If $\tan (\pi \cos \theta)=\cot (\pi \sin \theta)$,then a value of $\cos \left(\theta-\frac{\pi}{4}\right)$ among the following is

$A$ lead bullet of mass $10 \ g$ travelling at $300 \ m/s$ strikes against a block of wood and comes to rest. Assuming $50\%$ of heat is absorbed by the bullet,the increase in its temperature is (Specific heat of lead $= 150 \ J/kg \cdot ^{\circ}C$) (in $^{\circ}C$)

The time period of a second's pendulum is $2\, s$. The spherical bob,which is empty from inside,has a mass of $50\, g$. This is now replaced by another solid bob of the same radius but having a mass of $100\, g$. The new time period will be .... $s$.

If a body is projected with an angle $\theta$ to the horizontal,then

$A$ thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $\omega$. Two objects each of mass $m$ are attached gently to the opposite ends of a diameter of the ring. The ring will now rotate with an angular velocity:

The points whose position vectors are $2i+3j+4k$,$3i+4j+2k$,and $4i+2j+3k$ are the vertices of