An angle $\theta, 0<\theta<\frac{\pi}{2}$, which increases twice as fast as its sine, is

Match the LIST-I with LIST-II

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The function $f: R \rightarrow R$ (where $R$ is set of real numbers) defined as

$f(x) = x^2 + 2x$ is

The function $f(x)=\sin^4(x)+\cos^4(x)$, on interval $\left(0,\dfrac{\pi}{2}\right)$, has

For the function $f(x) = \dfrac{3}{2}x^4 - 4x^3 - 45x^2 + 49$. Which of the following statements is/are TRUE ?

A. $f(x)$ is increasing in $(-\infty, -3)$

B. $f(x)$ is increasing in $(-3, 0) \cup (5, \infty)$

C. $f(x)$ is decreasing in $(-\infty, -3) \cup (0, 5)$

D. $f(x)$ is decreasing in $(-\infty, -3) \cup (5, \infty)$

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Interval in which the function $f$ given by $f(x) = \tan x - 4x$, $x \in (0, \frac{\pi}{2})$ is strictly decreasing is

Sand is pouring from a pipe at the rate of 12 $cm^3$/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm ?