Among the 5 married couples, if the names of 5 men are matched with the names of their wives randomly, then the probability that no man is matched with the name of his own wife is ?

If all the numbers which are greater than 6000 and less than 10000 are formed with the digits $ 3,5,6,7,8 $ without repetition of the digits, then the difference between the number of odd numbers and the number of even numbers among them is:

Given the position vectors of points $A$ and $B$ as $ \mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k} $ and $ \mathbf{B} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} $, and $C$ divides $AB$ in the ratio 3:2. If $ \mathbf{D} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} $ is the position vector of point $D$, find the unit vector in the direction of $ \mathbf{CD} $:

If $\theta$ is the angle between the vectors $4\mathbf{i} - \mathbf{j} + 2\mathbf{k}$ and $ \mathbf{i} + 3\mathbf{j} - 2\mathbf{k}$, then $\sin 2\theta$ is:

If $ 0 \leq x \leq \frac{\pi}{2} $, then \[ \lim\limits_{x \to a} \frac{2\cos x - 1}{2\cos x - 1} \] Options:

If the function

\[ f(x) = \begin{cases} \frac{(e^x - 1) \sin kx}{4 \tan x}, & x \neq 0 \\ P, & x = 0 \end{cases} \]

is differentiable at $ x = 0 $, then:

If the real-valued function

\[ f(x) = \sin^{-1}(x^2 - 1) - 3\log_3(3^x - 2) \]

is not defined for all $ x \in (-\infty, a] \cup (b, \infty) $, then what is $ 3^a + b^2 $?

If $ \theta $ is the acute angle between the curves $ y^2 = x $ and $ x^2 + y^2 = 2 $, then $ \tan \theta $ is:

When $ |x| < 2 $, the coefficient of $ x^2 $ in the power series expansion of

\[ \frac{x}{(x-2)(x-3)} \]

is:

The length of the latus rectum of the ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ $ (a>b) $ is $ \frac{8}{3} $. If the distance from the center of the ellipse to its focus is $ \sqrt{5} $, then $ \sqrt{a^2 + 6ab + b^2} = \text{?} $

The slope of the tangent drawn from the point $ (1,1) $ to the hyperbola $ 2x^2 - y^2 = 4 $ is:

A, B, C, D are square matrices such that A + B is symmetric, A - B is skew-symmetric, and D is the transpose of C.

\[ A = \begin{bmatrix} -1 & 2 & 3 \\ 4 & 3 & -2 \\ 3 & -4 & 5 \end{bmatrix} \]

and

\[ C = \begin{bmatrix} 0 & 1 & -2 \\ 2 & -1 & 0 \\ 0 & 2 & 1 \end{bmatrix} \]

then the matrix $ B + D $ is:

If $ \alpha,\,\beta,\,\gamma $ are the roots of the equation \[ \begin{vmatrix} x & 2 & 2\\ 2 & x & 2\\ 2 & 2 & x \end{vmatrix} = 0 \]

The variance of the first 10 natural numbers which are multiples of 3 is:

A circle passes through the points $ (2, 0) $ and $ (1, 2) $. If the power of the point $ (0, 2) $ with respect to this circle is 4, then the radius of the circle is

The domain of the real valued function: \[ f(x) \;=\; \sqrt[3]{\frac{x \,-\, 2}{2x^2 \,-\, 7x \,+\,5}} \;+\; \log\bigl(x^2 \,-\, x \,-\, 2\bigr). \]

A rectangle is formed by the lines \[ x = 4, \quad x = -2, \quad y = 5, \quad y = -2 \] and a circle is drawn through the vertices of this rectangle. The pole of the line \[ y + 2 = 0 \] with respect to this circle is: