21.

Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X.

Find:$\int \frac{2 \cos \mathrm{x}}{(1-\sin \mathrm{x}) (1 + \sin { }^2 \mathrm{x}) }\mathrm{dx}$

Suppose a girl throws a die. If she gets 1 or 2 she tosses a coin three times and notes the number of tails. If she gets 3,4,5 or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw 3,4,5 or 6 with the ride ?

Let $\overrightarrow{\mathrm{a}} = 4\hat{\mathrm{i}} + 5\hat{\mathrm{j}} - \hat{\mathrm{k}} , \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 5\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}} = 3\hat{\mathrm{i}} + \hat{\mathrm{j}}- \hat{\mathrm{k}}$. Find a vector $\overrightarrow{\mathrm{d}}$ which perpendicular to both  $\overrightarrow{\mathrm{c}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{d}}. \overrightarrow{\mathrm{a}} = 21$

Let A = { x ∈ Z: 0 ≤  x≤ 12} show that R = {(a,b):a,b ∈  A, |a-b|} is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also, write the equivalence class [2].

Show that the function f: R → R defined by$\mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{x}}{{\mathrm{x}}^2 + 1},\forall \mathrm{x}\in \mathrm{R}$ si neither one- one nor onto. Also, if g: R → R is defined as g(x) = 2x -1 find fog (x)

If $A = \left|\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right|, find A^{-1}.$ Use it solve the system of equations.

2x - 3y + 5z = 11

3x + 2y - 4z = -5

x + y- 2z =-3

Using elementary row transformations, find the inverse of the matrix $\mathrm{A} = \left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& -5\end{array}\right]$

If $({\mathrm{x}}^2 + {\mathrm{y}}^2{)}^2 = \mathrm{xy}, \mathrm{find} \frac{\mathrm{dy}}{\mathrm{dx}}$

If $\mathrm{x} =\mathrm{a} (2\mathrm{\theta } - \sin 2\mathrm{\theta }) \mathrm{and} \mathrm{y} = \mathrm{a}(1- \cos 2\mathrm{\theta }), \mathrm{find}\frac{\mathrm{dy}}{\mathrm{dx}} \mathrm{when} \mathrm{\theta } =\frac{\mathrm{\pi }}{3}$