# 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.