# 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.