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] is
1 | |
2 | |
3 | |
4 |
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.
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 |
If 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} is
\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right] | |
\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right] | |
\left[\begin{array}{ll} 2 & 1 \\ 4 & 3 \end{array}\right] | |
\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right] |
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]
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
Horizontal line | |
Straight line at 45 degree angle | |
Bell-shaped | |
S-shaped |
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.

The CORRECT qualitative nature of the influence line diagram for shear force at G is

The CORRECT qualitative nature of the influence line diagram for shear force at G is
![]() | |
![]() | |
![]() | |
![]() |
Question 4 Explanation:

As per Muller Breslau principle ILD for stress function (shear -V_{G}) will be a combination of curves (3^{\circ} curves).
Question 5 |
Gypsum is typically added in cement to
prevent quick setting | |
enhance hardening | |
increase workability | |
decrease heat of hydration |
Question 5 Explanation:
The Gypsum is added to cement at the end of grinding clinker it is added to prevent quick setting.
There are 5 questions to complete.