GATE CE 2009


Question 1
A square matrix B is skew-symmetric if
A
B^{T}=-B
B
B^{T}=B
C
B^{-1}=B
D
B^{-1}=B^{T}
Engineering Mathematics   Linear Algebra
Question 1 Explanation: 
A square matrix B is defined as skew-symmetric if and only if B^{T}=-B
Question 2
For a scalar function f (x,y,z) = x^{2}+3y^{2}+2z^{2}, the gradient at the point P(1,2,-1) is
A
2\vec{i}+6\vec{j}+4\vec{k}
B
2\vec{i}+12\vec{j}-4\vec{k}
C
2\vec{i}+12\vec{j}+4\vec{k}
D
\sqrt{56}
Engineering Mathematics   Calculus
Question 2 Explanation: 
\begin{aligned} f&=x^{2}+3y^{2}+2z^{2} \\ \Delta f&=grad\: f=i\frac{\partial f}{\partial x}+j\frac{\partial f}{\partial y}+k\frac{\partial f}{\partial z} \\ &=i\left ( 2x \right )+j\left ( 6y \right )+k\left ( 4z \right ) \\ &\text{The gradient at P(1, 2, -1) is}\\ &=i\left ( 2\times 1 \right )+j\left ( 6\times 2 \right )+k\left ( 4\times -1 \right ) \\ &=2i+12j-4k \end{aligned}


Question 3
The analytic function f(z)=\frac{z-1}{z^{2}+1} has singularity at
A
1 and -1
B
1 and i
C
1 and -i
D
i and -i
Engineering Mathematics   Calculus
Question 3 Explanation: 
f\left ( z \right )=\frac{z-1}{z^{2}+1}=\frac{z-1}{z^{2}-i^{2}}=\frac{z-1}{\left ( z-i \right )\left ( z+i \right )}
\therefore The singularities arc at z=i and -i
Question 4
A thin walled cylindrical pressure vessel having a radius of 0.5 m and wall thickness of 25 mm is subjected to an internal pressure of 700 kPa. The hoop stress developed is
A
14MPa
B
1.4MPa
C
0.14MPa
D
0.014MPa
Solid Mechanics   Torsion of Shafts and Pressure Vessels
Question 4 Explanation: 
\begin{aligned} \text { Hoop stress } &=\frac{p d}{2 t}=\frac{700 \times 10^{3} \times 2 \times 0.5}{2 \times 25 \times 10^{-3}} \\ &=14 \times 10^{6} \mathrm{Pa}=14 \mathrm{MPa} \end{aligned}
Question 5
The modulus of rupture of concrete in terms of its characteristic cube compressive strength (f_{ck}) in MPa according to IS 456:2000 is
A
5000f_{ck}
B
0.7f_{ck}
C
5000\sqrt{f_{ck}}
D
0.7\sqrt{f_{ck}}
RCC Structures   Working Stress and Limit State Method




There are 5 questions to complete.

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