# GATE CE 2016 SET-1

 Question 1
Newton-Raphson method is to be used to find root of equation $3x-e^{x}+\sin x=0$. If the initial trial value for the root is taken as 0.333, the next approximation for the root would be _________
 A 0.33 B 0.54 C 0.36 D 0.76
Engineering Mathematics   Numerical Methods
Question 1 Explanation:
According to Newton-Raphson Method:
\begin{aligned} x_{N+1}&=X_{N}-\frac{f\left ( X_{N} \right )}{{F}'\left ( X_{N} \right )} \\ f\left ( x \right )&=3x-e^{x}+\sin x \\ {f}'\left ( x \right )&=3-e^{x}+\cos x \\ \Rightarrow \;\; X_{1}&=X_{0}-\frac{f\left ( 0.333 \right )}{{f}'\left ( 0.333 \right )} \\ &=0.333-\frac{3\times 0.333-e^{0.333}+\sin 0.333}{3-e^{0.333}+\cos 0.333} \\ \therefore \;\; X_{1}&=0.36 \end{aligned}
 Question 2
The type of partial differential equation $\frac{\partial^2 P}{\partial x^2} + \frac{\partial^2 P}{\partial y^2} +3\frac{\partial^2 P}{\partial x \partial y } +2\frac{\partial P}{\partial x}-\frac{\partial P}{\partial y}=0$ is
 A elliptic B parabolic C hyperbolic D none of these
Engineering Mathematics   Ordinary Differential Equation
Question 2 Explanation:
Comparing the given equation with the general form of second order partial differential equation, we have A=1, B=3, C=1 $\Rightarrow\; B^{2}-4AC=5\gt0$
$\therefore$ PDE is Hyperbola.
 Question 3
If the entries in each column of a square matrix M add up to 1, then an eigen value of M is
 A 4 B 3 C 2 D 1
Engineering Mathematics   Linear Algebra
Question 3 Explanation:
Consider the '$2\times 2$' square matrix $M=\begin{bmatrix} a & b\\ c & d \end{bmatrix}$
$\Rightarrow \;\; \lambda ^{2}-\left ( a+d \right )\lambda +\left ( ad-bc \right )=0 \;\;...(i)$
Putting $\lambda =1$, we get
0=0 which is true.
$\therefore \;\;\lambda =1$ satisfied the eq.(i) but $\lambda =2,3,4$ does not satisfy the eq.(i). For all possible values of a, d.
 Question 4
Type II error in hypothesis testing is
 A acceptance of the null hypothesis when it is false and should be rejected B rejection of the null hypothesis when it is true and should be accepted C rejection of the null hypothesis when it is false and should be rejected D acceptance of the null hypothesis when it is true and should be accepted
Engineering Mathematics   Probability and Statistics
 Question 5
The solution of the partial differential equation $\frac{\partial u}{\partial t}=\alpha \frac{\partial^2 u}{\partial x^2}$ is of the form
 A $C\cos \left ( kt \right )\left \lfloor C_{1}e^{(\sqrt{k/\alpha })x} + C_{2}e^{-(\sqrt{k/\alpha })x} \right \rfloor$ B $Ce^{kt}\left \lfloor C_{1}e^{(\sqrt{k/\alpha })x} + C_{2}e^{-(\sqrt{k/\alpha })x} \right \rfloor$ C $Ce^{kt}\left \lfloor C_{1}\cos (\sqrt{k/\alpha } )x+ C_{2}\sin {(-\sqrt{k/\alpha })x} \right \rfloor$ D $C\sin (kt)\left \lfloor C_{1}\cos (\sqrt{k/\alpha } )x+ C_{2}\sin {(-\sqrt{k/\alpha })x} \right \rfloor$
Engineering Mathematics   Ordinary Differential Equation
Question 5 Explanation:
The PDE $\frac{\partial u}{\partial t}=\alpha \frac{\partial^2 u}{\partial x^2} \;\;...(i)$
Solution of (i) is,
$u\left ( x,t \right )=\left ( A\cos px+B\sin px \right )Ce^{-p^{2}\alpha t}$
Put $-p^{2}\alpha =k$
$\Rightarrow \;\; p=\sqrt{-\frac{k}{\alpha }}=\sqrt{\frac{k}{\alpha }}i$
Putting value of p in eq.(i),
\begin{aligned}u\left ( x,t \right )&=\left ( A\cos \sqrt{\frac{k}{\alpha }}x+b\sin h\sqrt{\frac{k}{\alpha }}x \right )Ce^{kt} \\ &=Ce^{kt}\left [ A\left \{ \frac{e^{\sqrt{\frac{k}{\alpha }}x}+e^{-\sqrt{\frac{k}{\alpha }}x}}{2} \right \}+B\left \{ \frac{e^{\sqrt{\frac{k}{\alpha }}x}-e^{-\sqrt{\frac{k}{\alpha }}x}}{2} \right \} \right ] \\ &=Ce^{kt}\left [ e^{\sqrt{\frac{k}{\alpha }}x}\left \{ \frac{A+B}{2} \right \}+e^{-\sqrt{\frac{k}{\alpha }}x}\left \{ \frac{A-B}{2} \right \} \right ] \\ &=Ce^{kt}\left [ c_{1}e^{\sqrt{\frac{k}{\alpha }}x}+c_{2}e^{-\sqrt{\frac{k}{\alpha }}x} \right ] \end{aligned}
 Question 6
Consider the plane truss with load P as shown in the figure. Let the horizontal and vertical reactions at the joint B be $H_{B}$ and $V_{B}$, respectively and $V_{C}$ be the vertical reaction at the joint C. Which one of the following sets gives the correct values of $V_{B}, H_{B}$ and $V_{C}$ ?
