# GATE CE 2008

 Question 1
The product of matrices $(PQ)^{-1}P$ is
 A $P^{-1}$ B $Q^{-1}$ C $P^{-1}Q^{-1}P$ D $PQP^{-1}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
\begin{aligned} \left ( PQ \right )^{-1}P&=\left (Q^{-1}P^{-1} \right )P \\ &=\left ( Q^{-1}\ \right )\left ( P^{-1}P \right ) \\ &=\left ( Q^{-1} \right )\left ( I \right )=Q^{-1} \end{aligned}
 Question 2
The general solution of $\frac{d^{2}y}{dx^{2}}+y=0$ is
 A $y = P \cos x + Q \sin x$ B $y = P \cos x$ C $y = P \sin x$ D $y = P \sin^2 x$
Engineering Mathematics   Ordinary Differential Equation
Question 2 Explanation:
\begin{aligned} \frac{d^{2}y}{dx^{2}}+y&=0 \\ D^{2}+1&=0 \\ D=\pm i &=0\pm 1i \\ &\text{General solution is,} \\ y&=e^{0x}\left [ C_{1}\cos \left ( 1\times x \right )+C_{2}\sin \left ( 1\times x \right ) \right ] \\ &=C_{1}\cos x+C_{2}\sin x \\ &=P\cos x+Q\sin x \end{aligned}
where P and Q are some constants.

 Question 3
A mild steel specimen is under uniaxial tensile stress. Young's modulus and yield stress for mild steel are $2 \times 10^{5}$ MPa and 250 MPa respectively. The maximum amount of strain energy per unit volume that can be stored in this specimen without permanent set is
 A 156 $Nmm/mm^{3}$ B 15.6 $Nmm/mm^{3}$ C 1.56 $Nmm/mm^{3}$ D 0.156 $Nmm/mm^{3}$
Solid Mechanics   Properties of Metals, Stress and Strain
Question 3 Explanation:
The strain energy per unit volume may be given as
\begin{aligned} u &=\frac{1}{2} \times \frac{\sigma_{y}^{2}}{E}=\frac{1}{2} \times \frac{(250)^{2}}{2 \times 10^{5}} \\ &=0.156 \mathrm{N}\; \mathrm{mm} / \mathrm{mm}^{3} \end{aligned}
 Question 4
A reinforced concrete structure has to be constructed along a sea coast. The minimum grade of concrete to be used as per IS : 456-2000 is
 A M15 B M20 C M25 D M30
RCC Structures   Working Stress and Limit State Method
Question 4 Explanation:
As per clause 8.2.8 of 1S 456: 2.000 concrete in sea water or exposed directly along the sea coast shall be atleast M20 grade in the case of plain concrete and M30 in case of reinforced concrete.
 Question 5
In the design of a reinforced concrete beam the requirement for bond is not getting satisfied. The economical option to satisfy the requirement for bond is by
 A bounding of bars B providing smaller diameter bars more in number C providing larger diameter bars less in number D providing same diameter bars more in number
RCC Structures   Shear, Torsion, Bond, Anchorage and Development Length
Question 5 Explanation:
Bond stress $\left(\tau_{b d}\right)=\frac{\text { Tensile force }}{(n \pi \phi) \sigma_{s t}}$
$\tau_{b d}$ should be less than permissible value, if it is greater than $\left(\tau_{b d}\right)_{\text {permissible }}$ then best economical solution is to reduce the diameter of bar and increase its number.

There are 5 questions to complete.