The area of the region bounded by the lines \(\frac{x}{7\sqrt{3}a} + \frac{y}{b} = 4\), \(x = 0\) and \(y = 0\) is :

The rate of change (in cm\(^2\)/s) of the total surface area of a hemisphere with respect to radius r at \(r = \sqrt[3]{1.331}\) cm is :

If \(A\) is a square matrix and \(I\) is an identity matrix such that \(A^2 = A\), then \(A(I - 2A)^3 + 2A^3\) is equal to:

Let \([x]\) denote the greatest integer function. Then match List-I with List-II : List-I | List-II (A) |x-1|+|x-2| | (I) is differentiable everywhere except at x = 0 (B) x-|x| | (II) is continuous everywhere (C) x-[x] | (III) is not differentiable at x = 1 (D) x |x| | (IV) is di…

Match List-I with List-II List-I | List-II (A) Integrating factor of \(x\,dy - (y + 2x^2)dx = 0\) | (I) \(\frac{1}{x}\) (B) Integrating factor of \((2x^2 - 3y)dx = x\,dy\) | (II) \(x\) (C) Integrating factor of \((2y + 3x^2)dx + x\,dy = 0\) | (III) \(x^2\) (D) Integrating factor…

If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three vectors such that \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), where \(\vec{a}\) and \(\vec{b}\) are unit vectors and \(|\vec{c}| = 2\), then the angle between the vectors \(\vec{b}\) and \(\vec{c}\) is :

If the function \(f: \mathbb{N} \to \mathbb{N}\) is defined as \(f(n) = n - 1\) if \(n\) is even, and \(f(n) = n + 1\) if \(n\) is odd, then (A) f is injective (B) \(f\) is into (C) \(f\) is surjective (D) f is invertible Choose the correct answer from the options given below :

The value of the integral \(\int_{\log_e 2}^{\log_e 3} \frac{e^{2x} - 1}{e^{2x} + 1}\,dx\) is :

\(\int_0^{\pi/2} \frac{1 - \cot x}{\operatorname{cosec} x + \cos x}\,dx =\)