# 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.