Question

If $A_{i j}$ is the cofactor of the element $a_{i j}$ of the determinant $\begin{vmatrix}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{vmatrix}$ , then write the value of $a_{32}A_{32}.$

Evaluate : $\textstyle\int\!{\frac{\ d x}{\sin^{2}x\cos^{2}x}}$ .

Evaluate $\textstyle\int\!{\frac{(1+\log x)^{2}}{x}}\,d x\,.$

P and Q are two points with position vectors $3\stackrel{\rightarrow}{a}-2\stackrel{\rightarrow}{b}$ and $\stackrel{\rightarrow}{a}+\stackrel{\rightarrow}{b}$ respectively. Write the position vector of a point R which divides the line segment PQ externally in the ratio 2 : 1.

Let A = $A\ =\ {\left(\begin{array}{l l}{2}&{-1}\\ {3}&{4}\end{array}\right)}$ , B = $A\ =\ {\left(\begin{array}{l l}{5}&{2}\\ {7}&{4}\end{array}\right)}$ , C = $A\ =\ {\left(\begin{array}{l l}{2}&{5}\\ {3}&{8}\end{array}\right)}$ , find a matrix D such that CD - AB = 0.

Find : $\textstyle\int\!{\frac{1}{x(1+\log x)}}\,d x$

Find a vector in the direction of $\stackrel{\rightarrow}{a}=\stackrel{\wedge}{i}-2\;\stackrel{\wedge}{j}$ that has magnitude 7 units.