Question

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

Find the differential equation representing the family of curves $V={\frac{A}{r}}+B,$ where A and B are arbitrary constants.

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

If A is a matrix of order 3 x 4 and B is a matrix of order 4 x 3, then find order of matrix (AB.)

Find a vector $\stackrel{\rightarrow}{a}\$ of magnitude ${\mathfrak{5}}{\sqrt{2}}$ making an angle of $\frac{\pi}{4}$ with x-axis, $\frac{\pi}{2}$ with y-axis and an acute angle θ with z-axis.

Write the degree of the differential equation : $\left({\frac{d^{2}y}{d x^{2}}}\right)^{2}-2{\frac{d^{2}y}{d x^{2}}}-{\frac{d y}{d x}}+1=0.$

If the following function f(x) is continuous at x = 0, then write the value of k.