If \(P=\left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]\) is the adjoint of a \(3 \times 3\) matrix A and \(|A|=4\), then value of \(\alpha\) 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 unit vector perpendicular to each of the vectors \(\bar{a}+\bar{b}\) and \(\bar{a}-\bar{b}\), where \(\bar{a}=\hat{i}+\hat{j}+\hat{k}\) and \(\overline{\mathrm{b}}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}\) is

\(\int \frac{\left(x^2+1\right)}{(x+1)^2} d x=\)

If \(f(x)=\frac{x+2}{18},-2 < x < 4\)
\(=0 \quad, \quad\) otherwise,
is the p. d. f. of a r. v. X, then the value of \(\mathrm{P}(|\mathrm{X}| < 2)\) is

If the lengths of the sides of triangle are \(3,5,7\), then the largest angle of the triangle is

Area of the region bounded by $y=\mathrm{c}\mathrm{o}\mathrm{s}x,x=0,x=\pi$ and x-axis is …sq. units.

Which of the following is the communication channel in case of radio communication?

When cell of e.m.f. ' \(E_1{ }^{\prime}\) is connected to potentiometer wire, the balancing length is ' \(\ell_1{ }^{\prime}\). Another cell of e.m.f. ' \(E_2{ }^{\prime}\left(E_1>E_2\right)\) is connected so that two cells oppose each other, the balancing length is ' \(\ell_2{ }^{\prime}\). The ratio \(\mathrm{E}_1: \mathrm{E}_2\) is

The maximum volume of a right circular cylinder if the sum of its radius and height is
\(6 \mathrm{~m}\) is

If there is an addition of a single chromosome to the total set of chromosomes, it is called __________ .

An elastic material with Young's modulus 'Y' is subjected to a tensile stress ' \(\mathrm{S}\) '. The elastic energy stored per unit volume of the material will be

A solution is \(0 \cdot 25 \%\) by mass. What is the weight of solvent containing \(1.25 \mathrm{~g}\) solute?