# GATE EC 2008

 Question 1
All the four entries of the 2x2 matrix $P=\begin{bmatrix} P_{11} &P_{12} \\ P_{21} & P_{22} \end{bmatrix}$ are nonzero, and one of its eigenvalue is zero. Which of the following statements is true?
 A $P_{11}P_{22}-P_{12}P_{21}=1$ B $P_{11}P_{22}-P_{12}P_{21}=-1$ C $P_{11}P_{22}-P_{12}P_{21}=0$ D $P_{11}P_{22}+P_{12}P_{21}=0$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
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.
 Question 2
The system of linear equations
4x + 2y = 7
2x + y = 6
has
 A a unique solution B no solution C an infinite number of solutions D exactly two distinct solutions
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
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.

 Question 3
The equation sin(z) = 10 has
 A no real or complex solution B exactly two distinct complex solutions C a unique solution D an infinite number of complex solutions
Engineering Mathematics   Complex Analysis
Question 3 Explanation:
sinz can have value between -1 to +1. Thus no solution.
 Question 4
For real values of x, the minimum value of the function f(x)=exp(x)+exp(-x) is
 A 2 B 1 C 0.5 D 0
Engineering Mathematics   Calculus
Question 4 Explanation:
$f(x)=e^{x}+e^{-x}=e^{x}+\frac{1}{e^{x}}$
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, $\left(e^{x}+e^{-x}\right)_{\min }=2$
 Question 5
Which of the following functions would have only odd powers of x in its Taylor series expansion about the point x = 0 ?
 A $sin(x^{3})$ B $sin(x^{2})$ C $cos(x^{3})$ D $cos(x^{2})$
Engineering Mathematics   Calculus
Question 5 Explanation:
$\begin{array}{l} \sin x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\ldots . \\ \cos x=1-\frac{x^{2}}{2}+\frac{x^{4}}{4}-\frac{x^{6}}{6}+\ldots . \end{array}$

There are 5 questions to complete.