Let $R$ be a reflexive relation on a finite set $A$ having $n$ elements,and let there be $m$ ordered pairs in $R$. Then

The energy spectrum of $\beta$-particles [number $N(E)$ as a function of $\beta$-energy $E$] emitted from a radioactive source is:

$A$ plane passes through the point $(3, 5, 7)$. If the direction ratios of its normal are equal to the intercepts made by the plane $x + 3y + 2z = 9$ with the coordinate axes,then the equation of that plane is

Monocyclic compounds $\text{P , Q , R}$ and $S$ are the major products formed in the reaction sequences given below.
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The product having the highest number of unsaturated carbon atom$(s)$ is-

Consider the two following statements $A$ and $B$,and identify the correct choice given in the answers: $(A)$ Duddell's thermo galvanometer is suitable to measure direct current only. $(B)$ Thermopile can measure temperature differences of the order of $10^{-3} {}^{\circ}C$.

If $\int \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) d x=f(x)-\log \left(1+x^2\right)$ then $f(x)$ is equal to