If \(A, B\) and \(C\) are mutually exclusive and exhaustive events of a random experiment such that \(P(B)=\frac{3}{2} P(A)\) and \(P(C)=\frac{1}{2} P(B)\), then \(P(A \cup C)\) equals to

\(\int_{-1}^{1}|1-x| d x\) is equal to

If \(\mathbf{a} \hat{\mathfrak{i}}=\mathbf{a}(\hat{\hat{i}}+\hat{\hat{j}})=\mathbf{a}(\hat{\mathbf{i}}+\hat{\hat{b}}+\hat{\mathbf{k}})=1\), then \(\mathbf{a}=\)

\(11^{3}+12^{3}+13^{3}+\ldots . .+20^{3}\) is

\(\int \dfrac{\log x}{(1 + x)^2}\, dx\)

If \(\lim _{x \rightarrow \infty}\left(1+\dfrac{a}{x}+\dfrac{b}{x^2}\right)^{2 x}=e^2\), then

The expression \(\frac{1}{\sqrt{(3 x+1)}}\left[\left(\frac{1+\sqrt{3 x+1}}{2}\right)^{7}-\left(\frac{1-\sqrt{3 x+1}}{2}\right)^{7}\right]\) is a polynomial in \(x\) of degree

In the metallurgy of aluminium, when the fused matrix is electrolysed, the overall reduction reaction is

Match \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D with the appropriate functions given.
\(\begin{array}{|l|l|l|l|} \hline & \text { Column I } & & \text { Functions } \\ \hline \text { A } & \text { Oxidoreductase } & \text { P } & \text { Malfunctioning leads to Addison's disease. } \\ \hline \text { B } & \text { Phosphodiester bonds } & \text { Q } & \text { Regulates responses to external stimuli. } \\ \hline \text { C } & \text { Adrenal cortex } & \text { R } & \text { Catalyses glycolysis. } \\ \hline \text { D } & \text { Epinephrine. } & \text { S } & \text { Links nucleotides together. } \\ \hline \end{array}\)

If \(Q(1,0,1)\) is the image of the point \(P(a, b, c)\) in the line \(\dfrac{x+1}{2}=\dfrac{y-3}{-2}=\dfrac{z}{-1}\) then \(a+b+c\) is equal to

The chemical name of anisole is

Compounds \(\mathrm{A}\) and \(\mathrm{B}\), having the same molecular formula \((\mathrm{C}_4 \mathrm{H}_8 \mathrm{O})\), react separately with \(\mathrm{CH}_3 \mathrm{MgBr}\), followed by reaction with dil. \(\mathrm{HCl}\) to form compounds \(\mathrm{X}\) and \(\mathrm{Y}\) respectively. Compound \(\mathrm{Y}\) undergoes acidic dehydration in presence of Conc. \(\mathrm{H}_2 \mathrm{SO}_4\) much more readily than \(\mathrm{X}\). Compound \(\mathrm{Y}\) also reacts with Lucas reagent, much more readily than \(\mathrm{X}\), with appearance of turbidity. Identify \(\mathrm{X}\) and \(\mathrm{Y}\).

Which of the following is most stable?