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

If \(A, B\) and \(C\) are three singular matrices given by \(A = \begin{bmatrix} 1 & 4 \\ 3 & 2a \end{bmatrix}\), \(B = \begin{bmatrix} 3b & 5 \\ 4 & 2 \end{bmatrix}\) and \(C = \begin{bmatrix} a+b+c & c+1 \\ a+c & c \end{bmatrix}\), then the value of abc is:

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 \(\tan^{-1}\left(\frac{2}{3^{-x}+1}\right) = \cot^{-1}\left(\frac{3}{3^{x}+1}\right)\), then which one of the following is true?

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…

If \((\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 27\) and \(|\vec{a}| = 2|\vec{b}|\), then \(|\vec{b}|\) 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 :

The angle between two lines whose direction ratios are proportional to \(1, 1, -2\) and \((\sqrt{3} - 1), (-\sqrt{3} - 1), -4\) is:

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 :