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
 Question 6
In the theory of plastic bending of beams, the ratio of plastic moment to yield moment is called
 A shape factor B plastic section modulus C modulus of resilience D rigidity modulus
Design of Steel Structures   Plastic Analysis
 Question 7
For limit state of collapse, the partial safety factors recommended by IS 456:2000 for estimating the design strength of concrete and reinforcing steel are respectively
 A 1.15 and 1.5 B 1.0 and 1.0 C 1.5 and 1.15 D 1.5 and 1.0
RCC Structures   Working Stress and Limit State Method
 Question 8
The point within the cross sectional plane of a beam through which the resultant of the external loading on the beam has to pass through to ensure pure bending without twisting of the cross-section of the beam is called
 A moment centre B centroid C shear centre D elastic center
Solid Mechanics   Theory of Columns and Shear Centre
 Question 9
The square root of the ratio of moment of inertia of the cross-section to its cross-sectional area is called
 A second moment of area B slenderness ratio C section modulus D radius of gyration
Design of Steel Structures   Compression Member
Question 9 Explanation:
$r=\sqrt{\frac{\text{Moment of inertia}}{\text{Cross-sectional area}}}$