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

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

The integrating factor of the differential equation \(\cos x\frac{dy}{dx} + y\sin x = 1\) is

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

The order and degree of the differential equation \(\left[\left(\frac{d^2y}{dx^2}\right) - 3\right]^{\frac{1}{3}} = 2\left(\frac{dy}{dx}\right)^{\frac{1}{4}}\) are

[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?

\(\int\sqrt{1 - 49x^2}\,dx\) is equal to

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 be…

The shortest distances of the point \((1,2,3)\) from \(x, y, z\) axes respectively are