The equation of the tangent to the curve $\sqrt{\frac{\mathrm{x}}{\mathrm{a}}} + \sqrt{\frac{\mathrm{y}}{\mathrm{b}}}$ = 1 at the point (x₁, y₁) is $\frac{\mathrm{x}}{\sqrt{{\mathrm{ax}}_1}} + \frac{\mathrm{y}}{\sqrt{{\mathrm{by}}_1}} = \mathrm{k}$. Then, the value of k is

• 2

• 1

• 3

• 3

The slope of the normal to the curve x = t² + 3t - 8 and y = 2t² - 2t - 5 at the point (2, - 1) is

• $\frac{6}{7}$

• $- \frac{6}{7}$

• $\frac{7}{6}$

• - $\frac{7}{6}$

If the slope of y = 3x² + ax³ is maximum at x = $\frac{1}{2}$, then the value of a is

• 2

• 1

• - 1

• - 2

534.

If y = 4x - 5 is a tangent to the curve y = px³ + q at (2, 3), then (p + q) is equal to

• - 5

• 5

• - 9

• 9

The point on the curve y = 5 + x - x² at which the normal makes equal intercepts is

• (1, 5)

• (0, - 1)

• (- 1, 3)

• (0, 5)

If the point (a, b) on the curve y = $\sqrt{\mathrm{x}}$ is close to the point (1, 0), then the value of ab is

• $\frac{1}{2}$

• $\frac{\sqrt{2}}{2}$

• $\frac{1}{4}$

• $\frac{\sqrt{2}}{4}$

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