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: 
Working axial load
\begin{aligned} &=\frac{\text { Factored axial load }}{1.5} \\ &=\frac{2000}{1.5}=1333.33 \mathrm{kN} \end{aligned}
The allowable bearing pressure on concrete may be given as,
\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}
The permissible stress in direct compression in various grades of concrete as per IS 456: 2000 are tabulated below:


The permissible stress in concrete should be more than the allowable bearing pressure. Thus the minimum grade of concrete which should be used is M20
Question 6
Consider a reinforcing bar embedded in concrete. In a marine environment this bar undergoes uniform corrosion, which leads to the deposition of corrosion products on its surface and an increase in the apparent volume of the bar. This subjects the surrounding concrete to expansive pressure. As a result, corrosion induced cracks appear at the surface of concrete. Which of the following statements is TRUE?
A
Corrosion causes circumferential tensile stresses in concrete and the cracks will be parallel to the corroded reinforcing bar.
B
Corrosion causes radial tensile stresses in concrete and the cracks will be parallel to the corroded reinforcng bar.
C
Corrosion causes circumferential tensile stresses in concrete and the cracks will be perpendicular to the direction of the corroded reinforcing bar.
D
Corrosion causes radial tensile stresses in concrete and the cracks will be perpendicular to the direction of the corroded reinforcing bar.
RCC Structures   Shear, Torsion, Bond, Anchorage and Development Length
Question 6 Explanation: 
Corrosion in steel occupies a volume several times greater than volume of reinforcing steel hence exerts radial pressure on adjoining concrete layer. As a consequence, cracking results along reinforcing bars and concrete starts spalling.
Question 7
The results for sieve analysis carried out for three types of sand, P, Q and R, are given in the figure. If the fineness modulus values of the three sands are given as FMP, FMQ and FMR, it can be stated that
A
FM_{Q}=\sqrt{FM_{P} \times FM_{R}}
B
FM_{Q}=0.5(FM_{P}+FM_{R})
C
FM_{P}\gt FM_{Q}\gt FM_{R}
D
FM_{P}\lt FM_{Q}\lt FM_{R}
Geotechnical Engineering   Classification of Soils and Clay Minerals
Question 7 Explanation: 
The term fineness modulus is used to indicate an index number which roughly proportional to average size of particle in entire quantity of aggregate
F_{M Q}=\sqrt{F_{M P} \times F_{M R}} (Geometric mean)
Question 8
The cross-section of a thermo-mechanically treated (TMT) reinforcing bar has
A
soft ferrite-pearlite throughout
B
hard martensite throughout
C
a soft ferrite-pearlite core with a hard martensitic rim
D
a hard martensitic core with a soft pearlite-bainitic rim
RCC Structures   Working Stress and Limit State Method
Question 9
Consider a simply supported beam with a uniformly distributed load having a neutral axis (NA) as shown.

For points P (on the neutral axis) and Q (at the bottom of the beam) the state of stress is best represented by which of the following pairs?
A
A
B
B
C
C
D
D
Solid Mechanics   Principal Stress and Principal Strain
Question 9 Explanation: 


Point P:
Point P lies on NA, hence bending stress is zero at point P.
Point P also lies at mid span, so shear force, V=0
\Rightarrow Shear stress, \tau=0
\therefore State of stress of point P will be


Point Q :
At point Q flexural stress is maximum and nature of which is tensile due to downward loading. Point Q lies at the extreme of beam, therefore, shear stress at point Q is zero.
\therefore State of stress of point Q will be


Question 10
For a saturated sand deposit, the void ratio and the specific gravity of solids are 0.70 and 2.67, respectively. The critical (upward) hydraulic gradient for the deposit would be
A
0.54
B
0.98
C
1.02
D
1.87
Geotechnical Engineering   Seepage Analysis
Question 10 Explanation: 
Critical hydraulic gradient is given by
i_{c r}=\frac{G-1}{1+e}=\frac{2.67-1}{1+0.70}=0.98
There are 10 questions to complete.

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