Application of Derivatives

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

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

• $\sqrt{3}$

• $\frac{\sqrt{2}}{3}$

If a particle moves such that the displacement is proportional to the square of the velocity acquired, then its acceleration is :

• proportional to s²

• proportional to $\frac{1}{{\mathrm{s}}^2}$

• proportional to s

• a constant

The maximum value of xy when x + 2y = 8 is :

• 20

• 16

• 24

• 8

The function f(x) = tan^-1(sin(x) + cos(x)), x > 0 is always an increasing function on the interval :

• $\left(0, \mathrm{\pi }\right)$

• $\left(0, \frac{\mathrm{\pi }}{2}\right)$

• $\left(0, \frac{\mathrm{\pi }}{4}\right)$

• $\left(0, \frac{3\mathrm{\pi }}{4}\right)$

The radius of a cylinder is increasing at the rate of 3 m/s and its altitude is decreasing at the rate of 4 m/s. The rate of change of volume when radius is 4 m and altitude is 6 m, is :

• $80 \mathrm{\pi } \mathrm{cu} \mathrm{m}/\mathrm{s}$

• $144 \mathrm{\pi } \mathrm{cu} \mathrm{m}/\mathrm{s}$

• $- 80 \mathrm{\pi } \mathrm{cu} \mathrm{m}/\mathrm{s}$

• $64 \mathrm{\pi } \mathrm{cu} \mathrm{m}/\mathrm{s}$

A ladder 10 m long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of 3 emfs. The height of the upper end while it is descending at the rate of 4 emfs, is :

• 4$\sqrt{3}$

• 5$\sqrt{3}$

• 5$\sqrt{2}$

• 6 m

The equation of the tangent to the curve $\mathrm{y} = {\left(1 + \mathrm{x}\right)}^{\mathrm{y}} + {\sin }^{-1}\left({\sin }^2\left(\mathrm{x}\right)\right)$

• x - y + 1 = 0

• x + y + 1 = 0

• 2x - y + 1 = 0

• x + 2y + 2 = 0