Two vibrating tuning forks produce progressive waves given by $Y_1 = 4 \sin(500 \pi t)$ and $Y_2 = 2 \sin(506 \pi t)$. The number of beats produced per minute is:

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $(x_i, y_i)$,for $i=1, 2, 3, 4$,then $y_1+y_2+y_3+y_4$ equals

$\sin ^{-1} \frac{4}{5} + 2 \tan ^{-1} \frac{1}{3}$ is equal to

In a certain test there are $n$ questions. In the test,${2^{n - i}}$ students gave wrong answers to at least $i$ questions,where $i = 1, 2, \dots, n$. If the total number of wrong answers given is $2047$,then $n$ is equal to:

$A$ square loop of area $25 \, cm^2$ has a resistance of $10 \, \Omega$. This loop is placed in a uniform magnetic field of magnitude $40 \, T$. The plane of the loop is perpendicular to the magnetic field. The work done in pulling the loop out of the magnetic field slowly and uniformly in one second will be:

$A$ straight wire of length $20 \ cm$ carrying a current of $\frac{3}{\pi^2} \ A$ is bent in the form of a circle. The magnetic field at the centre of the circle is