# GATE CE 2004

 Question 1
Real matrices $[A]_{3 \times 1},[B]_{3 \times 3},[C]_{3 \times 5}$, $[D]_{5 \times 3},[E]_{5 \times 5}, [F]_{5\times 1}$ are given. Matrices $[B]$ and $[E]$ are symmetric.
Following statements are made with respect to these matrices.
I. Matrix product $\left [ F \right ]^{T}$ $\left [ C \right ]^{T} [B] [C] [F]$ is a scalar.
II. Matrix product $\left [ D \right ]^{T} [F] [D]$ is always symmetric.

With reference to above statements, which of the following applies ?
 A Statement I is true but II is false B Statement I is false but II is true C Both the statement are true D Both the statement are false
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Statement 1 is true.
Consider statement 2
$\left [ D \right ]^{T}\left [ F \right ]\left [ D \right ]$ is always symmetric.
Now since
\begin{aligned} \left [ {D}'FD \right ]{}'&= {\left [ FD \right ]}'{\left ( {D}' \right )}' \\ &={\left ( FD \right )}'D \\ &={D}'{F}'D \\ &={D}'FD \end{aligned}
(Since F may or may not be symmetric)
$\therefore \;\; {D}'FD$ is not always symmetric is true.
$\therefore$ 1 is true and 2 is false.
 Question 2
The summation of series $S=2+\frac{5}{2}+\frac{8}{2^{2}}+\frac{11}{2^{3}}+....\infty$ is
 A 4.5 B 6 C 6.75 D 10
Engineering Mathematics   Partial Differential Equation
Question 2 Explanation:
S is an arithmetico-geometric series and can be summed up as follows,
\begin{aligned} S&=2+\left ( \frac{5}{2}+\frac{8}{2^{2})}+\frac{11}{2^{3}}+...\infty \right ) \\ &=2+x\;\; (say)\\ \text{where } x&=\frac{5}{2}+\frac{8}{2^{2}}+\frac{11}{2^{3}}+...\infty \;\; ...(i)\\ \text{Multiplying by }&\text{1/2, we get, } \\ \frac{1}{2}x&=\frac{5}{2^{2}}+\frac{8}{2^{3}}+\frac{11}{2^{4}}+...\infty \;\;...(ii) \\ \text{Substracting (ii) } & \text{from (i), we get, }\\ x-\frac{1}{2}x&=\frac{5}{2}+\frac{\left ( 8-5 \right )}{2^{2}} \\ &+\frac{\left ( 11-8 \right )}{2^{3}} +\frac{\left ( 13-11 \right )}{2^{4}}+...\infty \\ \frac{x}{2}&=\frac{5}{2}+\left ( \frac{3}{2^{2}}+\frac{3}{2^{3}}+\frac{3}{2^{4}}+...\infty \right ) \\ \frac{x}{2}&=\frac{5}{2}+\frac{\frac{3}{2^{2}}}{\left [ 1-\frac{1}{2} \right ]} \\ \frac{x}{2}&=\frac{5}{2}+\frac{3}{2}=4 \\ \therefore \;\; x&=8\\ \text{and } S&=2+x=10\end{aligned}
 Question 3
The value of the function $f(x)=\lim_{x\rightarrow 0}\frac{x^{3}+x^{2}}{2x^{3}-7x^{2}}$ is
 A 0 B $-\frac{1}{7}$ C $\frac{1}{7}$ D $\infty$
Engineering Mathematics   Calculus
Question 3 Explanation:
$f\left ( x \right )=\lim_{x\rightarrow 0}\left [ \frac{x^{3}+x^{2}}{2x^{3}-7x^{2}} \right ]$
Since this has $\frac{0}{0}$ form, limit can be found by repeated application of L'Hospitals rule.
\begin{aligned}f\left ( x \right )&=\lim_{x\rightarrow 0}\left [ \frac{3x^{2}+2x}{6x^{2}-14x} \right ] \\ &=\lim_{x\rightarrow 0}\left [ \frac{6x+2}{12x-14} \right ] \\ &=\left [ \frac{6\times 0+2}{12\times 0-14} \right ] \\ &=-\frac{1}{7} \end{aligned}
 Question 4
For the plane truss shown in the figure, the number of zero force members for the given loading is A 4 B 8 C 11 D 13
Structural Analysis   Trusses
Question 4 Explanation: If three members meet at a joint and two of them are collinear, then the third member will carry zero force provided that there is no external load at the joint.
Thus using above statement we arrive at 8 zero force members which are highlighted by 'o' sign.
 Question 5
The unit load method used in structural analysis is
 A applicable only to statically indeterminate structures B another name for stiffness method C an extension of Maxwell's reciprocal theorem D derived from Castigliano's theorem
Structural Analysis   Methods of Structural Analysis
Question 5 Explanation:
Unit load method is used to find deflection at any point of structure. It is derived from Castigliano's theorem.
$\Delta=\frac{\partial U}{\partial Q}=\int M \frac{\partial M}{\partial Q} \frac{d x}{E I}=\int \frac{M m d x}{E I}$
 Question 6
For linear elastic systems, the type of displacement function for the strain energy is
 A linear B quadratic C cubic D quartic
Solid Mechanics   Properties of Metals, Stress and Strain
Question 6 Explanation:
$\text { Strain Energy }=\frac{1}{2} \times \sigma \times \varepsilon=\frac{1}{2} \mathrm{E} \varepsilon^{2}$
Since strain is directly proportional to displacement so strain energy is directly proportional to quadratic equation of displacement.
 Question 7
For a linear elastic structural system, minimization of potential energy yields
 A compatibility conditions B constitutive relations C equilibrium equations D strain-displacement relations
Structural Analysis   Methods of Structural Analysis
Question 7 Explanation:
In a linear elastic structural system for potential energy (U) to be minimum,
$\frac{\partial U}{\partial P}=0$
The above equation represents the compatibility condition at the point where load P is acting.
 Question 8
In the limit state design method of concrete structures, the recommended partial material safety factor ($\gamma _m$) for steel according to IS:456-2000 is
 A 1.5 B 1.15 C 1 D 0.87
RCC Structures   Working Stress and Limit State Method
 Question 9
For avoiding the limit state of collapse, the safety of RC structures is checked for appropriate combinations of Dead Load (DL), Imposed Load or Live Load (IL), Wind Load (WL) and Earthquake Load (EL). Which of the following load combinations is NOT considered ?
 A 0.9DL + 1.5WL B 1.5DL + 1.5WL C 1.5DL + 1.5WL + 1.5EL D 1.2DL + 1.2IL + 1.2WL
RCC Structures   Working Stress and Limit State Method
 Question 10
Rivet value is defined as
 A lesser of the bearing strength of rivet and the shearing strength of the rivet B lesser of the bearing strength of rivet and the tearing strength of thinner plate C greater of the bearing strength of rivet and the shearing of the rivet D lesser of the shearing strength of the rivet and the tearing strength of thinner plate
Design of Steel Structures   Structural Fasteners
There are 10 questions to complete. 