Question

If $\left(\begin{array}{l l }2x & 3\end{array}\right)$ $\left(\begin{array}{l l }1 & 2 \\ -3 & 0\end{array}\right)$ $\left(\begin{array}{l }x \\ 3\end{array}\right)$ = 0,

Evaluate $\textstyle\int\!{\frac{e^{2\,x}\,-\,e^{-2\,x}}{e^{2\,x}\,+\,e^{-2\,x}}}\,d x\,.$

If $\left(\begin{array}{l l l }2 & 1 & 3\end{array}\right)$ $\left(\begin{array}{l l l }-1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right)$ $\left(\begin{array}{l }1 \\ 0 \\ -1\end{array}\right)$ = A , then write the order of matrix A.

Find a unit vector parallel to the sum of vectors $\stackrel{\wedge}{i}+\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ and $2\stackrel{\wedge}{i}-3\stackrel{\wedge}{j} +5\stackrel{\wedge}{k}$

L and M are two points with position vectors $2\stackrel{\rightarrow}{a}-\stackrel{\rightarrow}{b}$ and $\stackrel{\rightarrow}{a}+2\stackrel{\rightarrow}{b}$ respectively. Write the position vectors of a point N which divides the line segment LM in the ratio 2:1 externally.

If $\begin{vmatrix} 2x & x+3 \\ 2(x+1) & x+1 \end{vmatrix} = \begin{vmatrix} 1 & 5 \\ 3 & 3 \end{vmatrix}$, write the value of x.

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.