If $\int \frac{\mathrm{d} x}{1+3 \sin ^2 x}=\frac{1}{2} \tan ^{-1}(\mathrm{f}(x))+\mathrm{c}$, where c is a constant of integration, then $\mathrm{f}(x)$ is equal to
A random variable X has the following probability distribution
| $X$ | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| $p(x)$ | $\mathrm{k^2}$ | $\mathrm{2k}$ | $\mathrm{k}$ | $\mathrm{2k}$ | $\mathrm{5k^2}$ |
Then $\mathrm{p}(x \geq 2)$ is equal to
Let $f:[-1,3] \rightarrow \mathbb{R}$ be defined as
$$\left\{\begin{array}{lc} |x|+[x], & -1 \leqslant x<1 \\ x+|x|, & 1 \leqslant x<2 \\ x+[x], & 2 \leqslant x \leqslant 3 \end{array}\right.$$
where $[t]$ denotes the greatest integer function. Then $f$ is discontinuous at
Let $\overline{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{b}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overline{\mathrm{c}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}$ be three vectors. A vector $\bar{v}$ 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