Question

Find the differential equation of the family of lines passing through the origin.

If A is square matrix of order 2 and $|a d j A|=9,$ , find | A |.

Write the differential equation formed from the equation y = mx + c, where m and c are arbitrary constants.

Write the value of : $\textstyle\int{\frac{\sec^{2}x}{\operatorname{cosec}^{2}x}}d x.$

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.

Write the number of vectors of unit length perpendicular to both the vector $\stackrel{\rightarrow}{a}=\stackrel{\wedge}{2i}+\stackrel{\wedge}{j}+\stackrel{\wedge}{2k}$ and ${\vec{b}}={\hat{j}}+{\hat{k}}\,.$

Evaluate : $\textstyle\int\!{\frac{e^{\tan^{-1}x}}{1+x^{2}}}\,d x$ .