The function \(f(x) = e^{|x|}\) is (A) continuous everywhere on \(R\) (B) not continuous at \(x = 0\) (C) Differentiable everywhere on \(R\) (D) not differentiable at \(x = 0\) (E) continuous and differentiable on \(R\) Choose the most appropriate answer from the options given below:

\(\int_{1/3}^{1}\frac{(x - x^3)^{1/3}}{x^4}\,dx =\)

[Figure: Mapping diagram of a relation f from set A = {a, b, c} to set B = {1, 2, 3, 4}. Arrows: a to 1, a to 3, b to 2, c to 4. So element a has two arrows (to 1 and to 3).] Which of the following is true on the basis of above diagram?

If \(0 < x < \pi\) and the matrix \(\begin{bmatrix}4\sin x & -1\\ -3 & \sin x\end{bmatrix}\) is singular, then the values of \(x\) are:

If the points \((2, -3)\), \((\lambda, -1)\) and \((0, 4)\) are collinear, then the value of \(\lambda\) is

The smaller of the areas enclosed by the circle \(x^2 + y^2 = 4\) and the line \(x + y = 2\) is

The value of \(\sin\left[2\cot^{-1}\left(\frac{-5}{12}\right)\right]\) is:

\(\sin^{-1}(1 - x) - 2\sin^{-1}x = \frac{\pi}{2}\), then \(x\) is equal to (A) 0 (B) 1 (C) \(\frac{1}{2}\) (D) 2 Choose the most appropriate answer from the options given below:

Let \(y = m\sin rx + n\cos rx\). What is the value of \(\frac{d^2y}{dx^2}\)?