The angle between two vectors $a=\hat i-\hat k$  and $b=\hat i-\hat j+\hat k$ is:
1. $\cos^{-1}\frac13$
2. $\cos^{-1}\frac{-1}3$
3. $\sin^{-1}\frac{-1}3$
4. $90^\circ$

A tangential force acting on the top of a sphere of mass $m$ kept on a rough horizontal plane as shown in the figure. 

If the sphere rolls without slipping, then the acceleration with which the centre of the sphere moves is:
1. $\frac{10~F}{7~M}$
2. $\frac{F}{2~M}$
3. $\frac{3~F}{7~M}$
4. $\frac{7~F}{2~M}$

The density of a rod having length $l$ varies as $\rho=c+dx,$ where $x$ is the distance from the left end. The distance from origin to the centre of mass is:
                 
1. $\dfrac{3cl+2Dl^2}{3(2c+Dl)}$
2. $\dfrac{3cl+4Dl^2}{3(2c+Dl)}$
3. $\dfrac{2cl+3Dl^2}{3(2c+1)}$
4. $\dfrac{cl+3Dl^2}{3(2c+Dl)}$

A block slides down on an incline of angle 30° with an acceleration $\frac g4.$ Find the kinetic friction coefficient.
1. $\frac{1}{2\sqrt2}$
2. $0.6$
3. $\frac{1}{2\sqrt3}$
4. $\frac{1}{\sqrt2}$

A wire of resistance 10 Q is bent to form a complete circle. Find its resistance between two diametrically opposite points. 
   
1. $5~\Omega$
2. $2.5~\Omega$
3. $1.25~\Omega$
4. $\frac{10}{3}~\Omega$