Equation of the plane which passes through the point of intersection of lines \(\frac{{x - 1}}{3} = \frac{{y - 2}}{1} = \frac{{z - 3}}{2}\) and \(\frac{{x - 3}}{1} = \frac{{y - 1}}{2} = \frac{{z - 2}}{3}\) and has the largest distance from the origin is

If the coefficients of \(x\) and \(x^2\) in \((1+x)^p(1-x)^q\) are \(4\) and \(-5\) respectively, then \(2 p+3 q\) is equal to

Let \( \bar x , M\) and \(\sigma^2\) be respectively the mean, mode and variance of \(n\) observations \(x_1 , x_2,...,x_n\) and \(d_i\, = - x_i - a, i\, = 1, 2, .... , n\), where \(a\) is any number.
Statement \(I\): Variance of \(d_1, d_2,.....d_n\) is \(\sigma^2\).
Statement \(II\) : Mean and mode of \(d_1 , d_2, .... d_n\) are \(-\bar x -a\) and \(- M - a\), respectively

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number \(2\) times is equal to the probability of getting an even number \(3\) times, then the probability of getting an odd number for odd number of times is

If \(|\vec{a}|=2,|\vec{b}|=5\) and \(|\vec{a} \times \vec{b}|=8\), then \(|\vec{a} \cdot \vec{b}|\) is equal to :

Fractional part of the number \(\frac{4^{2022}}{15}\) is equal to

If \(P\) is a \(3 \times 3\) real matrix such that \(P ^{ T }=a P +( a -1) I\), where \(a > 1\), then \(..........\)

Let \(A=\left[\begin{array}{l}a_{1} \\ a_{2}\end{array}\right]\) and \(B=\left[\begin{array}{l}b_{1} \\ b_{2}\end{array}\right]\) be two \(2 \times 1\) matrices with real entries such that \(A = XB,\) where \(X=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1 & -1 \\ 1 & k\end{array}\right],\) and \(k \in R\). If \(a _{1}^{2}+ a _{2}^{2}=\frac{2}{3}\left( b _{1}^{2}+ b _{2}^{2}\right)\) and \(\left( k ^{2}+1\right) b _{2}^{2} \neq-2 b _{1} b _{2}\) then the value of \(k\) is ....... .

For the function \(f(x) = e^{\sin|x|} - |x|\), \(x \in \mathbb{R}\), consider the following statements:
Statement I: \(f\) is differentiable for all \(x \in \mathbb{R}\).
Statement II: \(f\) is increasing in \(\left(-\pi, -\dfrac{\pi}{2}\right)\).
In the light of the above statements, choose the correct answer from the options given below:

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to

If the domain of the function \(f(\mathrm{x})=\frac{\cos ^{-1} \sqrt{x^{2}-x+1}}{\sqrt{\sin ^{-1}\left(\frac{2 x-1}{2}\right)}}\) is the interval \((\alpha, \beta]\), then \(\alpha+\beta\) is equal to:

Let P be a point in the plane of the vectors \(\overrightarrow{AB}=3\hat{i}+\hat{j}-\hat{k}\) and \(\overrightarrow{AC}=\hat{i}-\hat{j}+3\hat{k}\) such that P is equidistant from the lines AB and AC. If \({|\overrightarrow{AP}|}=\frac{\sqrt{5}}{2}\) then the area of the triangle ABP is :

Let a circle \(C\) in complex plane pass tltrough the points \(z _{1}=3+4 i , z _{2}=4+3 i\) and \(z _{3}=5 i\). If \(z \left(\neq z _{1}\right)\) is a point on \(C\) such that the line through \(z\) and \(z _{1}\) is perpendicular to the line through \(z _{2}\) and \(z _{3}\), then \(\arg ( z )\) is equal to