 A $V_{B}=0;H_{B}=0;V_{C}=P$ B $V_{B}=P/2;H_{B}=0;V_{C}=P/2$ C $V_{B}=P/2;H_{B}=P\left (\sin 60^{^{\circ} }\ \right);V_{C}=P/2$ D $V_{B}=P;H_{B}=P\left (\cos 60^{^{\circ} }\ \right);V_{C}=0$
Structural Analysis   Trusses
Question 6 Explanation: \begin{aligned} V_{B}+V_{C} &=P &\ldots(i)\\ \Sigma M_{B} &=0\\ V_{C} \times 2 L-P \times 2 L&=0 \\ \Rightarrow \quad V_{C}&=P \\ \text{From(i), }\quad V_{B}&=0 \\ \Sigma F_{x}&=0 \\ \Rightarrow \quad H_{B}&=0 \end{aligned}
 Question 7
In shear design of an RC beam, other than the allowable shear strength of concrete $(\tau _{c})$ there is also an additional check suggested in IS 456-2000 with respect to the maximum permissible shear stress $(\tau _{cmax})$. The check for $(\tau _{cmax})$ is required to take care of
 A additional shear resistance from reinforcing steel B additional shear stress that comes from accidental loading C possibility of failure of concrete by diagonal tension D possibility of crushing of concrete by diagonal compression
RCC Structures   Shear, Torsion, Bond, Anchorage and Development Length
 Question 8
The semi-compact section of a laterally unsupported steel beam has an elastic section modulus, plastic section modulus and design bending compressive stress of $500 \; cm^3, 650 \; cm^3$ and 200MPa, respectively. The design flexural capacity (expressed in kNm) of the section is ____.
 A 1000 B 100 C 120 D 1200
Design of Steel Structures   Beams
Question 8 Explanation:
Design flexural capacity,
$M_{d}= \beta _{b}Z_{p}\frac{f_{y}}{\gamma _{m0}}$
$\beta _{b}= 1.0$ (for plastic and compact sections)
$\; \; \; \; = \frac{Z_{e}}{Z_{p}}$ (for semi-compact sections)
$Z_{p}=\,$ Plastic section modulus
$Z_{e}=\,$ elastic section modulus
$f_{y}=\,$ yeild stress of steel provided
$\gamma_{m0}=\,$ material factor of safety of steel against yeilding
As, design bending stress is directly given,
\begin{aligned} M_{d}&= \beta _{b}Z_{p}f_{d} \\ &= \frac{Z_{e}}{Z_{p}}\times Z_{p}\times f_{d} \\ &= Z_{e}\times f_{d} \\ &= 500\times 10^{3}\times 200\times 10^{-6} \\ &= 100 kN-m \end{aligned}
 Question 9
Bull's trench kiln is used in the manufacturing of
 A Lime B cement C bricks D none of these
Construction Materials and Management
Question 9 Explanation:
Bull Trench kiln is a continuous kiln generally oval in plan. It is 50 to 100 m long and 1.5-2.5 m deep below ground level. It is devided into 8-12 sections which is used for manufacturing of bricks.
 Question 10
The compound which is largely responsible for initial setting and early strength gain of Ordinary Portland Cement is
 A $C_{3}A$ B $C_{3}S$ C $C_{2}S$ D $C_{4}AF$
Construction Materials and Management
Question 10 Explanation:
Tricalcium Silicate $\left ( C_{3}S \right )$ hardens rapidly and is largely responsible for initial set and early strength. In general, the early strength of potland cement concrete is higher with incresed percentage of $\left ( C_{3}S \right )$
There are 10 questions to complete. 