If \(2 x=-1+\sqrt{3} i\), then the value of \(\left(1-x^{2}+x\right)^{6}-\left(1-x+x^{2}\right)^{6}=\)

From a lot of 20 items containing 4 defective items, 2 items are drawn at random. If \(X\) denotes the number of defective items, then find the mean of the probability distribution of \(X\).

If \((1+i)^{200}=2^{98}(a+i b)\) then the value of \(a+b\) is

\(\text { The value of } \lim _\limits{x \rightarrow 1} \dfrac{x^{15}-1}{x^{10}-1}=\)

The number of solutions for the equation \(\sin 2 x+\cos 4 x=2\) is

A relation \(R\) is defined from \(\{2,3,4\}\) to \(\{3,6,7,10\}\). If \(x R y \Leftrightarrow x\) and \(y\) are co prime numbers. Then range of \(R\) is

Let \(A\) and \(G\) denote the arithmetic mean and geometric mean of positive real numbers \(5^x\) and \(5^{1-x}\). Then the minimum value of the expression \(5^x+5^{1-x}\) where \(x \in R\) is

The distance of the point \((3,4)\) from the line \(3 x+2 y+7=0\) measured along the line parallel to \(y-2 x+7=0\) is equal to

The stress verses strain graph of four metals A,B,C and D are as shown. Which of the graph represent the metal having greatest Young's modulus?

The potential difference between the points \(\mathrm{P}\) and \(\mathrm{Q}\) of the arrangement shown in figure is

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 dimensional formula for inductance is

For the same resonant frequency, if \(L\) is changed from \(L\) to \(\dfrac{L}{3}\), then capacitance should change from \(C\) to