Question

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

Evaluate $\textstyle\int$ sec² (7 – 4x) dx.

Give an example of a skew symmetric matrix of order 3.

In the interval ${\frac{\pi}{2}}<x<\pi$ , find the value of x for which that matrix $\left(\begin{array}{l l }2sinx & 3 \\ 1 & 2sinx\end{array}\right)$ is singular.

If $\begin{vmatrix} \sin\alpha & \cos\beta \\ \cos\alpha & \sin\beta \end{vmatrix} =-\ {\begin{array}{l}{1}\\ {2}\end{array}}\$ , write the value of x.

Find the position vector of a point which divides the join of points with position vectors $\;\stackrel{\rightarrow}{(a}-2\stackrel{\rightarrow}{b})$ and $\;\stackrel{\rightarrow}{(2a}+\stackrel{\rightarrow}{b})$ externally in the ratio 2 : 1.

Give an example of vectors $\stackrel{\rightarrow}{a}\$ and $\stackrel{\rightarrow}{b}\$ such that $\vert{\vec{b}}\vert=$ $\vert{\vec{b}}\vert$ but $\stackrel{\rightarrow}{a}\neq \stackrel{\rightarrow}{b}\;.$