\(\int \sin \sqrt{x} \mathrm{~d} x=\ldots+C\)
(where \(C\) is a constant of integration.)

If the general solution of the equation \(\frac{\tan 3 x-1}{\tan 3 x+1}=\sqrt{3}\) is \(x=\frac{\mathrm{n} \pi}{\mathrm{p}}+\frac{7 \pi}{\mathrm{q}}, \mathrm{n}, \mathrm{p}, \mathrm{q}, \in \mathrm{Z}\), then \(\frac{p}{q}\) is

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

\(\int \frac{\sin x}{\sqrt{5 \sin ^2 x+6 \cos ^2 x}} \mathrm{~d} x=\)

If \(\frac{\pi}{2} < \theta \leq\) and \(|\overline{\mathrm{a}}|=5,|\overline{\mathrm{b}}|=13,|\overline{\mathrm{a}} \times \overline{\mathrm{b}}|=25\), then the value of \(\bar{a} \cdot \bar{b}\) is

If \(3 \sin \alpha=5 \sin \beta\), then \(\tan \left(\frac{\alpha+\beta}{2}\right) \div \tan \left(\frac{\alpha-\beta}{2}\right)=\)

The area of the region bounded by the parabola \(x^{2}=16 y, y=1, y=4\) and the
Y-axis lying in the first quartrant is

The difference between the maximum values of \({ }^6 \mathrm{C}_{\mathrm{r}}\) and \({ }^n \mathrm{C}_{\mathrm{r}} 16\), then \(\mathrm{n}=\)

What is the product obtained when \(\mathrm{Br}_{2}\) water reacts with glucose?

Calculate the oxidation number of Cr in \(\mathrm{CrO}_4^{2-}\) ion and \(\mathrm{K}_2 \mathrm{Cr}_2 \mathrm{O}_7\) respectively.

For a body performing simple harmonic motion, its potential energy is \(E_x\) at displacement \(x\) and \(\mathrm{E}_{\mathrm{y}}\) at displacement y from mean position. The potential energy \(\mathrm{E}_0\) at displacement \((\mathrm{x}+\mathrm{y})\) is

An international treaty known as Montreal Protocol was signed to control emission of

A body performs linear simple harmonic motion of amplitude 'A". At what
displacement from the mean position, the potential energy of the body is one
fourth of its total energy?