A unit vector that is perpendicular to the vector \(2 \overline{\mathrm{i}}-\overline{\mathrm{j}}+2 \overline{\mathrm{k}}\) and coplanar with the vectors \(\overline{\mathrm{i}}+\overline{\mathrm{j}}-\overline{\mathrm{k}}\) and \(2 \overline{\mathrm{i}}+2 \overline{\mathrm{j}}-\overline{\mathrm{k}}\) is

The number of four digit numbers that can be formed using the digits \(1,2,3,4,5,6,7,8,9\) which are divisible by 3 , when repetition of digits is allowed any number of times, is

If \(A=\left[\begin{array}{lll}x & 2 & 1 \\ 2 & x & 1 \\ 2 & 1 & 0\end{array}\right]\) and \(\operatorname{det}\left(A^3\right)=125\), then \(x=\)

The coefficient of \(x^5\) in the expansion of \(\left(1+x^2\right)^5(1+x)^4\), is :

If \(\alpha=\lim _{x \rightarrow 0} \frac{x \cdot 2^x-x}{1-\cos x}\) and \(\beta=\lim _{x \rightarrow 0} \frac{x \cdot 2^x-x}{\sqrt{1+x^2}-\sqrt{1-x^2}}\), then

Let \(z=x+i y\) be a complex number with \(x, y \in Z\). Then, the area (in sq units) of the rectangle whose vertices are the roots of the equation \(\bar{z} \cdot z^3+z \cdot \bar{z}^3=350\) is

The number of ways of distributing eight identical rings to three different girls so that every girl gets at least one ring is

The hybridisations of \(\mathrm{N}\)-orbitals in \(\mathrm{NO}_3^{-}, \mathrm{NO}_2^{-}\) and \(\mathrm{NH}_4^{+}\)respectively are

The sensitivity of a galvanometer is \(60 \mathrm{div} / \mathrm{A}\). When a shunt is used, its sensitivity becomes \(10 \mathrm{div} / \mathrm{A}\). If the resistance of the galvanometer is \(20 \Omega\), then the value of shunt used is

The work done by a force $\overrightarrow{F}=-5x^4 \hat{\mathrm{i}} \mathrm{N}$ in displacing a body from $x=2 \mathrm{m}$ to $x=-2 \mathrm{m}$ is

\[
f(x)= \begin{cases}(1+3 x)^{\frac{4}{x},} & \text { if } x \neq 0, \\ a, & \text { if } x=0 .\end{cases}
\]
If \(f\) is continuous, then \(\log a=\)

Which of the following reactions are not feasible?
I. \(\mathrm{CH}_3 \mathrm{CH}=\mathrm{CH}_2 \frac{\mathrm{HCl}}{\left(\mathrm{C}_6 \mathrm{H}_5 \mathrm{CO}\right)_2 \mathrm{O}_2} \longrightarrow \mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CH}_2 \mathrm{Cl}\)
II. \(\mathrm{C}_6 \mathrm{H}_6+\mathrm{CH}_3 \mathrm{CH}_2 \mathrm{CH}_2 \mathrm{Cl} \stackrel{\mathrm{AlCl}_3}{\longrightarrow} \mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}\left(\mathrm{CH}_3\right)_2\)
III. \(\mathrm{CH}_3 \mathrm{CHClCH}_2 \mathrm{Cl} \underset{\Delta}{\stackrel{\mathrm{KOH}}{\longrightarrow}} \mathrm{CH}_3 \mathrm{C} \equiv \stackrel{+}{\mathrm{CH}}\)
IV. \(\mathrm{CH}_3 \mathrm{CH}=\mathrm{CHCH}_3 \stackrel{\mathrm{KMnO}_4 / \mathrm{H}^{+}}{\longrightarrow} \mathrm{CH}_3 \mathrm{COOH}\)

If the probability distribution of a random variable \(X\) is as follows, then \(P(x \leq 2)=\)

\(x_i\)	0	1	2	3	4
\(P\left(X = x_i\right)\)	3K	5K	\(3k^2\)	\(4k^2 + k\)	\(3k^2\)