The decay rate for a certain mass of a radioactive substance measured at different times varies with time as shown in the figure. The count rate at $t = 8 \, hr$ will be:

$A$ very small hole in an electric furnace is used for heating metals. The hole nearly acts as a black body. The area of the hole is $200 ~mm^2$. To keep a metal at $727^{\circ} C$,the heat energy flowing through this hole per second,in joules,is $\left(\sigma = 5.67 \times 10^{-8} ~W m^{-2} K^{-4}\right)$.

The electric field in an electromagnetic wave is given by $E = 50 \sin \frac{2 \pi}{\lambda}(ct - x) \ N/C$. Find the energy contained in a cylindrical region of cross-section $10 \ cm^2$ and length $50 \ cm$ along the $x$-axis.

$A$ series $LCR$ circuit is connected across a source of alternating $emf$ of changing frequency and resonates at frequency $f_0$. Keeping capacitance constant,if the inductance $(L)$ is increased by $\sqrt{3}$ times and resistance $(R)$ is increased by $1.4$ times,the resonant frequency now is

Let $a, b$,and $c$ be such that $\frac{1}{(1-x)(1-2x)(1-3x)} = \frac{a}{1-x} + \frac{b}{1-2x} + \frac{c}{1-3x}$. Then $\frac{a}{1} + \frac{b}{3} + \frac{c}{5}$ is equal to

Which of the following is used for making optical instruments?