GATE ME 2014 SET-3

 Question 1
Consider a 3x3 real symmetric matrix S such that two of its eigenvalues are $a\neq 0,\; b\neq 0$ with respective eigenvectors $\begin{bmatrix} x_{1}\\ x_{2}\\ x_{3} \end{bmatrix}$,$\begin{bmatrix} y_{1}\\ y_{2}\\ y_{3} \end{bmatrix}$. If $a\neq b$ then $x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}$ equals
 A a B b C ab D 0
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
$3\times 3$ real symmetric matrix such that two of its
eigen value are a \neq 0 b \neq 0 with respective eigen
vectors $\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right]\left[\begin{array}{l}y_{1} \\ y_{2} \\ y_{3}\end{array}\right] \text{ if } a \neq b$ then
$x_{1} y_{1}+x_{2} y_{2}+x_{3} y_{3}=0$ because they are orthogonal.
$\therefore \qquad x^{T} y =0 \quad(\text { since } a \neq b)$
$\left[x_{1} x_{2} x_{3}\right]\left[\begin{array}{c}y_{1} \\y_{2} \\y_{3}\end{array}\right]=0$
$x_{1} y_{1}+x_{2} y_{2}+x_{3} y_{3}=0$
 Question 2
If a function is continuous at a point,
 A the limit of the function may not exist at the point B the function must be derivable at the point C the limit of the function at the point tends to infinity D the limit must exist at the point and the value of limit should be same as the value of the function at that point
Engineering Mathematics   Calculus
Question 2 Explanation:
f(x) is continuous at any point
$\text{if} \quad \lim_{x \rightarrow a^{-}} f(x)=\lim_{x \rightarrow a^{+}} f(x)=f(a)$

 Question 3
Divergence of the vector field $x^{2}z\hat{i}+xy\hat{j}-yz^{2}\hat{k}$ at $(1,-1,1)$ is
 A 0 B 1 C 5 D 6
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} \vec{F} &=x^{2} z \hat{i}+x y \hat{j}-y z^{2} \hat{k} \\ \nabla \cdot \vec{F} &=\frac{\partial}{\partial x}\left(x^{2} z\right)+\frac{\partial}{\partial y}(x y)-\frac{\partial}{\partial z}\left(y z^{2}\right) \\ \nabla \cdot \vec{F} &=2 x z+x-2 y z \\ \left.\therefore \nabla \cdot \vec{F}\right|_{1,-1,1} &=2 \times 1 \times 1+1-2 \times-1 \times 1 \\ &=2+1+2=5 \end{aligned}
 Question 4
A group consists of equal number of men and women. Of this group 20% of the men and 50% of the women are unemployed. If a person is selected at random from this group, the probability of the selected person being employed is _______
 A 0.25 B 1 C 4.5 D 0.65
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
Total $\%$ of employed person
\begin{aligned} &=100-\frac{20+50}{2} \% \\ &=100-35 \%=65 \% \\ \text { Required prob } &=\frac{65}{100}=0.65 \end{aligned}
 Question 5
The definite integral $\int_{1}^{3}\frac{ 1}{x}\mathrm{d}x$ is evaluated using Trapezoidal rule with a step size of 1. The correct answer is _______
 A 1.165 B 2.365 C 8.254 D 9.548
Engineering Mathematics   Numerical Methods
Question 5 Explanation:

\begin{aligned} I &=\int_{1}^{3} \frac{1}{x} \mathrm{d} x=\frac{h}{2}\left[\left(y_{0}+y_{2}\right)+2 y_{1}\right] \\ &=\frac{1}{2}[1+0.33+2 \times 0.5]=\frac{2.33}{2}=1.165 \end{aligned}

There are 5 questions to complete.