\(\int e^x \left(\frac{2x+1}{2\sqrt{x}}\right) dx =\)

The area of the region enclosed between the curves \(4x^2 = y\) and \(y = 4\) is:

If \(f(x)\), defined by \(f(x) = kx + 1\) if \(x \le \pi\) and \(f(x) = \cos x\) if \(x > \pi\), is continuous at \(x = \pi\), then the value of \(k\) is :

The feasible region represented by the constraints \(4x + y \ge 80, x + 5y \ge 115, 3x + 2y \le 150, x, y \ge 0\) of an LPP is [Figure: graph with x and y axes showing the three lines. Marked points: (0,80), (0,75), (2,72), (0,23), (15,10), (40,15), (20,0), (50,0), (115,0) and th…

If \(P = \begin{bmatrix} -1 \\ 2 \\ 1 \end{bmatrix}\) and \(Q = \begin{bmatrix} 2 & -4 & 1 \end{bmatrix}\) are two matrices, then \((PQ)'\) will be :

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

\(\Delta = \begin{vmatrix} 1 & \cos x & 1 \\ -\cos x & 1 & \cos x \\ -1 & -\cos x & 1 \end{vmatrix}\) (A) \(\Delta = 2(2 - \cos^2 x)\) (B) \(\Delta = 2(2 - \sin^2 x)\) (C) Minimum value of \(\Delta\) is 2 (D) Maximum value of \(\Delta\) is 4

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 :

\(f(x) = \sin x + \frac{1}{2}\cos 2x\) in \([0, \frac{\pi}{2}]\) (A) \(f'(x) = \cos x - \sin 2x\) (B) The critical points of the function are \(x = \frac{\pi}{6}\) and \(x = \frac{\pi}{2}\) (C) The minimum value of the function is 2 (D) The maximum value of the function is…