# 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.