If the probability density function of a random variable is given by $$ f(x) = \begin{cases} 12x^2(1 - x), & 0 \leq x \leq 1 \\ 0, & \text{elsewhere} \end{cases} $$ then the mean and variance are respectively:
The area of a parallelogram whose diagonals are given by $ \vec{u} + \vec{v} $ and $ \vec{v} + \vec{w} $, where: $ \vec{u} = 2\hat{i} - 3\hat{j} + \hat{k}, \quad \vec{v} = -\hat{i} + \hat{k}, \quad \vec{w} = 2\hat{j} - \hat{k} $ is:
The direction ratios of the normal to the plane passing through the points $ (1, 2, -3), \quad (1, -2, 1) \quad \text{and parallel to the line} \quad \frac{x - 2}{2} = \frac{y + 1}{3} = \frac{z}{4} \text{ is:} $
For what values of $ \lambda $ and $ \mu $, the following system of equations has a unique solution? $ 2x + 3y + 5z = 9 $$ 7x + 3y - 2z = 8 $ $ 2x + 3y + \lambda z = \mu $
Let the function $ f(x) $ be defined as follows: $$ f(x) = \begin{cases} (1 + | \sin x |)^{\frac{a}{|\sin x|}}, & -\frac{\pi}{6}<x<0 \\b, & x = 0 \\ \frac{\tan 2x}{\tan 3x}, & 0<x<\frac{\pi}{6} \end{cases} $$ Then the values of $ a $ and $ b $ are:
If l, m, n are the pth, qth and rth terms of a G.P. respectively and l, m, n > 0, then
\[ \begin{vmatrix} \log_l p & 1 \\ \log_m q & 1 \\ \log_n r & 1 \end{vmatrix} \]