The principal value of \(\sin ^{-1}\left(-\frac{1}{2}\right)\) is

The length of the perpendicular from the point \(P(a, b)\) to the line \(\frac{x}{a}+\frac{y}{b}=1\) is

\(\int_{5}^{10} \frac{1}{(x-1)(x-2)} d x\) is equal to

\(\int \frac{5\left(x^6+1\right)}{x^2+1} \mathrm{~d} x=\) (Where \(C\) is a constant of integration.)

\(\int \frac{x^3}{(x+1)^2} \mathrm{~d} x=\)

If p : switch \(\mathrm{S}_1\) is closed, q : switch \(\mathrm{S}_2\) is closed, r : switch \(\mathrm{S}_3\) closed, then the symbolic form of the following switching circuit is equivalent to Switching Circuit :

If \(\bar{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}, \quad \bar{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\), \(\overline{\mathrm{c}}=\mathrm{c}_1 \hat{i}+\mathrm{c}_2 \hat{\mathrm{j}}+\mathrm{c}_3 \hat{\mathrm{k}}\) and \(\left[\begin{array}{lll}3 \overline{\mathrm{a}}+\overline{\mathrm{b}} & 3 \overline{\mathrm{~b}}+\overline{\mathrm{c}} & 3 \overline{\mathrm{c}}+\overline{\mathrm{a}}\end{array}\right]\) \(=\lambda\left|\overline{\mathrm{a}} \cdot \hat{\mathrm{i}} \overline{\mathrm{a}} \cdot \hat{\mathrm{j}} \overline{\mathrm{a}} \cdot \hat{\mathrm{k}} \ \overline{\mathrm{b}} \cdot \hat{\mathrm{i}} \overline{\mathrm{b}} \cdot \hat{\mathrm{j}} \overline{\mathrm{b}} \cdot \hat{\mathrm{k}} \ \overline{\mathrm{c}} \cdot \hat{\mathrm{i}} \overline{\mathrm{c}} \cdot \hat{\mathrm{j}} \overline{\mathrm{c}} \cdot \hat{\mathrm{k}}\right|\) then the value of \(\lambda\) is

\(A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1\end{array}\right]\) and \(A^{-1}=\frac{1}{2}\left[\begin{array}{ccc}1 & -1 & 1 \\ -8 & 6 & 2 c \\ 5 & -3 & 1\end{array}\right]\), then values of \(a\) and \(\mathrm{c}\) are respectively

A ray of light is incident at polarising angle such that its deviation is \(24^{\circ}\), then angle of incidence is

The value of \(\int_{\frac{1}{3}}^1\left(x-x^3\right)^{\frac{1}{3}} \mathrm{~d} x\) is

If the sum of the mean and the variance of a binomial distribution for 5 trials is \(1 \cdot 8\),
then \(\mathrm{p}=\)

\(\int \frac{x^3-7 x+6}{x^2+3 x} \mathrm{~d} x=\)

Sodium and copper have work functions \(2.3 \mathrm{~ev}\) and \(4.5 \mathrm{~ev}\) respectively. The ratio of threshold wavelength of sodium to that of copper is nearest to