A point on \(X O Z\) -plane divides the join of \((5,-3,-2)\) and \((1,2,-2)\) at

If a function \(f: R \rightarrow R\) is defined by \(f(x)=\frac{4 x}{5}+3\), then \(f^{-1}(x)=\)

A fair die with numbers 1 to 6 on their faces is thrown. Let \(\mathrm{X}\) denote the number of factors of the number, on the uppermost face, then the probability distribution of \(\mathrm{X}\) is

If \(x+\mathrm{y}=6, x \geqslant 0, y \geqslant 0\), then the maximum value of \(x^2 \mathrm{y}\) is

If \(\sin ^{-1} x+\cos ^{-1} y=\frac{3 \pi}{10}\), then the value of \(\cos ^{-1} x+\sin ^{-1} y\) is

The order of the differential equation, whose general solution is given by
\(y=\left(\mathrm{c}_1+\mathrm{c}_2\right) \cos \left(x+\mathrm{c}_3\right)-\mathrm{c}_4 \mathrm{e}^{x+\mathrm{c}^5}\)
where \(c_1, c_2, c_3, c_4\) and \(c_5\) are arbitrary constant, is

\(f(x)=\left\{\begin{array}{cc}
3-x, & -1 \leqslant x < 0 \\
1+\frac{5 x}{3}, & -3 \leqslant x \leqslant 2
\end{array}\right.\)
and \(g(x)=\left\{\begin{array}{rr}-x, & -2 \leqslant x \leqslant 3 \\ x, & 0 \leqslant x \leqslant 1\end{array}\right.\) then range of \((f o g)(x)\) is

Let \(\quad \overline{\mathrm{a}}=\alpha \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \quad \overline{\mathrm{b}}=3 \hat{\mathrm{i}}-\beta \hat{\mathrm{j}}+4 \hat{\mathrm{k}} \quad\) and \(\overline{\mathrm{c}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}\), where \(\alpha, \beta \in \mathbb{R}\), be three vectors. If the projection at \(\overline{\mathrm{a}}\) on \(\overline{\mathrm{c}}\) is \(\frac{10}{3}\) and \(\overline{\mathrm{b}} \times \overline{\mathrm{c}}=-6 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}\), then the value of \(\alpha^2+\beta^2-\alpha \beta\) is equal to

In a meeting \(70 \%\) of the members favour and \(30 \%\) oppose a certain proposal. A member is selected at random. We take \(\mathrm{X}=0\) if he opposed the proposal and we take \(\mathrm{X}=1\), if the member is in favour. Then variance of \(X\) is

Identify the catalyst used in following reaction at \(500^{\circ} \mathrm{C}\) ?
\(\mathrm{CO}+\mathrm{H}_2 \mathrm{O} \rightleftharpoons \mathrm{CO}_2+\mathrm{H}_2\)

The value of \(\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)\) at \(x=\frac{1}{5}\), where \(0 \leq \cos ^{-1} x \leq \pi\) and \(-\frac{\pi}{2} \leq \sin ^{-1} x \leq \frac{\pi}{2}\), is

Dissociation constant of 0.01 M weak acid is \(10^{-4}\). What is percent dissociation of acid?

What is degree of dissociation of \(\mathrm{CH}_3 \mathrm{COOH}\) if \(\wedge^{\circ}\left(\mathrm{CH}_3 \mathrm{COO}^{-}\right)=50 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}\), \(\wedge^{\circ}\left(\mathrm{H}^{+}\right)=350 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}\) and molar conductivity of \(5 \times 10^{-2} \mathrm{M} \mathrm{CH}_3 \mathrm{COOH}\) is \(20 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}\) ?