\(\int \sin ^{-1}\left(\sqrt{\frac{x}{a+x}}\right) d x=\)

If \(2 \sin \mathrm{h}^{-1}\left(\frac{a}{\sqrt{1-a^2}}\right)=\log \left(\frac{1+x}{1-x}\right)\), then \(x\) is equal to

The vector equation of any plane passing through the line of intersection of the planes \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_1=\mathrm{q}_1\) and \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_2=\mathrm{q}_2\) is given by \(\overrightarrow{\mathrm{r}}\left(\overrightarrow{\mathrm{m}}_1+\lambda \overrightarrow{\mathrm{m}}_2\right)=\mathrm{q}_1+\lambda \mathrm{q}_2\) for \(\lambda \in \mathbb{R}\). The vector equation of a plane passing through the point \(2 \hat{i}-3 \hat{j}+\hat{k}\) and the line of intersection of the planes r. \((\hat{i}-2 \hat{j}+3 \hat{k})=5\) and \(\vec{r} \cdot(3 \hat{i}+\hat{j}-2 \hat{k})=7\)

\(\sum_{k=1}^6\left(\sin \frac{2 \pi k}{7}-i \cos \frac{2 \pi k}{7}\right)=\)

If \(\mathbf{a}\) is a vector of magnitude \(7, \mathbf{b}\) is a vector of magnitude 8 , then the maximum value of \(\mathbf|{a} \cdot \mathbf{b}|\) is

One of the latus recta of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) subtends an angle \(2 \operatorname{Tan}^{-1}\left(\frac{3}{2}\right)\) at the centre of the hyperbola. If \(\mathrm{b}^2=36\) and e is the eccentricity of the given hyperbola, then \(\sqrt{\mathrm{a}^2+\mathrm{e}^2}=\)

Let \(n=1 !+4 !+7 !+\ldots+400 !\). Then ten's digit of \(n\) is

Which of the following act as intracellular messengers?

\(\int_0^{\pi / 2} \frac{\sin ^3 x \cos x d x}{\sin ^4 x+\cos ^4 x}=\)

In an \(A C\)-circuit containing only capacitance, the current

If $a,b,c$ are respectively the $5^{\mathrm{th}},8^{\mathrm{th}},{13}^{\mathrm{th}}$ terms of an arithmetic progression, then $\left|\begin{array}{ccc}a& 5& 1\\ b& 8& 1\\ c& 13& 1\end{array}\right|=$

Direction ratios of normal to a plane passing through \((1,0,0)\) and \((0,1,0)\) which makes an angle of \(\frac{\pi}{4}\) with the plane \(x+y-3=0\) are

The number of points \(P(x, y)\) with natural numbers as coordinates that lie inside the quadrilateral formed by the lines \(2 x+y=2\), \(x=0, y=0\) and \(x+y=5\) is