The probability function of a random variable \(X\) is given by \(P(X=k)=c k^2\), where \(c\) is a constant and \(k \in\{0,1,2,3,4\}\). If \(\sigma^2\) is the variance of \(X\) and \(\mu\) is the mean of \(X\), then \(\sigma^2+\mu^2=\)

If \(\int \sin ^5 x d x=\frac{-\cos ^5 x}{5}+a \cos ^3 x+b \cos x+c\), then \(a+b=\)

Find the angle between the planes \(x+2 y+2 z-5=0\) and \(3 x+3 y+2 z-8=0\)

The number of four-digit numbers formed by using the digits \(0,2,4,5\) and which are not divisible by 5 , is

Assertion (A): If \(B\) is a \(3 \times 3\) matrix and \(|B|=6\), then \(|\operatorname{Adj}(B)|=36\)
Reason (R): If \(B\) is a square matrix of order \(n\), then \(|\operatorname{Adj}(B)|=|B|^{\mathbf{n}}\)

\(\lim _{x \rightarrow 0} x^3\left\{\sqrt{x^2+\sqrt{x^4+1}}-\sqrt{2} x\right\}=\)

At \(\mathrm{x}=0, f(x)=\cos x-1+\frac{x^2}{2}-\frac{x^3}{3}\)

If \(\bar{a}, \bar{b}\) and \(\bar{c}\) are vectors such that \(|\bar{a}|=\left|\frac{\bar{b}}{2}\right|=\left|\frac{\bar{c}}{3}\right|=1 ; \bar{b}\) and \(\bar{c}\) are perpendicular; and the projections of \(\bar{b}\) and \(\bar{c}\) on \(\bar{a}\) are equal then \(|\bar{a}-\bar{b}+\bar{c}|=\)

Frank and Hertz experiment proves that

Through the point (4, 5), a straight line is drawn making positive intercepts on the coordinate axes. The area of the triangle thus
formed is least, when the ratio of the intercepts on the X and Y axes is

The minimum value of \(|z-1|+|z-5|\) is

The angle between the lines \(2 x+11 y-7=0\) and \(x+3 y+5=0\) is

From the following data
\[
\begin{aligned}
& \mathrm{CH}_3 \mathrm{OH}(l)+\frac{3}{2} \mathrm{O}_2(g) \longrightarrow \mathrm{CO}_2(g)+2 \mathrm{H}_2 \mathrm{O}(l) \\
& \Delta_r H^{\circ}=-726 \mathrm{~kJ} \mathrm{~mol}^{-1} \\
& \mathrm{H}_2(g)+\frac{1}{2} \mathrm{O}_2(g) \longrightarrow \mathrm{H}_2 \mathrm{O}(l) \text {; } \\
& \Delta_r \mathrm{H}^{\circ}=-286 \mathrm{~kJ} \mathrm{~mol}^{-1} \\
& \mathrm{C}(\text { graphite })+\mathrm{O}_2(\mathrm{~g}) \longrightarrow \mathrm{CO}_2(\mathrm{~g}) \text {; } \\
& \Delta_r \mathrm{H}^{\circ}=-393 \mathrm{~kJ} \mathrm{~mol}^{-1} \\
&
\end{aligned}
\]
The standard enthalpy of formation of \(\mathrm{CH}_3 \mathrm{OH}(l)\) in \(\mathrm{kJ} \mathrm{mol}^{-1}\) is