$\lim _{x \rightarrow 0} \frac{\sqrt{1+x^2}-\sqrt{1-x+x^2}}{3^x-1}$ is equal to

$A$ bullet fired into a fixed target loses half of its velocity after penetrating $3 \ cm$. How much further will it penetrate before coming to rest,assuming that it faces constant resistance to motion? (in $cm$)

The period of $f(x)=\cos \left(\frac{x}{3}\right)+\sin \left(\frac{x}{2}\right)$ is (in $\pi$)

If the harmonic mean of the roots of the equation $\sqrt{2}x^2 - bx + (8 - 2\sqrt{5}) = 0$ is $4$,then the value of $b$ is:

$A$ rubber cube of side $5 ~cm$ has one face fixed,while a tangential force $1800 ~N$ is applied on its opposite face. If the modulus of rigidity of rubber is $2.4 \times 10^6 ~N/m^2$,then the lateral displacement of the strained face is . . . . . . (in $mm$)

$A$ ball is thrown on a lawn in such a way that it initially slides with a speed $v_0$ without rolling. It gradually picks up rotational motion. Find the speed of the ball at which there will be rolling without slipping.