The set expression \(A \cup (B \cap (A' \cup B'))\) is equivalent to

The ratio of the coefficient of \(x^3\) to the term independent of \(x\) in the expansion of \(\left(2 x+\dfrac{1}{x^2}\right)^{12}\) is

If \(|\mathbf{a} \times \mathbf{b}|=5\) and \(|\mathbf{a} \cdot \mathbf{b}|=3\), then \(|\mathbf{a}|^{2}|\mathbf{b}|^{2}\) is equal to

If \(\lim _{x \rightarrow 0} \dfrac{\left(1+a^3\right)+8 e^{1 / x}}{1+\left(1-b^3\right) e^{1 / x}}=2\), then

A coin is tossed until a head appears or until the coin has been tossed three times. Given that 'head' does not appear on the first toss, what is the probability that the coin is tossed thrice?

A spherical snowball is melting such that its volume is decreasing at the rate of \(1 \mathrm{~cm}^3 / \mathrm{min}\). The rate at which the diameter is decreasing when the diameter is 10 cm is

If \(\sin x+\sin ^2 x=1\) then \(\cos ^8 x+2 \cos ^6 x+\cos ^4 x\) is equal to

The separation between C and O atoms in CO is 0.12 nm . The distance of C atom from the centre of mass is

In which one of the case the chemical reaction given in Column I does not match with the name of the reaction given in Column II?
\(\begin{array}{|l|l|l|} \hline \text{S.No.} & \text{Column I - Reactions} & \text{Column II - Name of reaction.} \\ \hline \text{A} & \mathrm{C}_6 \mathrm{H}_5 \mathrm{CHO}+ \text{Conc. NaOH, heat} \rightarrow \mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_2 \mathrm{OH}+\mathrm{C}_6 \mathrm{H}_5 \mathrm{COONa} & \text{Cannizzaro's reaction} \\ \hline \text{B} & \mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_3+ \text{ (i) } \mathrm{CrO}_2 \mathrm{Cl}_2 / \mathrm{CS}_2 \text{ (ii) } \mathrm{H}_3 \mathrm{O}^{+} \rightarrow \mathrm{C}_6 \mathrm{H}_5 \mathrm{CHO} & \text{Etard reaction.} \\ \hline \text{C} & \mathrm{C}_6 \mathrm{H}_6+\mathrm{CO} / \mathrm{HCl} \text{ (anh. } \mathrm{AlCl}_3 \rightarrow \mathrm{C}_6 \mathrm{H}_5 \mathrm{CHO} & \text{Kolbe's reaction} \\ \hline \text{D} & \mathrm{RCN}+ \text{ (i) } \mathrm{SnCl}_2+\mathrm{HCl} \text{ (ii) } \rightarrow \mathrm{RCHO} & \text{Stephen's reaction.} \\ \hline \end{array}\)

If three numbers \(a, b, c\) constitute both an A.P and G.P, then

A string vibrates with a frequency of \(200 \mathrm{~Hz}\). When its length is doubled and tension is altered, it begins to vibrate with a frequency of \(300 \mathrm{~Hz}\). The ratio of the new tension to the original tension is

Evaluate \(\int \frac{1}{\sin x+\sqrt{3} \cos x} d x\).

A parallel plate capacitor has a capacity \(C\). The separation between the plates is doubled and a dielectric medium is inserted between the plates. If the capacity is \(3 C\), then the dielectric constant of the medium will be