Question

Write the integrating factor of the following differential equation. $(1+y^{2})+(2x y-\mathrm{cot}\;y){\frac{d y}{d x}}\;=0$

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

Let R={(a,a³): a is a prime number less than 5} be a relation. Find the range of R.

Evaluate $\textstyle\int{\frac{\cos{\sqrt{x}}}{\sqrt{x}}}\,d x\ .$

Write the inverse of the matrix $\left(\begin{array}{l l }cosθ & -sinθ \\ sinθ & cosθ\end{array}\right)$ .

If $(2x\ \ 4){\binom{x}{-8}}=0,$ find the positive value of x.

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.