\(f(x) = \sin x + \frac{1}{2}\cos 2x\) 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 \(\frac{3}{4}\) Choose the correct answer from the options below:

\(\Delta = \begin{vmatrix} 1 & \cos x & 1 \\ -\cos x & 1 & \cos x \\ -1 & -\cos x & 1 \end{vmatrix}\) (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

The direction cosines of the line which is perpendicular to the lines with direction ratios \(1, -2, -2\) and \(0, 2, 1\) are:

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 :

Let \(X\) denote the number of hours you play during a randomly selected day. The probability that \(X\) can take values \(x\) has the following form, where \(c\) is some constant. \(P(X = x) = 0.1\) if \(x = 0\); \(cx\) if \(x = 1\) or \(x = 2\); \(c(5 - x)\) if \(x = 3\) or…

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 :

\(\sin y = x\sin(a + y)\), then \(\frac{dy}{dx}\) is :

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

The unit vector perpendicular to each of the vectors \(\vec{a} + \vec{b}\) and \(\vec{a} - \vec{b}\), where \(\vec{a} = \hat{i} + \hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}\), is: