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 :

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…

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

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:

If the random variable X has the following distribution : X | 0 | 1 | 2 | otherwise P(X) | k | 2k | 3k | 0 Match List-I with List-II : List-I | List-II (A) k | (I) \(\frac{5}{6}\) (B) P(X < 2) | (II) \(\frac{4}{3}\) (C) E(X) | (III) \(\frac{1}{2}\) (D) P(1 \(\le\) X \(\le\) 2) |…

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 :

For a square matrix \(A_{n\times n}\) (A) \(|\operatorname{adj} A| = |A|^{n-1}\) (B) \(|A| = |\operatorname{adj} A|^{n-1}\) (C) \(A(\operatorname{adj} A) = |A|\) (D) \(|A^{-1}| = \frac{1}{|A|}\) Choose the correct answer from the options given below :

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 matrix \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\) is a : (A) scalar matrix (B) diagonal matrix (C) skew-symmetric matrix (D) symmetric matrix