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




There are 5 questions to complete.

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