# GATE CE 2011

 Question 1
[A] is a square matrix which is neither symmetric nor skew-symmetric and $[A]^{T}$ is its transpose. The sum and difference of these matrices are defined as $[S] = [A]+[A]^{T}$ and $[D] = [A]-[A]^{T}$, respectively. Which of the following statements is TRUE?
 A Both [S] and [D] are symmetric B Both [S] and [D] are skew-symmetric C [S] is skew-symmetric and [D] is symmetric D [S] is symmetric and [D] is skew-symmetric
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Since
\begin{aligned} \left ( A+{A}' \right )&=A^{t}+\left ( A^{t} \right )^{t} \\ &=A^{t}+A\\ \text{i.e. } S^{t}&=S \\ \therefore\;\; & \text{ S is symmetric}\\ \text{Since }\left ( A-A^{t} \right )^{t}&=A^{t}-\left ( A^{t} \right )^{t} \\ &=A^{t}-A \\ &=-\left ( A-A^{t} \right ) \\ \text{i.e. } D^{t}&=-D \end{aligned}
So D is Skew-Symmetric.
 Question 2
The square root of a number N is to be obtained by applying the Newton Raphson iterations to the equation $x^2-N=0$. If i denotes the iteration index, the correct iterative scheme will be
 A $x_{i+1}=\frac{1}{2}(x_{i}+\frac{N}{x_{i}})$ B $x_{i+1}=\frac{1}{2}(x_{i}^{2}+\frac{N}{x_{i}^{2}})$ C $x_{i+1}=\frac{1}{2}(x_{i}+\frac{N^{2}}{x_{i}})$ D $x_{i+1}=\frac{1}{2}(x_{i}-\frac{N}{x_{i}})$
Engineering Mathematics   Numerical Methods
Question 2 Explanation:
\begin{aligned} x_{i+1}&=x_{i}-\frac{f\left ( x_{i} \right )}{{f}'\left ( x_{i} \right )} \\ &=x_{i}-\left ( \frac{x_{i}^{2}-N}{2x_{i}} \right ) \\ &=\frac{x_{i}^{2}+N}{2x_{i}} \\ &=\frac{1}{2}\left [ x_{i}+\frac{N}{x_{i}} \right ] \end{aligned}

 Question 3
There are two containers, with one containing 4 Red and 3 Green balls and the other containing 3 Blue and 4 Green balls. One ball is drawn at random from each container. The probability that one of the balls is Red and the other is Blue will be
 A $1/7$ B $9/49$ C $12/49$ D $3/7$
Engineering Mathematics   Probability and Statistics
Question 3 Explanation:
P(one ball is Red & another is blue)
=p(first is Red and second is Blue)
$=\frac{4}{7}\times \frac{3}{7} =\frac{12}{49}$
 Question 4
For the fillet weld of size 's' shown in the figure the effective throat thickness is A 0.61s B 0.65s C 0.7s D 0.75s
Design of Steel Structures   Structural Fasteners
Question 4 Explanation:
Effective throat thickness is the shortest distance from the root of fillet weld to the face of the line joining the toes. The effective throat thickness should not be less than 3 mm.
Effective throat thickness =Ks
where s is the size of weld in mm and K is a constant. The value of K depends upon the angle between the fusion faces. Question 5
A 16 mm thick plate measuring 650 mm x 420 mm is used as a base plate for an ISHB 300 column subjected to a factored axial compressive load of 2000 kN. As per IS 456-2000, the minimum grade of concrete that should be used below the base plate for safely carrying the load is
 A M15 B M20 C M30 D M40
RCC Structures   Footing, Columns, Beams and Slabs
Question 5 Explanation:
\begin{aligned} &=\frac{\text { Factored axial load }}{1.5} \\ &=\frac{2000}{1.5}=1333.33 \mathrm{kN} \end{aligned}
\begin{aligned} \sigma_{\text {all }} &=\frac{\text { Direct load }}{\text { Area of base plate }} \\ &=\frac{1333.33 \times 10^{3}}{650 \times 420}=4.88 \mathrm{N} / \mathrm{mm}^{2} \end{aligned} 