In a triangle, if \(b=20, c=21\) and \(\sin A=\frac{3}{5}\), then \(a\) is equal to :

\[
\int \frac{\sqrt{x^2+1}\left[\log \left(x^2+1\right)-2 \log x\right]}{x^4} d x=
\]

The function \(f(x)=\frac{e^{|x|}-e^{-x}}{e^x+e^{-x}}+\cos ^3\left(\frac{x}{2}\right)\) from \(R\) to itself is

If \(f(x)=\int \frac{d x}{\left(x^2+2\right)}\) and \(f(\sqrt{2})=0\), then \(f(0)=\)

If the power of the point $(1,6)$ with respect to the circle $x^2+y^2+4x-6y-a=0$ is $-16$ then ${}^{\text{'}}a^{\text{'}}$ equals

If \(l, m, n\) are in arithmetic progression, then the straight line \(l x+m y+n=0\) will pass through the point

\(f(x)=\left\{\begin{array}{cc}\frac{x-4}{|x-4|}+a, & x < 4 \\ a+b, & x=4 \\ \frac{x-4}{|x-4|}+b, & x>4\end{array}\right.\)
If \(f(x)\) given above is continuous at \(x=4\), then find the values of ' \(a\) ' and ' \(b\) '.

With respect to chlorobenzene, which of the following statements is not correct?

When ' \(\mathrm{X}\) ' \(\mathrm{g}\) of graphite is completely burnt in a bomb calorimeter in excess of \(\mathrm{O}_2\) at \(298 \mathrm{~K}\) and \(1 \mathrm{~atm}\) pressure as given in the equation
\(\mathrm{C}\) (graphite) \(+\mathrm{O}_2(\mathrm{~g}) \rightarrow \mathrm{CO}_2(\mathrm{~g})\)
The temperature of calorimeter raised from \(298 \mathrm{~K}\) to \(302 \mathrm{~K}\). If the heat capacity of the calorimeter and molar enthalpy change for the reaction at \(1 \mathrm{~atm}\) and \(298 \mathrm{~K}\) are \(20.7 \mathrm{~kJ} \mathrm{~K}^{-1}\) and \(-248.4 \mathrm{~kJ} \mathrm{~mol}^{-1}\), ' \(\mathrm{X}^{\prime}\) in \(\mathrm{g}\) is

If a point \(P\) moves such that its distances from the point \(A(1,1)\) and the line \(x+y+2=0\) are equal, then the locus of \(P\) is

The function \(f(x)=2 x^3-9 a x^2+12 a^2 x+1(a>0)\) attains its maximum and minimum at \(p\) and \(q\) respectively and \(p^2=q\). Then, \(a=\)

The process of recovering the modulating signal from the modulated carrier wave is called

In a steady state, the temperature at the end $A$ and end $B$ of a $20 \mathrm{cm}$ long rod $AB$ are $100°\mathrm{C}$ and $0°\mathrm{C}$. The temperature of a point, $9 \mathrm{cm}$ from $A$ is