Assume \(P, Q, R\) and \(S\) are matrices of order \(2 \times m\), \(k \times n\), \(m \times 2\) and \(2 \times 3\) respectively. The restrictions on \(k, m\) and \(n\), so that \(PQ + RS\) is defined are

The system of equations \(3x + 4y = 5, 6x + 7y = -8\) is written in matrix form as

If \(2\begin{bmatrix}a & d\\ b & c\end{bmatrix} + 3\begin{bmatrix}1 & -1\\ 0 & 2\end{bmatrix} = 3\begin{bmatrix}3 & 5\\ 4 & 6\end{bmatrix}\), then the value of \(|a + b - c - d|\) is

Consider the function \(f(x) = x^{\frac{1}{x}}\). Its

The given function \(f(x) = x^5 - 5x^4 + 5x^3 - 1\) has/have (A) Local maxima at \(x = 1\) (B) Local maximum value is 0 (C) Local minimum at \(x = 3\) (D) Local minimum value is -28 (E) The point of inflexion is \(x = 1\) Choose the correct answer from the options given below

Match List-I with List-II List-I | List-II (A) If \(x = t^2\) and \(y = t^3\) then \(\frac{d^2y}{dx^2}\) at \(t = 1\) | (i) -2 (B) If \(f(x) = \sqrt{x} + 1\), the \(f^{\prime}(1)\) | (ii) -1 (C) The minimum value of \(f(x) = 9x^2 + 12x + 2\) is | (iii) \(\frac{3}{4}\) (D) The poi…

The area enclosed by the curve \(y^2 = 4ax\) and its latus rectum is

\(\int \frac{xe^x}{(x+1)^2}\,dx =\)

The solution of the differential equation \((x + 1)\frac{dy}{dx} = 1 + y\) is