$A$ conducting rod $PQ$ of length $5\,m$ oriented as shown in the figure is moving with a velocity $\vec{v} = (2\,m/s)\hat{i}$ without any rotation in a uniform magnetic field $\vec{B} = (3\hat{j} + 4\hat{k})\,T$. The induced $EMF$ in the rod is........$V$.

$A$ plane passing through $(-1, 2, 3)$ and whose normal makes equal angles with the coordinate axes is

The first life on Earth was .....

Two identical bodies are made of a material for which the heat capacity increases with temperature. One of these is held at a temperature of $100^{\circ} C$,while the other one is kept at $0^{\circ} C$. If the two are brought into contact,then assuming no heat loss to the environment,the final temperature that they will reach is

Consider a water jar of radius $R$ that has water filled up to height $H$ and is kept on a stand of height $h$ (see figure). Through a hole of radius $r$ $(r << R)$ at its bottom,the water leaks out and the stream of water coming down towards the ground has a shape like a funnel as shown in the figure. If the radius of the cross-section of the water stream when it hits the ground is $x$,then:

If $\alpha$ and $\beta$ are the roots of the equation $ax^2+bx+c=0$ and the equation having roots $\frac{1-\alpha}{\alpha}$ and $\frac{1-\beta}{\beta}$ is $px^2+qx+r=0$,then $r=$