If \(x = e^{y + e^{y + \cdots\infty}}\), \(x > 0\), then \(\frac{dy}{dx}\) is equal to

If \(x = 2\sin\theta\) and \(y = 2\cos\theta\), then the value of \(\frac{d^2y}{dx^2}\) at \(\theta = 0\) 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:

A line makes the angle \(\theta\) with each of the \(x\) and \(z\) axes. If the angle \(\beta\) which it makes with \(y\)-axis is such that \(\sin^2\beta = 3\sin^2\theta\), then the value of \(\cos^2\theta\) is

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

If the vertices of a triangle \(ABC\) are \(A(1,2,1)\), \(B(4,2,3)\) and \(C(2,3,1)\), then the equation of the median passing through the vertex A, is

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 R is a relation on Z (set of all integers) defined by \(xRy\), if \(|x - y| \leq 1\), then (1) \(R\) is reflexive (2) \(R\) is symmetric (3) \(R\) is transitive (4) \(R\) is not symmetric (5) \(R\) is not transitive Choose the most appropriate answer from the options given bel…

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