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.

Find the maximum value of $\begin{vmatrix}1 & 1 & 1 \\ 1 & 1+sinθ & 1 \\ 1 & 1 & 1+cosθ\end{vmatrix}$

Find the sum of the vectors $\stackrel{\rightarrow}{a}=\hat{i}-2\hat{j}+\hat{k}\;,$ $\vec{b}=-2\hat{i}+4\hat{j}+5\hat{k}$ and $\stackrel{\rightarrow}{c}=\hat{i}-6\hat{j}-7\hat{k}.$

If A is a square matrix such that A² = A, then write the value of 7A – (I + A)³, where I is an identity matrix.

Find : $\textstyle\int{\frac{\sin^{2}\!x-\cos^{2}\!x}{\sin^{2}\!x\cos^{2}\!x}}\,d x$

Evaluate: $\textstyle\int\cot{x}(cosec x-1)e^{x}d x$ .

State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.