GATE CE 2005

Question 1
Consider the matrices X_{(4,3)}, \; Y_{(4,3)} and P_{(2,3)}. The order of [P(X^{T}Y)^{-1}P^{T}]^{T} will be
A
\left ( 2\times 2 \right )
B
\left ( 3\times 3 \right )
C
\left ( 4\times 3 \right )
D
\left ( 3\times 4 \right )
Engineering Mathematics   Linear Algebra
Question 1 Explanation: 
With the given order we can say that order of matrices are as follows:
\begin{aligned} X^{T} &\rightarrow 3\times 4 \\ Y & \rightarrow 4\times 3 \\ X^{T}Y & \rightarrow 3\times 3 \\ \left ( X^{T}Y \right )^{-1} & \rightarrow 3\times 3 \\ P & \rightarrow 2\times 3 \\ P^{T} & \rightarrow 3\times 2 \\ P\left ( X^{T}Y \right )^{-1}P^{T} & \rightarrow \left ( 2\times 3 \right )\left ( 3\times 3 \right )\left ( 3\times 2 \right ) \\ & \rightarrow 2\times 2 \\ \therefore\;\; \left ( P\left ( X^{T}Y \right )^{-1}P^{T} \right )^{T} & \rightarrow 2\times 2 \end{aligned}
Question 2
Consider a non-homogeneous system of linear equations representing mathematically an over-determined system. Such a system will be
A
consistent having a unique solution
B
consistent having a unique solution
C
inconsistent having a unique solution
D
inconsistent having no solution
Engineering Mathematics   Linear Algebra
Question 2 Explanation: 
In an over determined system having more equations than variables, it is necessary to have consistent unique solution, by definition.
Question 3
Which one of the following is NOT true for complex number Z_1 \; and \; Z_2?
A
\frac{Z_{1}}{Z_{2}}=\frac{Z_{1}\bar{Z_{2}}}{|Z_{2}|^{2}}
B
|Z_{1}+Z_{2}|\leq |Z_{1}|+|Z_{2}|
C
|Z_{1}-Z_{2}|\leq |Z_{1}|-|Z_{2}|
D
|Z_{1}+Z_{2}|^{2}+|Z_{1}-Z_{2}|^{2} =2 |Z_{1}|^{2}+2|Z_{2}|^{2}
Engineering Mathematics   Calculus
Question 3 Explanation: 
(A) is true since
\frac{Z_{1}}{Z_{2}}=\frac{Z_{1}\bar{Z_{2}}}{Z_{2}\bar{Z_{2}}}=\frac{Z_{1}\bar{Z_{2}}}{\left | Z_{2} \right |^{2}}
(B) is true by triangle inequality of complex number.
(C) is not true since \left | Z_{1}-Z_{2} \right |\geq \left | Z_{1} \right |-\left | Z_{2} \right |
(D) is true since
\begin{aligned}\left | Z_{1}+Z_{2} \right |^{2}&=\left ( Z_{1}+Z_{2} \right )\left ( \overline{Z_{1}+Z_{2}} \right ) \\ &=\left ( Z_{1}+Z_{2} \right )\left ( \bar{Z_{1}}+\bar{Z_{2}} \right ) \\ &=Z_{1}\bar{Z_{1}}+Z_{2}\bar{Z_{2}}+Z_{2}\bar{Z_{1}}+Z_{1}\bar{Z_{2}} \;\;...(i)\\ \left | Z_{1}-Z_{2} \right |^{2}&=\left ( Z_{1}+Z_{2} \right )\left ( \overline{Z_{1}-Z_{2}} \right ) \\ &=\left ( Z_{1}-Z_{2} \right )\left ( \bar{Z_{1}}-\bar{Z_{2}} \right ) \\ &=Z_{1}\bar{Z_{1}}+Z_{2}\bar{Z_{2}}-Z_{2}\bar{Z_{1}}-Z_{1}\bar{Z_{2}}\;\;...(ii)\end{aligned}
Adding (i) and (ii) we get,
\left | Z_{1}+Z_{2} \right |^{2}+\left | Z_{1}-Z_{2} \right |^{2}=2Z_{1}\bar{Z_{2}}+2Z_{2}\bar{Z_{2}} =2\left | Z_{1} \right |^{2}+2\left | Z_{2} \right |^{2}
Question 4
Which one of the following statement is NOT true ?
A
The measure of skewness is dependent upon the amount of dispersion
B
In a symmetric distribution, the values of mean, mode and median are the same
C
In a positively skewed distribution : mean \gt median \gt mode
D
In a negatively skewed distribution : mode \gt mean \gt median
Engineering Mathematics   Probability and Statistics
Question 4 Explanation: 
A, B, C are true
(D) is not true since in a negatively skewed distribution, mode \gt median \gt mean
Question 5
IS:1343-1980 limits the minimum characteristic strength of pre-stressed concrete for post tensioned work and pretension work as
A
25 MPa, 30 MPa respectively
B
25 MPa, 35 MPa respectively
C
30 MPa 35 MPa respectively
D
30 MPa, 40 MPa respectively
RCC Structures   Prestressed Concrete Beams
Question 6
The permissible stress in axial tension \sigma _{st} in steel member on the net effective area of the section shall not exceed the following value (f_{y} is the yield stress)
A
0.80 f_{y}
B
0.75 f_{y}
C
0.60 f_{y}
D
0.50 f_{y}
Design of Steel Structures   Tension Member
Question 7
The partial factor of safety for concrete as per IS:456-2000 is
A
1.5
B
1.15
C
0.87
D
0.446
RCC Structures   Working Stress and Limit State Method
Question 8
The symmetry of stress tensor at a point in the body under equilibrium is obtained from
A
conserved of mass
B
force equilibrium equations
C
moment equilibrium equations
D
conservation of energy
Solid Mechanics   Properties of Metals, Stress and Strain
Question 8 Explanation: 


