# GATE CE 2001

 Question 1
The number of boundary conditions required to solve the differential equation $\frac{\partial^2 \phi }{\partial x^2}+\frac{\partial^2 \phi }{\partial y^2}=0$ is
 A 2 B 0 C 1 D 4
Engineering Mathematics   Ordinary Differential Equation
 Question 2
Value of the integral $I=\int_{0}^{\pi /4}cos^{2}xdx$ is
 A $\frac{\pi }{8}+\frac{1}{4}$ B $\frac{\pi }{8}-\frac{1}{4}$ C $-\frac{\pi }{8}-\frac{1}{4}$ D $-\frac{\pi }{8}+\frac{1}{4}$
Engineering Mathematics   Ordinary Differential Equation
Question 2 Explanation:
\begin{aligned} I&=\int_{0}^{\pi /4}\cos ^{2}xdx \\ &=\int_{0}^{\pi /4}\frac{1+\cos 2x}{2}dx \\ &=\int_{0}^{\pi /4}\frac{dX}{2}+\int_{0}^{\pi /4}\frac{\cos 2x}{2}dx \\ &=\frac{1}{2}\left [ x \right ]_{0}^{\pi /4}+\frac{1}{2}\left [ \frac{\sin 2x}{2} \right ]_{0}^{\pi /4} \\ &=\frac{\pi }{8}+\frac{1}{4}\left [ \sin \frac{\pi }{2}-\sin x \right ] \\ &=\frac{\pi }{8}+\frac{1}{4}\times 1 =\frac{\pi }{8}+\frac{1}{4} \end{aligned}
 Question 3
Limit of the following series as x approaches $\frac{\pi }{2}$ is $f(x)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}$
 A $\frac{2\pi }{3}$ B $\frac{\pi }{2}$ C $\frac{\pi }{3}$ D 1
Engineering Mathematics   Partial Differential Equation
Question 3 Explanation:
$\lim_{x\rightarrow \frac{\pi }{2}}f\left ( x \right )=\lim_{x\rightarrow \frac{\pi }{2}}\sin x=1$
 Question 4
The degree of static indeterminacy, $N_s$, and the degree of kinematic indeterminacy, $N_k$ , for the plane frame shown below, assuming axial deformations to be negligible, are given by
 A $N_s$ = 6 and $N_k$ = 11 B $N_s$ = 6 and $N_k$ = 6 C $N_s$ = 4 and $N_k$ = 6 D $N_s$ = 4 and $N_k$ = 4
Structural Analysis   Determinacy and Indeterminacy
Question 4 Explanation:
Degree of static indeterminacy,
\begin{aligned} N_{s} &=3 m+r_{e}-3 j-r_{r} \\ &=3 \times 5+(3+2+2)-3 \times 6-0 \\ &=4 \end{aligned}
Degree of kinematic indeterminacy,
\begin{aligned} N_{k} &=3 j-r_{e}-m \\ &=3 \times 6-(3+2+2)-5=6 \\ N_{s} &=4 \text { and } N_{k}=6 \end{aligned}
 Question 5
The bending moment (in kNm units) at the mid span location X in the beam with overhangs shown below is equal to
 A 0 B -10 C -15 D -20
Solid Mechanics   Shear Force and Bending Moment
Question 5 Explanation:

\begin{aligned} \mathrm{R}+\mathrm{V}&=30 &\ldots(i)\\ \text { Now, } \mathrm{V} &\times 2-20 \times 3+10 \times 1=0 \\ \mathrm{V}&=\frac{60-10}{2}=25 \mathrm{kN}\\ \therefore \quad \mathrm{R}&=5 \mathrm{kN}\\ \mathrm{BM} \text { at }\quad X&=25 \times 1-20 \times 2=-15 \mathrm{kNm} \end{aligned}
 Question 6
Identify the FALSE statement from the following, pertaining to the effects due to a temperature rise $\Delta T$ in the bar BD alone in the plane truss shown below:
 A No reactions develop at supports A and D B The bar BD will be subject to a tensile force C The bar AC will be subject to a compressive force D The bar BC will be subject to a tensile force
Structural Analysis   Trusses
Question 6 Explanation:
As temperature increases the bar AD will tend to elongate but joint B and D will offer resistance.
Hence bar BD will be in compression.
 Question 7
Identify the correct deflection diagram corresponding to the loading in the plane frame shown below:
 A A B B C C D D
Structural Analysis   Methods of Structural Analysis
Question 7 Explanation:

Since, there is no bending moment in vertical leg so there will be no bending in member AB. And to maintain angle between AB and BC at joint 8, frame will sway as given below

 Question 8
Identify the FALSE statement from the following, pertaining to the methods of structural analysis
 A Influence lines for stress resultants in beams can be drawn using Muller Breslau's Principle B The Moment Distribution Method is a force method of analysis, not a displacement method C The Principle of Virtual Displacements can be used to establish a condition of equilibrium D The Substitute Frame Method is not applicable to frames subject to significant sideway
Structural Analysis   Methods of Structural Analysis
 Question 9
Identify the FALSE statement from the following, pertaining to the design of concrete structures
 A The assumption of a linear strain profile in flexure is made use of in working stress design, but not in ultimate limit state design. B Torsional reinforcement is not required to be provided at the corners of simply supported rectangular slabs, if the corners are free to lift up. C A rectangular slab, whose length exceeds twice its width, always behaves as a two way slab, regardless of the support conditions. D The 'load balancing' concept can be applied to select the appropriate tendon profile in a prestressed concrete beam subject to a given pattern of loads.

 Question 10
Identify the most efficient but joint (with double cover plates) for a plate in tension from the patterns (plan views) shown below, each comprising 6 identical bolts with the same pitch and gauge.
 A A B B C C D D
Design of Steel Structures   Structural Fasteners
Question 10 Explanation:
The most common type of rivet patterns are chain riveting and diamond riveting. Staggered pattern in option (A) yields more net area of the section and because of this reaon this pattern is most suitable for tension members. Staggered and diamond pattern are better as compared to the chain pattern.
There are 10 questions to complete.