If the straight line y - 2x+ 1 = 0 is the tangent to the curve xy + ax + by = 0 at x = i, then the values of a and b are respectively

• 1 and 2

• 1 and - 1

• - 1 and 2

• 1 and - 2

If the angle between the curves y = 2^x and y = 3^x is $\mathrm{\alpha }$, then the value of tan($\mathrm{\alpha }$) is equal to

• $\frac{\log \left({\displaystyle \frac{3}{2}}\right)}{1 + \log \left(2\right)\log \left(3\right)}$

• $\frac{6}{7}$

• $\frac{1}{7}$

• $\frac{\log \left({\displaystyle 6}\right)}{1 + \log \left(2\right)\log \left(3\right)}$

The function f(x) = 3x³ - 36x + 99 is increasing for

• $- \infty < \mathrm{x} < 2$

• $- 2 < \mathrm{x} < \infty$

• - 2 < x < 2

• x < - 2 or x > 2

The minimum value of the function $\mathrm{f}\left(\mathrm{x}\right) = \frac{1}{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}$ in the interval $\left[0, \frac{\mathrm{\pi }}{2}\right]$ is

• $\frac{\sqrt{2}}{2}$

• $- \frac{\sqrt{2}}{2}$

• $\frac{2}{\sqrt{3} + 1}$

• $-\frac{2}{\sqrt{3} + 1}$

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

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

• $\frac{1}{2}$

• $\frac{\sqrt{2}}{2}$

• $\frac{1}{4}$

• $\frac{\sqrt{2}}{4}$