Area of the region bounded by the curve \(y=\sin \left(\dfrac{x}{2}\right)\) between \(-4 \pi\) and 0 is

\(\text { Evaluate: } \cot ^{-1}\left(-\dfrac{3}{\sqrt{3}}\right)-\sec ^{-1}\left(-\dfrac{2}{\sqrt{2}}\right)-\operatorname{cosec}^{-1}(-1)-\tan ^{-1}(1)\)

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answer just by guessing, is

A and B play a game where each is asked to select a number from 1 to 25 . If the two numbers match, both will win a prize. The probability that they will NOT win a prize in a single trial is

The radius and centre of circle which equation is \(3 x^2+3 y^2+6 x+12 y+5=0\) is

The projection of \(=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) on \(\hat{v}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{R}}\) is

\(\mathrm{a}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \mathrm{b}=\hat{\mathrm{i}}-\hat{\mathrm{j}}\) 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

The function of \(\mathrm{AlCl}_{3}\) in Friedel-Craft's reaction is

In the group \(\left(G \otimes_{15}\right)\), where \(G=\{3,6,9,12\}\), \(\otimes_{15}\) is multiplication modulo 15 , the identity element is

Anisole can be prepared by the action of methyl iodide on sodium phenate. The reaction is called

When a body is earth connected, electrons from the earth flow into the body. This means the body is

\(\cot ^{-1}(21)+\cot ^{-1}(13)+\cot ^{-1}(-8)=\)

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?
\(\begin{array}{|l|l|l|} \hline \text{S.No.} & \text{Column I - Reactions} & \text{Column II - Name of reaction.} \\ \hline \text{A} & \mathrm{C}_6 \mathrm{H}_5 \mathrm{CHO}+ \text{Conc. NaOH, heat} \rightarrow \mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_2 \mathrm{OH}+\mathrm{C}_6 \mathrm{H}_5 \mathrm{COONa} & \text{Cannizzaro's reaction} \\ \hline \text{B} & \mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_3+ \text{ (i) } \mathrm{CrO}_2 \mathrm{Cl}_2 / \mathrm{CS}_2 \text{ (ii) } \mathrm{H}_3 \mathrm{O}^{+} \rightarrow \mathrm{C}_6 \mathrm{H}_5 \mathrm{CHO} & \text{Etard reaction.} \\ \hline \text{C} & \mathrm{C}_6 \mathrm{H}_6+\mathrm{CO} / \mathrm{HCl} \text{ (anh. } \mathrm{AlCl}_3 \rightarrow \mathrm{C}_6 \mathrm{H}_5 \mathrm{CHO} & \text{Kolbe's reaction} \\ \hline \text{D} & \mathrm{RCN}+ \text{ (i) } \mathrm{SnCl}_2+\mathrm{HCl} \text{ (ii) } \rightarrow \mathrm{RCHO} & \text{Stephen's reaction.} \\ \hline \end{array}\)