Question

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.

Write the projection of the vector $7\stackrel{\wedge}{i}+\stackrel{\wedge}{j} -4\stackrel{\wedge}{k}$ on the vector $2\stackrel{\wedge}{i}+6\stackrel{\wedge}{j} +3\stackrel{\wedge}{k}$.

If m and n are the order and degree, respectively of the differential equation $\textstyle y\left({\frac{d y}{d x}}\right)^{3}+x^{3}\left({\frac{d^{2}y}{d x^{2}}}\right)^{2}-x y$= sin x, then write the value of m + n.

Evaluate x if : $\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}$

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

Write the solution of the differential equation ${\frac{d y}{d x}}=2^{-y}.$

If A is a square matrix of order 3 such that $|a d j\,A|$ = 225, find $|A^{\prime}|.$