The angle between the vectors \(\mathrm{a}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\) and \(\mathrm{b}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}\) is

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

Which of the following relations on the set of real numbers \(\mathrm{R}\) is an equivalence relation?

In an entrance test, there are multiple choice questions. There are four possible answers to each question of which only one is correct. The probability that a student knows the answer to a question is \(90 \%\). If he gets the correct answer to a question, then the probability that he was guessing is

\(\left|\begin{array}{ccc}
\cos (\alpha+\beta) & -\sin (\alpha+\beta) & \cos 2 \beta \\
\sin \alpha & \cos \alpha & \sin \beta \\
-\cos \alpha & \sin \alpha & \cos \beta
\end{array}\right|
\) is independent of

If \(f(x)=\dfrac{(x+1)^7 \sqrt{1+x^2}}{\left(x^2-x+1\right)^6}\) then the value of \(f^{\prime}(0)\) is equal to

The sides of a triangle are \(x=2, y+1=0\) and \(x+2 y=4\). Its circumcentre is

When the temperature of a reaction \(\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}\) is increased from 300 K to 310 K the rate constant increases by \(12 \%\). What is the Activation energy of the reaction?

The feasible region corresponding to the linear constraints of a Linear Programming problem is given below.




Which of the following is NOT a constraint to the given LPP

Rank the following compounds in order of increasing basicity.

\(\mathrm{Cu}^{+}\) undergoes disproportionation, according to the equation,
\(
2 \mathrm{Cu}^{+} \rightleftharpoons \mathrm{Cu}^{2+}+\mathrm{Cu}
\)
The \(\mathrm{E}^{\circ}\) value for the reaction is:
\([\mathrm{E}_{\mathrm{Cu}^{2+} / \mathrm{Cu}}^{\mathrm{o}}=0.34 \mathrm{~V} \text { and } \mathrm{E}_{\mathrm{Cu}^{2+} / \mathrm{Cu}^{+}}^{\mathrm{O}}=0.15 \mathrm{~V}]\)

Choose the correct metal/ ion from the brackets which ..........
A. has chemical reactivity similar to that of the first few members of the Lanthanoids \((\mathrm{Zn}, \mathrm{Ca}, \mathrm{Fe}, \mathrm{Cu})\).
B. has stable \(4 \mathrm{f}^7\) electronic configuration, but acts as a strong reducing agent and converts to \(\mathrm{M}^{3+}\) state. \(\left(\mathrm{Eu}^{2+}, \mathrm{Ce}^{2+}, \mathrm{Pr}^{2+}, \mathrm{Dy}^{2+}\right)\)
C. is a colorless ion \(\left(\mathrm{Tm}^{3+}, \mathrm{Lu}^{3+}, \mathrm{Gd}^{3+}, \mathrm{Sm}^{3+}\right)\).
D. shows stable +2 oxidation state and is diamagnetic \((\mathrm{Ce}, \mathrm{Sm}, \mathrm{Ho}, \mathrm{Yb})\)

An aqueous solution of \(X\) on addition of hydrogen peroxide in ice cold conditions gives blue colour to the ethereal layer. Then, \(X\) can be