If y = f(x² + 2) and f'(3) = 5, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ at x = 1 is

• 5

• 25

• 15

• 10

Let, f(x) = x² + bx + 7. If f'(5) = 2$\mathrm{f}\text{'}\left(\frac{7}{2}\right)$, then the value of b is

• 4

• 3

• - 4

• - 3

If $\mathrm{y} = {\sin }^{-1}\left(2\mathrm{x}\sqrt{1 - {\mathrm{x}}^2}\right), - \frac{1}{\sqrt{2}} \le \mathrm{x} \le \frac{1}{\sqrt{2}}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{\mathrm{x}}{\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{1}{\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{2}{\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{2\mathrm{x}}{\sqrt{1 - {\mathrm{x}}^2}}$

A straight line parallel to the line 2x - y + 5 = 0 is also a tangent to the curve y² = 4x + 5. Then, the point of contact is

• (2, 1)

• (- 1, 1)

• (1, 3)

• (3, 4)

The function f(x) = 2x³ - 15x² + 36x + 6 is strictly decreasing in the interval

• (2, 3)

• $\left(- \infty , 2\right)$

• (3, 4)

• $\left(- \infty , 3\right) \cup \left(4, \infty \right)$

The slope of the tangent to the curve y²e^xy = 9e^{- 3}x² at (- 1, 3) is

• $\frac{- 15}{2}$

• $\frac{- 9}{2}$

• 15

• $\frac{15}{2}$

The radius of a cylinder is increasing at the rate of 5 cm/min so that its volume is constant. When its radius is 5 cm and height is 3 cm, then the rate of decreasing of its height is

• 6 cm/min

• 3 cm/min

• 4 cm/min

• 5 cm/min

The function $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{2{\mathrm{x}}^2 - 1, \mathrm{if} 1 \le \mathrm{x} \le 4 \\ 151 - 30\mathrm{x}, \mathrm{if} 4 < \mathrm{x} \le 5\right\} \end{gathered}$$ is not suitable to apply Rolle's theorem, since

• f(x) is not continuous on [1, 5]

• f(1) $\ne$ f(5)

• f(x) is continuous only at x = 4

• f(x) is not differentiable at x = 4

9.

The slope of the normal to the curve y = x² - $\frac{1}{{\mathrm{x}}^2}$ at (- 1, 0) is

• $\frac{1}{4}$

• $- \frac{1}{4}$

• 4

• - 4

The minimum value of sin(x) + cos(x) is

• $\sqrt{2}$

• $- \sqrt{2}$

• $\frac{1}{\sqrt{2}}$

• $- \frac{1}{\sqrt{2}}$