Application of Derivatives

The angle between the curves y = 4x + 4 and y² = 36(9 - x) is

• 30°

• 45°

• 60°

• 90°

If m and M respectively denote the minimum and maximum of f(x) = (x - 1)² + 3 for x $\in$ [- 3, 1], then the ordered pair (m, M) is equal to

• (- 3, 19)

• (3, 19)

• (- 19, 3)

• (- 19, - 3)

The length of the subtangent at (2, 2) to thecurve x⁵ = 2y⁴ is

• $\frac{5}{2}$

• $\frac{8}{5}$

• $\frac{2}{5}$

• $\frac{5}{8}$

There is an error of $\pm 0.04$cm in the measurement of the diameter of a sphere. When radius is 10cm, the percentage error in the volume of the sphere is

• $\pm 1.2$

• $\pm 1.0$

• $\pm 0.8$

• $\pm 0.6$

$\mathrm{The} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^3 +\hspace{0.17em}{\mathrm{ax}}^2 + \mathrm{bx} +\hspace{0.17em}\mathrm{c}, {\mathrm{a}}^2 \le 3\mathrm{b} \mathrm{has}$

• one maximum value

• one minimum value

• no extreme value

• one maximum and one minimum value

$\mathrm{The} \mathrm{maximum} \mathrm{value} \mathrm{of} \frac{\log \left(\mathrm{x}\right)}{\mathrm{x}}, 0 < \mathrm{x} < \infty \mathrm{is}$

• $\infty$

• e

• 1

• e ^{- 1}

$\mathrm{z} = \tan \left(\mathrm{y} +\hspace{0.17em}\mathrm{ax}\right) +\hspace{0.17em}\sqrt{\mathrm{y} - \mathrm{ax}} \Rightarrow {\mathrm{z}}_{\mathrm{xx}} - {\mathrm{a}}^2{\mathrm{z}}_{\mathrm{yy}} =?$

• 0

• 2

• ${\mathrm{z}}_{\mathrm{x}} + {\mathrm{z}}_{\mathrm{y}} = 0$

• ${\mathrm{z}}_{\mathrm{x}}{\mathrm{z}}_{\mathrm{y}}$

The height of the cone of maximum volume inscribed in a sphere of radius R is

• $\frac{\mathrm{R}}{3}$

• $\frac{{\displaystyle 2\mathrm{R}}}{{\displaystyle 3}}$

• $\frac{4\mathrm{R}}{3}$

• $\frac{4\mathrm{R}}{\sqrt{3}}$

If the distance s travelled by a particle in time t is given by s = t² - 2t + 5, then its acceleration is

• 0

• 1

• 2

• 3

The length of the sub tangent at any point (x₁, y₁) on the curve y = 5^x is

• $5^{{\mathrm{x}}_1}$

• ${\mathrm{y}}_1{5}^{{\mathrm{x}}_1}$

• ${\log }_{\mathrm{e}}\left(5\right)$

• $\frac{1}{{\log }_{\mathrm{e}}\left(5\right)}$