The maximum area of a rectangle that can be inscribed in a circle of radius 2 units is

• $8\mathrm{\pi } \mathrm{sq} \mathrm{units}$

• 4 sq units

• 5 sq units

• 8 sq units

If the length of the subtangent at any point to the curve xyⁿ = a is proportional to the abscissa, then n is

• any non-zero real number

• 2

• - 2

• 1

The function f(x) = $\frac{\mathrm{x}}{3} + \frac{3}{\mathrm{x}}$ decreases in the interval

• (- 3, 3)

• $\left(- \infty , 3\right)$

• $\left(3, \infty \right)$

• (- 9, 9)

The tangent to the curve y = x³ + 1 at (1, 2) makes an angle $\mathrm{\theta }$ with y-axis, then the value of $\tan \left(\mathrm{\theta }\right)$ is

• $- \frac{1}{3}$

• 3

• - 3

• $\frac{1}{3}$

The local minimum value of the function f' given by f(x) = 3 + $\left|\mathrm{x}\right|, \mathrm{x} \in \mathrm{R}$ is

• - 1

• 3

• 1

• 0

A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At that instant, when the radius of circular wave is 8 cm, how fast is the enclosed area

• $6\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $8\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $\frac{8}{3} {\mathrm{cm}}^2/\mathrm{s}$

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

267.

A gardeneris digging a plot of land. As he gets tired, he works more slowly. After 't' minutes he is digging at a rate of $\frac{2}{\sqrt{\mathrm{t}}}$ m²/min . How long will it take him to dig an area of 40 sq m ?

• 100 min

• 10 min

• 30 min

• 40 min

Area of the region bounded by two parabolas y = x² and x = y² is

• $\frac{1}{4}$

• $\frac{{\displaystyle 1}}{{\displaystyle 3}}$

• 4

• 3

A balloon which always remains spherical is being inflated by pumping in 10 cu cm ofgas persecond. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm.

• $\frac{1}{30\mathrm{\pi }}\mathrm{cm}/\mathrm{s}$

• $\frac{1}{90\mathrm{\pi }}\mathrm{cm}/\mathrm{s}$

• $\frac{1}{\mathrm{\pi }}\mathrm{cm}/\mathrm{s}$

• $\frac{1}{9\mathrm{\pi }}\mathrm{cm}/\mathrm{s}$

If x is real, then the minimum value of x² - 8x + 17 is

• 3

• 1

• 4

• 2