Let \(z\) be a complex number such that \(|z|+z=3+i, i=\sqrt{-1}\), then \(|z|\) is equal to

The value of \(\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \frac{1}{\sin 2 x\left(\tan ^5 x+\cot ^5 x\right)} \mathrm{dx}\) is

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar vectors and \(\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \overline{\mathrm{q}}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \overline{\mathrm{r}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \quad\) then \(2 \overline{\mathrm{a}} \cdot \overline{\mathrm{p}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{q}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{r}}=\)

The domain of the function \(f(x)=\frac{1}{\sqrt{x+|x|}}\) is

If \(\mathrm{p} \rightarrow(\sim \mathrm{p} \vee \mathrm{q})\) is false, then the truth values of \(\mathrm{p}\) and \(\mathrm{q}\) are respectively

The negation of the logical statement \((\mathrm{p} \vee \sim \mathrm{q}) \rightarrow(\mathrm{p} \wedge \sim \mathrm{q})\) is

\(\int_0^2 \frac{3 x+1}{x^2+4} d x=\)

Which from following compounds has highest boiling point?

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) are unit coplanar vectors, then the scalar triple product \(\left[\begin{array}{lll}2 \bar{a}-\bar{b} & 2 \bar{b}-\bar{c} & 2 \bar{c}-\bar{a}\end{array}\right]\) has the value

\(\lim _{x \rightarrow 0} \frac{\cos a x-\cos b x}{x^{2}}\) is equal to

Let \(\mathrm{A}=\left[\begin{array}{cc}1 & 2 \\ -5 & 1\end{array}\right]\) and \(\mathrm{A}^{-1}=x \mathrm{~A}+y \mathrm{I}_2\), (where \(\mathrm{I}_2\) is unit matrix of order 2), then

\(\quad \int 3^{3^x} \cdot 3^x d x=\)

Which functional group from following is considered as principal functional group if polyfunctional compound is to be named by IUPAC system?