If f(x) = ${\log }_{{\mathrm{x}}^3}\left({\log }_{\mathrm{e}}\left({\mathrm{x}}^2\right)\right)$, then f'(x) at x = e is

• $\frac{1}{3\mathrm{e}}\left(1 - {\log }_{\mathrm{e}}\left(2\right)\right)$

• $\frac{1}{3\mathrm{e}}\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right)$

• $\frac{1}{3\mathrm{e}}\left(- 1 + {\log }_{\mathrm{e}}\left(2\right)\right)$

• $- \frac{1}{3\mathrm{e}}\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right)$

12.

If f(x) = (x - 2)(x - 4)(x - 6) ... (x - 2n), then f'(2) is

• (- 1)ⁿ2^{n - 1} (n - 1)!

• (- 2)^{n - 1} (n - 1)!

• (- 2)ⁿ n!

• (- 1)^{n - 1}2ⁿ (n - 1)!

If $\mathrm{\theta }$ is semi vertical angle of a cone of maximum volume and given slant height, then tan($\mathrm{\theta }$) is equal to

• 2

• 1

• $\sqrt{2}$

• $\sqrt{3}$

A man of 2 m height walks at a uniform speed of 6 km/h away from a lamp post of 6 m height. The rate at which the length of his shadow increase, is

• 2 km/h

• 1 km/ h

• 3 km/h

• 6 km/h

If y = 4x-5 is a tangent to the curve y² = px³ + q at (2, 3), then

• p = 2, q = - 7

• p = - 2, q = 7

• p = - 2, q = - 7

• p = 2, q = 7

A missile is fired from the fround level rises x metre vertically upwards in t second, where x = 100t - $\frac{25}{2}$t². The maximum height recahed is

• 200 m

• 125 m

• 160 m

• 190 m

If the curves x² = 9A(9 - y) and x² = A(y + 1) intersect orthogonally, then the value of A is

• 3

• 4

• 5

• 7

If f (x) = 3x⁴ + 4x³ - 12x² + 12, then f(x) is

• increasing in (- $\infty$, - 2) and in (0, 1)

• increasing in (- 2, 0) and in (1, $\infty$)

• decreasing in (- 2, 0) and in (0, 1)

• decreasing in (- $\infty$, - 2) and in (1, $\infty$)

In a $∆\mathrm{ABC}$, if $\left(\sqrt{3} - 1\right)$a = 2b, A = 3B, then C is

• $60°$

• 120$°$

• 30°

• 45°

The x - axis, y - axis and a line passing through the point A (6, 0) from a $∆$ ABC. If $\angle$A = 30°, then the area of the $∆$, in sq unit is

• $6\sqrt{3}$

• $12\sqrt{3}$

• $4\sqrt{3}$

• 8$\sqrt{3}$