Suppose $\lambda$ is an eigenvalue of matrix A and $x$ is the corresponding eigenvector. Let $x$ also be an eigenvector of the matrix $\mathrm{B}=\mathrm{A}-2 \mathrm{I}$, where I is the identity matrix. Then, the eigenvalue of B corresponding to the eigenvector $x$ is equal to

Let $A=\left[\begin{array}{cc}1 & 1 \\ 1 & 3 \\ -2 & -3\end{array}\right]$ and $b=\left[\begin{array}{l}b_1 \\ b_2 \\ b_3\end{array}\right]$. For $\mathrm{Ax}=\mathrm{b}$ to be solvable, which one of the following options is the correct condition on $b_1, b_2$ and $b_3$ :

Which one of the following options is the correct Fourier series of the periodic function $f(x)$ described below:

$$ f(x)=\left\{\begin{array}{cl} 0 & \text { if }-2 < x < -1 \\ 2 k & \text { if }-1 < x < 1 \text {; period }=4 \\ 0 & \text { if }-1 < x < 2 \end{array}\right. $$

$X$ is the random variable that can take any one of the values, $0,1,7,11$ and 12 . The probability mass function for $X$ is

$$ \begin{aligned} & \mathrm{P}(X=0)=0.4 ; \mathrm{P}(X=1)=0.3 ; \mathrm{P}(X=7)=0.1 ; \\ & \mathrm{P}(X=11)=0.1 ; \mathrm{P}(X=12)=0.1 \end{aligned} $$

Then, the variance of $X$ is