value of $\frac{\log \left(\mathrm{x}\right)}{\mathrm{x}}$ is equal to :

• $\frac{2}{\mathrm{e}}$

• $\frac{1}{\mathrm{e}}$

• e

• 1

If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing, is

• a constant

• proportional to the radius

• inversely proportional to the radius

• inversely proportional to the surface area