Mathematics Tes...

Evaluate the integral \(\int_{0}^{1} \frac{\tan ^{-1} x}{1+x^{2}} d x\).

If \(A=\left\{x \in R : x^{2}=2\right\}\) and \(B=\left\{y \in R : y^{2}-5 y+6=0\right\}.\) Find \(n(A \times B)\).

Let \(\mathrm{A}=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]\). If \(\mathrm{M}\) and \(\mathrm{N}\) are two matrices given by \(\mathrm{M}=\sum_{\mathrm{k}=1}^{10} \mathrm{~A}^{2 \mathrm{k}}\) and \(\mathrm{N}=\sum_{\mathrm{k}=1}^{10} \mathrm{~A}^{2 \mathrm{k}-1}\) then \(\mathrm{MN}^2\) is :

A plane passes through the points \(\mathrm{A}(1,2,3), \mathrm{B}(2,3,1)\) and \(\mathrm{C}(2,4,2)\). If 0 is the origin and \(\mathrm{P}\) is \((2,-1,1)\), then the projection of OP on this plane is of length.

In any group, the number of improper subgroups is:

In linear programming, lack of points for a solution set is said to have:

Find the area of the region (in square unit) bounded by the curve \(y=x-2\) and \(x=0\) to \(x=4\).

Find the values of \({k}\) so the line \(\frac{2 {x}-2}{2 {k}}=\frac{4-{y}}{3}=\frac{{z}+2}{-1}\) and \(\frac{{x}-5}{1}=\frac{{y}}{{k}}=\frac{{z}+6}{4}\) are at right angles