If f'(x) > 0, $\mathrm{x} \in \mathrm{R}, \mathrm{f}\text{'}\left(3\right) = 0$ and g(x) = $\mathrm{f}\left({\tan }^2\left(\mathrm{x}\right) - 2\tan \left(\mathrm{x}\right) + 4\right), 0 < \mathrm{x} < \frac{\mathrm{\pi }}{2}$, then g(x) is increasing in

• $\left(0, \frac{\mathrm{\pi }}{4}\right)$

• $\left(\frac{\mathrm{\pi }}{6}, \frac{\mathrm{\pi }}{3}\right)$

• $\left(0, \frac{\mathrm{\pi }}{3}\right)$

• $\left(\frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{2}\right)$

The radius of a cylinder is increasing at the rate of 2 m/s and its height is decreasing at the rate of 3 m/s. When the radius is 3 m and height is 5 m, then the volume of the cylinder would change at the rate of

• $87\mathrm{\pi } {\mathrm{m}}^3/\mathrm{s}$

• $33\mathrm{\pi } {\mathrm{m}}^3/\mathrm{s}$

• $27\mathrm{\pi } {\mathrm{m}}^3/\mathrm{s}$

• $15\mathrm{\pi } {\mathrm{m}}^3/\mathrm{s}$

The values of a, if f(x) =$2{\mathrm{e}}^{\mathrm{x}} - {\mathrm{ae}}^{-\mathrm{x}} + \left(2\mathrm{a} +\hspace{0.17em}1\right)\mathrm{x} - 3$  increases x, are in

• $[0, \infty )$

• $(- \infty , 0]$

• $\left(- \infty , \infty \right)$

• $\left(1, \infty \right)$

A cylindrical tank of radius 2 m is being filled with rice at the rate of 314 cubic m/h. The depth ofthe rice is increasing at the rate of

• 25 cubic m/h

• 0.25 cubic m/h

• 1 cubic m/h

• $\frac{3}{4}$ cubic m/h

The distance of the point on the curve x² = 2y, which is nearest to the pont (0, 5) is

• 3

• 4

• $2\sqrt{2}$

• None of these

The minimum value of $\left(\mathrm{x} - \mathrm{\alpha }\right)\left(\mathrm{x} - \mathrm{\beta }\right)$ is

• 0

• $\mathrm{\alpha \beta }$

• $\frac{1}{4}{\left(\mathrm{\alpha } - \mathrm{\beta }\right)}^2$

• $- \frac{1}{4}{\left(\mathrm{\alpha } - \mathrm{\beta }\right)}^2$

The equation of tangent to the curve 6y = 7 - x³ at (1, 1) is

• 2x + y = 3

• x + 2y = 3

• x + y = 1

• x + y + 2 = 0

The maximum value of xy subject to x + y = 7 is

• 10

• 12

• $\frac{49}{4}$

• $\frac{55}{4}$

The family of curves in which the sub-tangent at any point to any curve is double the abscissa is given by

• x = Cy²

• y = Cx²

• x² = Cy²

• y² = Cx²

310.

If log (1 + x) -  $\frac{2\mathrm{x}}{2 + \mathrm{x}}$ is increasing, then

• $0 < \mathrm{x} < \infty$

• $- \infty < \mathrm{x} < 0$

• $- \infty < \mathrm{x} < \infty$

• $- 1 < \mathrm{x} < 2$