# GATE Civil Engineering 2020 SET-1

 Question 1
In the following partial differential equation, $\theta$ is a function of t and z, and D and K are functions of $\theta$

$D(\theta )\frac{\partial^2 \theta }{\partial z^2}+\frac{\partial K(\theta )}{\partial z}-\frac{\partial \theta }{\partial t}=0$

The above equation is
 A a second order linear equation B a second degree linear equation C a second order non-linear equation D a second degree non-linear equation
Engineering Mathematics   Ordinary Differential Equation
Question 1 Explanation:
$\because \;\;1^{st}$ term of given D. Equation contains product of dependent variable with it's derivative, so it is non-linear and also we have 2nd order derivative so it's order is two
i.e., 2nd order non linear equation.
 Question 2
The value of $\lim_{x \to \infty }\frac{x^2-5x+4}{4x^2+2x}$
 A 0 B $\frac{1}{4}$ C $\frac{1}{2}$ D 1
Engineering Mathematics   Calculus
Question 2 Explanation:
It is in $\left (\frac{\infty }{\infty } \right )$ from so by L-Hospital Rule
\begin{aligned} =&\lim_{x \to \infty }\left ( \frac{2x-5}{8x+2} \right )=\frac{\infty }{\infty }\\ =&\lim_{x \to \infty }\left ( \frac{2}{8} \right )=\frac{1}{4} \end{aligned}

 Question 3
The true value of ln(2) is 0.69. If the value of ln(2) is obtained by linear interpolation between ln(1) and ln(6), the percentage of absolute error (round off to the nearest integer), is
 A 35 B 48 C 69 D 84
Engineering Mathematics   Calculus
Question 3 Explanation:
True value of $\ln 2=0.69=T$
\begin{aligned} &x &&y=\ln x \\ &x_0=1 & &0 \\ &x_1=6& &1.79 \end{aligned}
Divided differentiation
\begin{aligned} \frac{1.79-0}{6-1}&=0.358=f[x_0,x_1] \\ \text{Approx:}\;\;\ln 2 &=f[x_0]+(x-x_0)f[x_0,x_1] \\ &= 0+(2-1)0.358\\ &= 0.358=A\\ \% \; error &= \frac{T-A}{T}\times 100=48.11\% \end{aligned}
 Question 4
The area of an ellipse represented by an equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is
 A $\frac{\pi a b}{4}$ B $\frac{\pi a b}{2}$ C $\pi a b$ D $\frac{4 \pi a b}{3}$
Engineering Mathematics   Ordinary Differential Equation
Question 4 Explanation: \begin{aligned} \text{Area} &=\int \int (1)dydx \\ &=\int_{x=-a}^{a}\int_{y=-\frac{b}{a}}^{+\frac{b}{a}}(1)dydx \\ &=4\int_{x=0}^{a} \int_{y=0}^{\frac{b}{a}\sqrt{a^2-x^2}}(1)dydx\\ &= 4 \int_{x=0}^{a}\int_{y=0}^{\frac{b}{a}\sqrt{a^2-x^2}} dx\\ &= \pi ab \end{aligned}
 Question 5
Consider the planar truss shown in the figure (not drawn to the scale) Neglecting self-weight of the members, the number of zero-force members in the truss under the action of the load P, is
 A 6 B 7 C 8 D 9
Structural Analysis   Trusses
Question 5 Explanation: As $\Delta _{AB}=0$, hence $F _{AB}=0$
Total number of zero force member = 8

There are 5 questions to complete.