Question 1 |
Solution for the system defined by the set of equations
4y + 3z = 8;
2x - z = 2 and
3x + 2y = 5 is
4y + 3z = 8;
2x - z = 2 and
3x + 2y = 5 is
x=0; y=1; z=4/3 | |
x=0; y=1/2; z=2 | |
x=1; y=1/2; z=2 | |
non-existent |
Question 1 Explanation:
The augmented matrix for given system is,
\left [ \left.\begin{matrix} 0 & 4 & 3\\ 2 & 0 & -1\\ 3 & 2 & 0 \end{matrix}\right|\begin{matrix} 8\\ 2\\ 5 \end{matrix} \right ]\xrightarrow[]{exchange 1st and 2nd row}\left [ \left.\begin{matrix} 2 &0 & -1\\ 0 & 4 & 3\\ 3 & 2 & 0 \end{matrix}\right|\begin{matrix} 2\\ 8\\ 5 \end{matrix} \right ]
then by Gauss elimination procedure
\left [ \left.\begin{matrix} 2 & 0 & -1\\ 0 & 4 & 3\\ 3 & 2 & 0 \end{matrix}\right|\begin{matrix} 2\\ 8\\ 5 \end{matrix} \right ]\xrightarrow[]{R_{3}-\frac{3}{2}R_{1}}\left [ \left.\begin{matrix} 2 &0 & -1\\ 0 & 4 & 3\\ 0 & 2 & 3/2 \end{matrix}\right|\begin{matrix} 2\\ 8\\ 2 \end{matrix} \right ]\xrightarrow[]{R_{3}-\frac{2}{4}R_{2}}\left [ \left.\begin{matrix} 2 & 0 & -1\\ 0 & 4 & 3\\ 0 & 0 & 0 \end{matrix}\right|\begin{matrix} 8\\ 8\\ -2 \end{matrix} \right ]
For last row we see 0=-2 which is inconsistent
\therefore Solution is non-existent for above system.
\left [ \left.\begin{matrix} 0 & 4 & 3\\ 2 & 0 & -1\\ 3 & 2 & 0 \end{matrix}\right|\begin{matrix} 8\\ 2\\ 5 \end{matrix} \right ]\xrightarrow[]{exchange 1st and 2nd row}\left [ \left.\begin{matrix} 2 &0 & -1\\ 0 & 4 & 3\\ 3 & 2 & 0 \end{matrix}\right|\begin{matrix} 2\\ 8\\ 5 \end{matrix} \right ]
then by Gauss elimination procedure
\left [ \left.\begin{matrix} 2 & 0 & -1\\ 0 & 4 & 3\\ 3 & 2 & 0 \end{matrix}\right|\begin{matrix} 2\\ 8\\ 5 \end{matrix} \right ]\xrightarrow[]{R_{3}-\frac{3}{2}R_{1}}\left [ \left.\begin{matrix} 2 &0 & -1\\ 0 & 4 & 3\\ 0 & 2 & 3/2 \end{matrix}\right|\begin{matrix} 2\\ 8\\ 2 \end{matrix} \right ]\xrightarrow[]{R_{3}-\frac{2}{4}R_{2}}\left [ \left.\begin{matrix} 2 & 0 & -1\\ 0 & 4 & 3\\ 0 & 0 & 0 \end{matrix}\right|\begin{matrix} 8\\ 8\\ -2 \end{matrix} \right ]
For last row we see 0=-2 which is inconsistent
\therefore Solution is non-existent for above system.
Question 2 |
The differential equation \frac{dy}{dx}=0.25y^2 is to be solved using the backward
(implicit) Euler's method with the boundary condition y=1 at x=0 and with
a step size of 1. What would be the value of y at x=1 ?
1.33 | |
1.67 | |
2 | |
2.33 |
Question 2 Explanation:
\frac{\mathrm{d} y}{\mathrm{d} x}=0.25y^{2} \;\;\left ( y=1\: at\: x=0 \right )
h=1
Iterative equation for backward (implicit) Euler methods for above equation would be,
\frac{y_{k+1}-y_{k}}{h}=0.25y^{2}_{k+1}
\Rightarrow \;\; y_{k+1}-y_{k}=0.25h y^{2}_{k+1}
\Rightarrow \;\; 0.25h y^{2}_{k+1}-y_{k+1}+y_{k}=0
putting k=0 in above equation
0.25h y^{2}_{1}-y_{1}+y_{0}=0
since y_{0}=1 and h=1
0.25 y_{1}^{2}-y_{1}+1=0
y_{1}=2
h=1
Iterative equation for backward (implicit) Euler methods for above equation would be,
\frac{y_{k+1}-y_{k}}{h}=0.25y^{2}_{k+1}
\Rightarrow \;\; y_{k+1}-y_{k}=0.25h y^{2}_{k+1}
\Rightarrow \;\; 0.25h y^{2}_{k+1}-y_{k+1}+y_{k}=0
putting k=0 in above equation
0.25h y^{2}_{1}-y_{1}+y_{0}=0
since y_{0}=1 and h=1
0.25 y_{1}^{2}-y_{1}+1=0
y_{1}=2
Question 3 |
The necessary and sufficient condition for a surface to be called as a free surface is
no stress should be acting on it | |
tensile stress acting on it must be zero | |
shear stress acting on it must be zero | |
no point on it should be under any stress |
Question 4 |
Mohr's circle for the state of stress defined by \begin{bmatrix} 30 &0 \\ 0&30 \end{bmatrix} MPa is a circle with
centre at (0, 0) and radius 30 MPa | |
centre at (0, 0) and radius 60 MPa | |
centre at (30, 0) and radius 30 MPa | |
centre at (30, 0) and zero radius |
Question 4 Explanation:
The maximum and minimum principal stresses are
same. So, radius of circle becomes zero and centre
is at (30, 0). The circle is represented by a point.
Question 5 |
The buckling load P=P_{cr} for the column AB shown in figure, as K_T approaches infinity,
becomes \alpha \frac{\pi ^{2}EI}{L^{2}}, where \alpha is equal to
0.25 | |
1 | |
2.05 | |
4 |
There are 5 questions to complete.