# GATE ME 2018 SET-1

 Question 1
Four red balls, four green balls and four blue balls are put in a box. Three balls are pulled out of the box at random one after another without replacement. The probability that all the three balls are red is
 A $\frac{1}{72}$ B $\frac{1}{55}$ C $\frac{1}{36}$ D $\frac{1}{27}$
Engineering Mathematics   Probability and Statistics
Question 1 Explanation: Probability that all the three balls are red is
$\begin{array}{l} =\mathrm{R} \cdot \mathrm{R} \cdot \mathrm{R} \\ =\frac{4}{12} \times \frac{3}{11} \times \frac{2}{10}=\frac{24}{1320}=\frac{1}{55} \end{array}$
 Question 2
The rank of the matrix $\begin{bmatrix} -4 & 1 & -1 \\ -1 & -1 & -1 \\ 7 & -3 & 1 \end{bmatrix}$ is
 A 1 B 2 C 3 D 4
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
\begin{aligned} &\left[\begin{array}{ccc}-4 & 1 & -1 \\-1 & -1 & -1 \\7 & -3 & 1\end{array}\right]\\ R_{1} \longleftrightarrow R_{2} \qquad&{\left[\begin{array}{ccc}-1 & -1 & -1 \\-4 & 1 & -1 \\7 & -3 & 1\end{array}\right]} \\ R_{2}-4 R_{1}, R_{3}+7 R_{1} \qquad&{\left[\begin{array}{ccc}-1 & -1 & -1 \\0 & 5 & 3 \\0 & -10 & -6\end{array}\right]} \\ R_{3}+2 R_{2} \qquad &{\left[\begin{array}{ccc}-1 & -1 & -1 \\0 & 5 & 3 \\0 & 0 & 0\end{array}\right]} \end{aligned}
No. of non zero rows =2
rank=2

 Question 3
According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value $\xi$ in this interval such that $\int_{a}^{b}f(x)dx$ =
 A f($\xi$)(b-a) B f(b)($\xi$-a) C f(a)(b-$\xi$) D 0
Engineering Mathematics   Calculus
Question 3 Explanation:
$\int_{a}^{b} f(\xi) d x=f(\xi)(b-a)$
 Question 4
F(z) is a function of the complex variable z = x + iy given by
$F(z)=iz+kRe(z)+ i \; lm(z)$
For what value of k will F(z) satisfy the Cauchy-Riemann equations?
 A 0 B 1 C -1 D y
Engineering Mathematics   Complex Variables
Question 4 Explanation:
\begin{aligned} F(z) &=i z+k \operatorname{Re}(z)+i \operatorname{Im}(z) \\ u+i v &=i(x+i y)+k x+i y \\ u+i v &=k x-y+i(x+y) \\ u &=k x-y, v=x+y \\ u_{x} &=k, u_{y}=-1 \\ V &=x+y \\ v_{x} &=1 \\ v_{y} &=1 \\ u_{x} &=v_{y} \\ k &=1 \end{aligned}
 Question 5
A bar of uniform cross section and weighing 100 N is held horizontally using two massless and inextensible strings S1 and S2 as shown in the figure. The tension of the strings are
 A $T_{1} = 100 N \; and \; T_{2} = 0 N$ B $T_{1} = 0 N \; and \; T_{2} = 100 N$ C $T_{1} = 75 N \; and \; T_{2} = 25 N$ D $T_{1} = 25 N \; and \; T_{2} = 75 N$
Engineering Mechanics   FBD, Equilirbium, Plane Trusses and Virtual work
Question 5 Explanation: \begin{aligned} T_{1}+T_{2} &=100 \mathrm{N} \qquad \ldots(i)\\ \Sigma M_{A} &=0 \\ T_{2} \cdot \frac{L}{2} &=100 \times \frac{L}{2} \\ \therefore \qquad T_{2} &=100 \mathrm{N} \\ T_{1} &=0 \mathrm{N} \end{aligned}

There are 5 questions to complete.