GATE Civil Engineering 2021 SET-1

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Question 1

The rank of matrix

\left[\begin{array}{llll} 1 & 2 & 2 & 3 \\ 3 & 4 & 2 & 5 \\ 5 & 6 & 2 & 7 \\ 7 & 8 & 2 & 9 \end{array}\right]

A	1
B	2
C	3
D	4

Engineering Mathematics Linear Algebra

Question 1 Explanation:

Using

R_{2} \rightarrow R_{2} \rightarrow 3 R_{1}, R_{3} \rightarrow R_{3}-5 R_{1}, R_{4} \rightarrow R_{4}-7 R_{1}

A=\left[\begin{array}{cccc} 1 & 2 & 2 & 3 \\ 0 & -2 & -4 & -4 \\ 0 & -4 & -8 & -8 \\ 0 & -6 & -12 & -12 \end{array}\right]

Using

R_{3} \rightarrow R_{3}-2 R_{2}, R_{4} \rightarrow R_{4}-3 R_{2}

A=\left[\begin{array}{cccc} 1 & 2 & 2 & 3 \\ 0 & -2 & -4 & -4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]

So,

\rho(A)=

No. of non-zero rows = 2.

Question 2

P=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]

and

Q=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]

then

Q^{T} P^{T}

A	$\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]$
B	$\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]$
C	$\left[\begin{array}{ll} 2 & 1 \\ 4 & 3 \end{array}\right]$
D	$\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right]$

Engineering Mathematics Linear Algebra

Question 2 Explanation:

\begin{array}{l} \quad P Q=\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]=\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right] \\ (P Q)^{\top}=\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right] \end{array}

Now using Reversal law

Q^{\top} P^{\top}=(P Q) T=\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right]

Question 3

The shape of the cumulative distribution function of Gaussian distribution is

A	Horizontal line
B	Straight line at 45 degree angle
C	Bell-shaped
D	S-shaped

Engineering Mathematics Probability and Statistics

Question 3 Explanation:

PDF:f(x)=\frac{1}{\sigma \sqrt{2 \pi}}e^{-(x-\mu )^2/(2\sigma ^2)}

CDF:F(x)=\frac{1}{2}\left [ 1+eff\left ( \frac{x-\mu }{\sigma \sqrt{2}} \right ) \right ]

Question 4

A propped cantilever beam EF is subjected to a unit moving load as shown in the figure (not to scale). The sign convention for positive shear force at the left and right sides of any section is also shown.