GATE CE 2006

Question 1
Solution for the system defined by the set of equations
4y + 3z = 8;
2x - z = 2 and
3x + 2y = 5 is
A
x=0; y=1; z=4/3
B
x=0; y=1/2; z=2
C
x=1; y=1/2; z=2
D
non-existent
Engineering Mathematics   Linear Algebra
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.
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 ?
A
1.33
B
1.67
C
2
D
2.33
Engineering Mathematics   Ordinary Differential Equation
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
Question 3
The necessary and sufficient condition for a surface to be called as a free surface is
A
no stress should be acting on it
B
tensile stress acting on it must be zero
C
shear stress acting on it must be zero
D
no point on it should be under any stress
Fluid Mechanics and Hydraulics   Hydrostatic Forces
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
A
centre at (0, 0) and radius 30 MPa
B
centre at (0, 0) and radius 60 MPa
C
centre at (30, 0) and radius 30 MPa
D
centre at (30, 0) and zero radius
Solid Mechanics   Principal Stress and Principal Strain
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
A
0.25
B
1
C
2.05
D
4
Solid Mechanics   Theory of Columns and Shear Centre
Question 6
A long shaft of diameter d is subjected to twisting moment T at its ends. The maximum normal stress acting at its cross-section is equal to
A
zero
B
\frac{16T}{\pi d^{3}}
C
\frac{32T}{\pi d^{3}}
D
\frac{64T}{\pi d^{3}}
Solid Mechanics   Torsion of shafts and Pressure Vessels
Question 6 Explanation: 
Cross-section of the shaft is the section perpendicular to the longitudinal axis of shaft. In case of twisting moment acting at its ends, normal stress is zero on the cross-section only shear stress acts.
Question 7
If the characteristic strength of concrete f_{ck} is defined as the strength below which not more than 50% of the test results are expected to fall, the expression for f_{ck} in terms of mean strength f_{m} and standard deviation S would be
A
f_{m}- 0.1645 S
B
f_{m} - 1.645 S
C
f_{m}
D
f_{m} + 1.645S
RCC Structures   Working Stress and Limit State Method
Question 8
The range of void ratio between which quick sand condition occurs in cohesionless granular soil deposits is
A
0.4 - 0.5
B
0.6 - 0.7
C
0.8 - 0.9
D
1.0 - 1.1
Geotechnical Engineering   Seepage Analysis
Question 8 Explanation: 
The specific gravity of cohesionless granular soils (sands) does not vary much and for all practical purposes it is taken to be 2.65. Critical hydraulic gradient should be nearly 1 for quick sand condition.
i.e.\quad i_{cr}=\frac{G-1}{1+e}
From the above equation, the void ratio range is found to be between 0.6 and 0.7.
Question 9
Figure given below shows a smooth vertical gravity retaining wall with cohesionless soil backfill having an angle of internal friction \phi. In the graphical representation of Rankine's active earth pressure for the retaining wall shown in figure, length OP represents
A
vertical stress at the base
B
vertical stress at a height H/3 from the base
C
lateral earth pressure at the base
D
lateral earth pressure at a height H/3 from the base
Geotechnical Engineering   Retaining Wall-Earth Pressure Theories
Question 9 Explanation: 


\begin{array}{l} O P=\text {Major stress}=\sigma_{1} \\ O Q=\text {Mirror stress}=\sigma_{3} \end{array}
Active condition:



OP = Vertical stress at base
=\sigma_{v} \cos \beta=\gamma+\cos \beta
OQ = Lateral earth pressure at base
=\mathrm{K}_{\mathrm{a}} \cdot \sigma_{\mathrm{v}} \cos \beta
Note: QR and QR' are failure plane for active case.
Question 10
Which of the following statement is NOT TRUE in the context of capillary pressure in solids ?
A
Water is under tension in capillary zone
B
Pore water pressure is negative in capillary zone
C
Effective stress increases due to the capillary zone
D
Capillary pressure is more in coarse grained soils
Geotechnical Engineering   Effective Stress and Permeability
Question 10 Explanation: 
Capillary rise is more in fine grained soils and as a result of this, the capillary pressure is also more than the coarse grained soils.
There are 10 questions to complete.

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