Mathematics

Problem

Match the following.

Column I	Column II
(A) The function $f(x) = \frac{e^{x \cos x} - 1 - x}{\sin x^2}$ is not defined at $x = 0$ . The value of $f(0)$ , so that $f$ is continuous at $x = 0$ , is	(p) -1
(B) The value of the definite integral $\int_{0}^{1} \frac{dx}{\sqrt{x} + \sqrt[3]{x}}$ is equal to $a + b \ln 2$ , where $a$ and $b$ are integers, then $(a + b)$ is equal to	(q) 0
(C) Given, $e^n \int_{0}^{n} \frac{\sec^2 \theta - \tan \theta}{e^\theta} d\theta = 1$ , then the value of $\tan n$ is equal to	(r) 1/2
(D) Let $a_n = \int_{n}^{n+1} \tan^{-1}(nx) dx$ and $b_n = \int_{n}^{n+1} \sin^{-1}(nx) dx$ , then $\lim_{n \to \infty} \frac{a_n}{b_n}$ has the value equal to	(s) 1