Taking moment equilibrium about the centre, we get
\begin{aligned} \tau_{y x} \times \frac{d}{2}+\tau_{y x} \times \frac{d}{2}&=\tau_{x y} \times \frac{d}{2}+\tau_{x y} \times \frac{d}{2} \\ \therefore \quad \tau_{x y}&=\tau_{y x} \end{aligned}
Question 9
The components of strain tensor at a point in the plane strain case can be obtained by measuring longitudinal strain in following directions
A
along any two arbitrary directions
B
along any three arbitrary directions
C
along two mutually orthogonal directions
D
along any arbitrary direction
Solid Mechanics   Properties of Metals, Stress and Strain
Question 9 Explanation: 
In case of plane strain condition, the components
of strain tensor at a point are \epsilon_{x}, \in_{y} \text{ and }\phi_{x y} .
Here we have three unknowns so we require 3 equations to find them and these unknowns can be find by the equations of longitudinal strain,
\epsilon_{1}=\epsilon_{x} \cos ^{2} \theta_{1}+\epsilon_{y} \sin ^{2} \theta_{1}+\frac{\phi_{x y}}{2} \sin 2 \theta_{1}
\epsilon_{2}=\epsilon_{x} \cos ^{2} \theta_{2}+\epsilon_{y} \sin ^{2} \theta_{2}+\frac{\phi_{x y}}{2} \sin 2 \theta_{2}
\epsilon_{3}=\epsilon_{x} \cos ^{2} \theta_{3}+\epsilon_{y} \sin ^{2} \theta_{3}+\frac{\phi_{x y}}{2} \sin 2 \theta_{3}
Therefore to find out component of strain tensor in plain strain condition, we measure longitudinal strains \left(\epsilon_{1}, \epsilon_{2}\text{ and }\epsilon_{3}\right) along any three arbitrary direction.
Question 10
Considering beam as axially rigid, the degree of freedom of a plane frame shown below is
A
9
B
8
C
7
D
6
Structural Analysis   Determinacy and Indeterminacy
Question 10 Explanation: 
Degree of kinematic indeterminacy or degree of freedom,
\begin{aligned} D_{k} &=3 j-r_{e}-m \\ &=3 \times 4-3-1 \\ &=8 \end{aligned}
There are 10 questions to complete.

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