If $\frac{x^2+x+1}{x^2+2x+1} = A + \frac{B}{x+1} + \frac{C}{(x+1)^2}$,then $A-B$ is equal to

$A$ drum of radius $R$ and mass $M$ rolls down without slipping along an inclined plane of angle $\theta$. The frictional force

$100$ coplanar forces,each equal to $10 \, N$,act on a body. Each force makes an angle of $\pi / 50$ with the preceding force. What is the resultant of the forces? $(N)$

Let $f$ and $g$ be continuous functions on $[0, a]$ such that $f(x) = f(a - x)$ and $g(x) + g(a - x) = 4$. Then $\int_{0}^{a} f(x)g(x) dx$ is equal to

If $Q$ denotes the set of all rational numbers and $f\left(\frac{p}{q}\right)=\sqrt{p^2-q^2}$ for any $\frac{p}{q} \in Q$,then observe the following statements.
$I$. $f\left(\frac{p}{q}\right)$ is real for each $\frac{p}{q} \in Q$
$II$. $f\left(\frac{p}{q}\right)$ is a complex number for each $\frac{p}{q} \in Q$.
Which of the following is correct?

$A$ metal crystallises in a face-centred cubic $(FCC)$ structure. If the edge length of its unit cell is '$a$',the closest approach between two atoms in the metallic crystal will be: