$A$ horizontal straight wire $20\, m$ long extending from east to west is falling with a speed of $5.0\, m/s$,at right angles to the horizontal component of the earth's magnetic field $0.30 \times 10^{-4}\, Wb/m^2$. The instantaneous value of the $e.m.f.$ induced in the wire will be.....$mV$.

The diagram shows a force-extension graph for a rubber band. Consider the following statements:
$(I)$ It will be easier to compress this rubber than expand it.
$(II)$ Rubber does not return to its original length after it is stretched.
$(III)$ The rubber band will get heated if it is stretched and released.
Which of these can be deduced from the graph?

The roots of the equation $x^3-3x-2=0$ are

$A$ function $y = f(x)$ satisfies $(x + 1) f'(x) - 2(x^2 + x) f(x) = \frac{e^{x^2}}{x + 1}$ for all $x > -1$. If $f(0) = 5$,then $f(x)$ is:

The figure shows a system of logic gates. From the study of the truth table,it can be found that to produce a high output $(1)$ at $R$,we must have

In a step-up transformer, the turn ratio is $1:2$. A Leclanche cell $(e.m.f. 1.5\,V)$ is connected across the primary. The voltage developed in the secondary would be.......$V$