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

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

The length of the subtangent to the curve x² + xy + y² = 7 at (1, - 3) is :

• 3

• 5

• $\frac{3}{5}$

• 15

Twenty two metres are available to fence a flower bed in the form of a circular sector. If the flower bed should have the greatest possible surface area, the radius of the circle must be :

• 4 m

• 3 m

• 6 m

• 5 m

569.

The value of x for which the polynomial 2x³ - 9x² + 12x + 4 is a decreasing function of x, is :

• - 1 < x < 1

• 0 < x < 2

• x > 3

• 1 < x < 2

The length of the longest size rectangle of maximum area that can be inscribed in a semicircle of radius 1, so that 2 vertices lie on the diameter, is :

• $\sqrt{2}$

• 2

• $\sqrt{3}$

• $\frac{\sqrt{2}}{3}$