# 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

There are 5 questions to complete.