If \(\omega_1\) and \(\omega_2\) are two non-zero complex numbers and \(\mathrm{a}, \mathrm{b}\) are non zero real numbers such that \(\left|\mathrm{a} \omega_1+\mathrm{b} \omega_2\right|=\left|\mathrm{a} \omega_1-\mathrm{b} \omega_2\right|\), then \(\frac{\omega_1}{\omega_2}\) is

If -1 is a twice repeated root of the equation \(a\left(x^3+x^2\right)+\) \(\mathrm{bx}+\mathrm{c}=0\), then \(\mathrm{a}: \mathrm{b}: \mathrm{c}=\)

The function \(f(x)=x e^{-x} \forall x \in \mathbb{R}\) attains a maximum value at \(x=k\), then \(k=\)

If \(\alpha, \beta\) are the roots of the equation \(x^2-2 x+4=0\) and for any \(n \in \mathrm{N}, \alpha^n+\beta^n=k \cos \frac{n \pi}{3}\) then \(k=\)

\(\int \frac{x^4-1}{x^2 \sqrt{x^4+x^2+1}} d x=\)

If the mean and variance of a binomial distribution are 4 and 2 respectively, then the probability of 2 successes of that binomial variate \(X\), is

The coefficient of \(x^f\) in the expansion of \(\frac{1}{\sqrt[3]{(1-2 x)^2}}\) is

If \(\vec{a}, \vec{b}\) are the two non collinear vectors. then \(|\vec{b}| \vec{a}+|\vec{a}| \vec{b}\) represents

Observe the following statements :
I. Bleaching powder is used in the preparation of chloroform.
II. Bleaching powder decomposes in the presence of \(\mathrm{CoCl}_2\) to liberate \(\mathrm{O}_2\).
III. Aqueous \(\mathrm{KHF}_2\) is used in the preparation of fluorine.
The correct combination is :

The power of the point \((-3,7)\) with respect to a circle, with centre \((3,7)\) and radius 2, is

Nitrobenzene on reduction using zinc in alkaline medium results in \(X\). The number of \(\operatorname{sigma}(\sigma)\) and pi \((\pi)\) bonds in \(X\) is

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\)
Assertion (A) : The identity
\(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for
\(\vec{a}\).
Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\),
\(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\)
Which of the following is correct?

If $\overline{V}=2\overline{i}+\overline{j}-\overline{k}, \overline{W}=\overline{i}+3\overline{k}$ and $\overline{U}$ is a unit vector, then the maximum value of $\left[\overline{U}\overline{V}\overline{W}\right]$ is