GATE EC 2008

Eigenvalues are the roots of the determinant formed by matrix [sI-P]
\begin{array}{l} [sI-P]=\left[\begin{array}{cc}s-P_{11} & -P_{12} \\-P_{21} & s-P_{22}\end{array}\right] \\\; [s I - P]=0 \\ \Rightarrow \quad \left(s-P_{11}\right)\left(s-P_{22}\right)-P_{12} P_{21}=0\\ \Rightarrow \quad s^{2}-\left(P_{11}+P_{22}\right) s+P_{11} P_{22}-P_{12} P_{21}=0 \end{array}
Since, one of the its eigenvalues is zero, therefore,
putting s=0
P_{11} P_{22}-P_{12} P_{21}=0
which is the desired condition.

The system can be written in matrix from as
\left[\begin{array}{ll}4 & 2 \\ 2 & 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}7 \\6 \end{array}\right]
The Augmentod matrix [AIB] is given by
\left[\begin{array}{ll|l}4 & 2 & 7 \\ 2 & 1 & 6\end{array}\right]
Performing Gauss elimination on this [A \mid B] as follows:
\left[\begin{array}{ll|l}4 & 2 & 7 \\2 & 1 & 6 \end{array}\right] \frac{R_{2}-\frac{2}{4} R_{1}}{=R_{2}-\frac{1}{2} R_{1}}\left[\begin{array}{ll|l}4 & 2 & 7 \\0 & 0 & 5 / 2 \end{array}\right]
Now Rank [A \mid B]=2
(The number of non-zero rows in [ A|B]
Rank [A]=1
(The number of non-zero rows in [A])
since, Rank [A \mid B] \neq \text{Rank}[A].
The system has no solution.

sinz can have value between -1 to +1. Thus no solution.

f(x)=e^{x}+e^{-x}=e^{x}+\frac{1}{e^{x}}
Arithmetic mean of e^{x} and \frac{1}{e^{x}} is
\mathrm{A.M},=\frac{e^{x}+\frac{1}{e^{x}}}{2}
Geometric mean of e^{x} and \frac{1}{e^{x}} is
G.M. =e^{x} \cdot \frac{1}{e^{x}}=1
It is known that A.M. \geq G.M
\frac{\left(e^{x}+\frac{1}{e^{x}}\right)}{2} \geq 1
e^{x}+\frac{1}{e^{x}} \geq 2
Therefore, \left(e^{x}+e^{-x}\right)_{\min }=2

\begin{array}{l} \sin x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\ldots . \\ \cos x=1-\frac{x^{2}}{2}+\frac{x^{4}}{4}-\frac{x^{6}}{6}+\ldots . \end{array}