If a number $ n $ is chosen at random from the set $\{11, 12, 13, ..., 30\}\,$ then, the probability that $ n $ is neither divisible by 3 nor divisible by 5, is

A five-digit number is formed by using the digits 1, 2, 3, 4, 5 with no repetition. The probability that the numbers 1 and 5 are always together, is

If two lines $ L_1 : \frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4} $ and $ L_2 : \frac{x-3}{1} = \frac{y-k}{2} = z $ intersect at a point, then $ 2k $ is equal to

The lines $ \frac{x-1}{2} = \frac{y-4}{4} = \frac{z-2}{3} \quad \text{and} \quad \frac{1-x}{1} = \frac{y-2}{5} = \frac{3-z}{a} \quad \text{are perpendicular to each other, then} \ a \ \text{equals to} $

The integral $ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^2 x \, dx \text{ is equal to} $

The integral $ \int \frac{4x^2}{\sqrt{1-16x^2}} \, dx \text{ is equal to} $

The integral $ \int \frac{x \, dx}{2(1+x^2)^{3/2}} $ is equal to

The plane $ x - 2y + z = 0 $ is parallel to the line

Given $ \mathbf{a} = 2\hat{i} + \hat{j} - \hat{k}, \quad \mathbf{b} = \hat{i} - \hat{j}, \quad \mathbf{c} = 5\hat{i} - \hat{j} + \hat{k},$ then the unit vector parallel to $\mathbf{a} + \mathbf{b} - \mathbf{c} $ but in the opposite direction is

The maximum value of $ Z = 12x + 13y $, subject to constraints $ x \geq 0, \quad y \geq 0, \quad x + y \leq 5, \quad 3x + y \leq 9 $ is

If the vectors $ \mathbf{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}, \quad \mathbf{b} = \hat{i} + 2\hat{j} - \hat{k}, \quad \mathbf{c} = m\hat{i} - \hat{j} + 2\hat{k} $ are coplanar, then the value of $ m $ is

The angle between the vectors $ a = \hat{i} + 2 \hat{j} + 2 \hat{k} \quad \text{and} \quad b = \hat{i} + 2 \hat{j} - 2 \hat{k} \quad \text{is} $

If $$ A = \begin{pmatrix} k + 1 & 2 \\4 & k - 1 \end{pmatrix}$$ is a singular matrix, then possible values of k are