Differentiate \(\log _a x\) with respect to \(a^x\)

The distance between the point \((1,2)\) from the line \(3 x+4 y=6\) measured parallel to the line \(3 x-4 y=5\).

The value of the expression \(\cos^{-1}\left(\cos\dfrac{7\pi}{6}\right) + \sin^{-1}\left(\sin\dfrac{22\pi}{3}\right) + \tan^{-1}\left(\tan\dfrac{4\pi}{5}\right)\) is:

If the distance ' \(S\) metres traversed by a particle in \(t\) seconds is given by \(S=t^{3}-3 t^{2}\), then the velocity of the particle when the acceleration is zero, in \(\mathrm{m} / \mathrm{sec}\) is

If \(S_{n}=\frac{1}{6 \cdot 11}+\frac{1}{11 \cdot 16}+\frac{1}{16 \cdot 21}+\ldots\) to \(n\) terms, then \(6 S_{n}=\)

How many numbers of \(6 -\)digits can be formed from the digits of the number \(112233\) ?

\(\mathrm{a}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \mathrm{b}=\hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\mathrm{c}=5 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\), then unit vector parallel to \(\mathrm{a}+\mathrm{b}-\mathrm{c}\) but in opposite direction is

If \(\mathrm{p}=\hat{i}+\hat{j}, \mathrm{q}=4 \hat{k}-\hat{j}\) and \(\mathrm{r}=\hat{i}+\hat{k}\), then the unit vector in the direction of \(3 \mathrm{p}+\mathrm{q}-2 \mathrm{r}\) is

A man grows into a giant such that his height increases to 8 times his original height. Assuming that his density remains same, the stress in the leg will change by a factor of

Match the reaction in Column I with the major product formed given in Column II.
\(\begin{array}{|l|l|l|} \hline & \text { Column I- Reaction/conditions } & & \text { Column II- Product } \\ \hline A & \mathrm{C}_6 \mathrm{H}_6 \xrightarrow[\substack{\text{(ii) NaOH} \\ \text{(iii) H^+}}]{\text{(i) Oleum}} & P & \text { Pentan-2-ol. } \\ \hline B & \text{Butanal } \xrightarrow[\text{(ii) } \mathrm{H}^+]{\text{(i) } \text{MeMgBr/ dry ether}} & Q & \text { 2-Hydroxybenzoic acid. } \\ \hline C & \text {2-Methylpropene } \xrightarrow[\text { (ii) } \mathrm{H}_2 \mathrm{O}_2 / \mathrm{NaOH}]{\text { (i) } \mathrm{B}_2 \mathrm{H}_6} & R & \text { Phenol } \\ \hline D & \text{Phenol } \xrightarrow[\text { (ii) } \mathrm{CO}_2, 400 \mathrm{~K}, 7 \mathrm{~atm} .]{\text { (i) } \mathrm{NaOH}} & S & \text { 2-Methylpropan-1-0 I } \\ \hline \end{array}\)

The radius of a nucleus as measured by electron scattering is \(4.8 \mathrm{~fm}\). The mass number of nucleus is most likely to be

Cationic complex is

Two point charges \(Q_{1}=2 \mu \mathrm{C}\) and \(Q_{2}=1 \mu \mathrm{C}\) are placed as shown. The coordinates of the point \(P\) are \((2 \mathrm{~cm}, 1 \mathrm{~cm})\). The electric intensity vector at \(P\) subtends an angle \(\theta\) with the positive \(X\)-axis. The value of \(\theta\) is given by