Application of Derivatives

P is the point of contact of the tangent form the origin to the curve y = log_e(x). The length of the perpendicular drawn form the origin to the normal at P is

• $\frac{1}{2\mathrm{e}}$

• $\frac{1}{\mathrm{e}}$

• $2\sqrt{{\mathrm{e}}^2 + 1}$

• $\sqrt{{\mathrm{e}}^2 +\hspace{0.17em}1}$

For the curve 4x⁵ = 5y⁴, the ratio of the cube of the subtangent at a point on the curve the square of the subnormal at the same point is

• $\frac{4^4}{5}$

• $\frac{5^4}{4}$

• $\frac{4^4}{5^4}$

• ${\left(\frac{5}{4}\right)}^4$

The set of real values of x for which f(x) = $\frac{\mathrm{x}}{\log \left(\mathrm{x}\right)}$ is increasing is

• $\left\{\mathrm{x} :\hspace{0.17em}\mathrm{x} \ge \mathrm{e}\right\}$

• empty

• {x : x < e}

• {1}

A sphere increases its volume at the rate of $\mathrm{\pi }$ cc/s. The rate at which its surface area increases when the radius is 1 cm is

• $2\mathrm{\pi } \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $\mathrm{\pi } \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $\frac{3\mathrm{\pi }}{2} \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $\frac{\mathrm{\pi }}{2} \mathrm{sq} \mathrm{cm}/\mathrm{s}$

If a ball is thrown vertically upwards and the height 's' reached in time 't' is given by s = 22t - 11t², then the total distance travelled by the ball is

• 44 units

• 33 units

• 11 units

• 22 units

The sum of two positive numbers is given. If the sum of their cubes is minimum, then

• they are equal

• one is twice the other

• they are unequal

• one is thrice the other

The perimeter of a sectoris a constant. Ifits area is to be maximum, then the sectorial angle is

• $\frac{{\mathrm{\pi }}^{\mathrm{c}}}{6}$

• $\frac{{\mathrm{\pi }}^{\mathrm{c}}}{4}$

• 4^c

• 2^c

The tangent to the curve xy = 25 at any point on it cuts the coordinate axes at A and B, then the area of the $∆$OAB is

• 50 sq units

• 25 sq units

• 75 sq units

• 100 sq units

The length of the subtangent, ordinate and the subnormal are in

• AP

• HP

• GP

• Arithmetico-geometric progression

The maximum value of xe^-x is

• e

• $\frac{1}{\mathrm{e}}$

• - e

• - $\frac{1}{\mathrm{e}}$