In an experiment,four quantities $a, b, c$ and $d$ are measured with percentage errors of $1\%, 2\%, 3\%$ and $4\%$ respectively. Quantity $P$ is calculated as $P = \frac{a^3 b^2}{cd}$. The percentage error in the measurement of $P$ is .......... $\%$

$A$ mercury drop of radius $1 ~cm$ is sprayed into $10^6$ drops of equal size. The energy expended in joules is (Surface tension of mercury is $460 \times 10^{-3} ~N/m$)

The transformed equation of $x^2+6xy+8y^2=10$ when the axes are rotated through an angle $\frac{\pi}{4}$ is:

Match the following (Take the relative strength of the strongest fundamental forces in nature as $1$):
| List-$I$ (Fundamental forces in nature) | List-$II$ (Relative strength) |
| :--- | :--- |
| $(A)$ Strong nuclear force | $(e)$ $10^{-2}$ |
| $(B)$ Weak nuclear force | $(f)$ $1$ |
| $(C)$ Electromagnetic force | $(g)$ $10^{10}$ |
| $(D)$ Gravitational force | $(h)$ $10^{-13}$ |
| | $(i)$ $10^{-39}$ |
The correct match is:

At $80\,^oC$,the vapour pressure of pure liquid $A$ is $520\,mmHg$ and that of pure liquid $B$ is $1000\,mmHg$. If a mixture of $A$ and $B$ boils at $80\,^oC$ and $1\,atm$ pressure,the amount of $A$ in the mixture is $........$ $mol\,\%$.

Cathode rays have