If \[ \int x \sin x \sec^3 x \, dx = \frac{1}{2} \left[ f(x) \sec^2 x + g(x) \left( \frac{\tan x}{x} \right) \right] + c, \] \(\text{then which of the following is true?}\)
Let \( \mathbf{A} = 2\hat{i} + \hat{j} - 2\hat{k} \) and \( \mathbf{B} = \hat{i} + \hat{j} \). If \( \mathbf{C} \) is a vector such that \( |\mathbf{C} - \mathbf{A}| = 3 \) and the angle between \( \mathbf{A} \times \mathbf{B} \) and \( \mathbf{C} \) is \( 30^\circ \), then \( [(\mathbf{A} \times \mathbf{B}) \times \mathbf{C}] = 3 \), the value of \( \mathbf{A} \cdot \mathbf{C} \) is equal to:
Consider the following frequency distribution table. \[ \begin{array}{|c|c|} \hline \textbf{Class Interval} & \textbf{Frequency} \\ \hline 10-20 & 180 \\ \hline 20-30 & f_1 \\ \hline 30-40 & 34 \\ \hline 40-50 & 180 \\ \hline 50-60 & 136 \\ \hline 60-70 & 50 \\ \hline 70-80 & f_2 \\ \hline \end{array} \] If the total frequency is 685 and the median is 42.6, then the values of \( f_1 \) and \( f_2 \) are
Consider the circuit shown below and find the minimum number of NAND gates required to design it.
Consider the following minterm expression for \( F \): \[ F(P, Q, R, S) = \Sigma(0, 2, 5, 7, 8, 10, 13, 15) \] \(\text{The minterms 2, 7, 8, and 13 are don't care terms. The minimal sum of products form for F is}\)
