A ring is dropped into a quiet pond and waves move in circles at a speed of 2 cm/second. At the instant, when the radius of the circular wave is 12 cm, how fast is the enclosed area increasing?

Match List-I with List-II

The function $f(x) = 2x^3 - 15x^2 + 36x + 5$ for $x \in [2,5]$ has

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Which of the following functions are increasing on $x \in \left(0, \frac{\pi}{2}\right)$?

(A) $f(x) = \sin x$

(B) $f(x) = \cos x$

(C) $f(x) = \tan x$

(D) $f(x) = \cos 3x$

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Let $f(x) = x^3 - 6x^2 + 9x - 8$ be a function, then which of the following statements are TRUE?

(A) $f'(x) = 3(x - 1)(x - 3)$

(B) The critical points of the function are $x = 1$ and $x = 3$

(C) $x = 1$ is the point of local minimum

(D) The local maximum value is $-4$

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The sum of two positive numbers is 60. If the sum of their squares in minimum, then the absolute value of the difference of their cubes is

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

The interval in which $f(x) = \frac{x}{2} + \frac{2}{x}$ is a decreasing function of $x$ is :