The function f (x) = x² e^{- 2x}, x > 0. Then the maximum value of f (x) is :

• $\frac{1}{\mathrm{e}}$

• $\frac{1}{2\mathrm{e}}$

• $\frac{1}{{\mathrm{e}}^2}$

• $\frac{4}{{\mathrm{e}}^4}$

The angle between the tangents at those points on the curve x = t² + 1 and y = t² - t - 6 where it meets x-axis is :

• $\pm {\tan }^{-1}\left(\frac{4}{29}\right)$

• $\pm {\tan }^{-1}\left(\frac{5}{49}\right)$

• $\pm {\tan }^{-1}\left(\frac{10}{49}\right)$

• $\pm {\tan }^{-1}\left(\frac{8}{29}\right)$

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

67.

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, $\infty$)

Gas is being pumped into a spherical balloon at the rate of 30 ft³/min. Then, the rate at which the radius increases when it reaches the value 15 ft is

• $\frac{1}{15\mathrm{\pi }} \mathrm{ft}/\min$

• $\frac{1}{30\mathrm{\pi }} \mathrm{ft}/\min$

• $\frac{1}{20} \mathrm{ft}/\min$

• $\frac{1}{25} \mathrm{ft}/\min$

A point on curve xy² = 1 which is at minimum distance from the origin is

• (1, 1)

• (1/4, 2)

• (2^1/6, 2^{- 1/3})

• (2^{- 1/3}, 2^1/6)