\(\int \frac{x}{1+x^4} \mathrm{~d} x=\)

If \(\bar{x}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \bar{y}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}\) and \(\overline{\mathrm{z}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}\) where \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar vectors, then value of \(\bar{x} \cdot(\overline{\mathrm{a}}+\overline{\mathrm{b}})+\bar{y} \cdot(\overline{\mathrm{~b}}+\overline{\mathrm{c}})+\overline{\mathrm{z}} \cdot(\overline{\mathrm{c}}+\overline{\mathrm{a}})\) is

The value of \(\operatorname{cosec}^{-1}(\sqrt{2})+\cos ^{-1}\left(\frac{-1}{2}\right)-\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)\) is equal to

\(A\) line \(L\) is passing through points \(A(1,3,2)\) and \(B(2,2,1)\). If mirror image of point \(\mathrm{P}(1,1,-1)\) in the line L is \((\mathrm{x}, \mathrm{y}, \mathrm{z})\) then \(x+\mathrm{y}+\mathrm{z}=\)

The equation of normal to the curve \(x^{2}+y^{2}=r^{2}\) at \(p(\theta)\) is

Radium decompose at the rate proportional to the amount present at any time. If \(\mathrm{P} \%\) of amount disappears in one year, then amount of radium left after 2 years is

\(\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} d x=\)
(where \(\mathrm{C}\) is a constant of integration)

The lines \(\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k} \quad\) and \(\frac{x-1}{\mathrm{k}}=\frac{y-4}{2}=\frac{z-5}{1}\) are coplanar if

A random variable \(\mathrm{X}\) has the following probability distribution

Then \(F(4)=\)

Which of the following is NOT a characterstic of chemisorption?

Ratio of kinetic energy at mean position to potential energy at \(A / 2\) of a partide performing SHM

\(\vec{a}=\hat{i}+\hat{j}+\hat{k}\), and \(\vec{b}=\hat{i}-\hat{j}+\widehat{k}\) and \(\vec{c}=\hat{i}-\hat{j}-\hat{k}\) are three vectors then vector \(\overline{\mathrm{r}}\) in the plane of \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\), whose projection on \(\overline{\mathrm{c}}\) is \(\frac{1}{\sqrt{3}}\), is given by

If \(E^{\circ}\left(\mathrm{Zn}_{(\mathrm{aq})}^{+2} \mid \mathrm{Zn}_{(\mathrm{s})}\right)=-0 \cdot 76 \mathrm{~V}\).
Calculate potential for \(\mathrm{Zn}_{(\mathrm{s})} \rightarrow \mathrm{Zn}_{(0.01 \mathrm{M})}^{+2}+2 \overline{\mathrm{e}}\) at 298 K ?