# GATE CE 2019 SET-2

 Question 1
Euclidean norm (length) of the vector $[4\; -2\; -6]^T$ is
 A $\sqrt{12}$ B $\sqrt{24}$ C $\sqrt{48}$ D $\sqrt{56}$
Engineering Mathematics   Calculus
Question 1 Explanation:
Let $X=\begin{bmatrix} 4\\ -2\\ -6 \end{bmatrix}$
\begin{aligned} \text{Norm} \; X&=||X|| \\ &=\sqrt{(4)^2 +(-2)^2 +(-6)^2} \\ &= \sqrt{16+4+36}\\ &= \sqrt{56} \end{aligned}
 Question 2
The Laplace transform of $\sinh (at)$ is
 A $\frac{a}{s^2-a^2}$ B $\frac{a}{s^2+a^2}$ C $\frac{s}{s^2-a^2}$ D $\frac{s}{s^2+a^2}$
Engineering Mathematics   Partial Differential Equation
Question 2 Explanation:
$L(\sinh (at))=\frac{a}{s^2-a^2}$
 Question 3
The following inequality is true for all x close to 0.
$2-\frac{x^2}{3} \lt \frac{xsinx}{1-cosx} \lt 2$
What is the value of $\lim_{x \to 0}\frac{xsinx}{1-cosx}$?
 A 0 B 1/2 C 1 D 2
Engineering Mathematics   Calculus
Question 3 Explanation:
$\lim_{x \to 0}\frac{x \sin x}{1-\cos x}$
$\lim_{x \to 0}\frac{\frac{\sin x}{x}}{\frac{1-\cos x}{x^2}}=\frac{1}{\frac{1}{2}}=2$
 Question 4
What is curl of the vectorfield $2x^2y i+5z^2j-4yzk$?
 A $6zi+4xj-2x^2k$ B $6zi-8xyj+2x^2yk$ C $-14zi+6yj+2x^2k$ D $-14zi-2x^2k$
Engineering Mathematics   Calculus
Question 4 Explanation:
\begin{aligned} \text{curl}\bar{F}&=\begin{vmatrix} \bar{i} & \bar{j} & \bar{k}\\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ 2x^2y & 5z^2 & -4yz \end{vmatrix} \\ &=\bar{i}(-4z-10z)-\bar{j}(0-0)+\bar{k}(0-2x^2) \\ &= 14zi-2x^2k \end{aligned}
 Question 5
A closed thin-walled tube has thickness, t, mean enclosed area within the boundary of the centerline of tube's thickness, $A_m$, and shear stress, $\tau$. Torsional moment of resistance, T, of the section would be
 A $0.5 \tau A_m t$ B $\tau A_m t$ C $2 \tau A_m t$ D $4 \tau A_m t$
Solid Mechanics   Torsion of Shafts and Pressure Vessels
Question 5 Explanation:

Shear Stress,
\begin{aligned} \tau &=\frac{T}{J}R=\frac{T}{2 \pi R^3 t}R \\ \tau &=\frac{T}{2 \pi R^2 t}=\frac{T}{2 A_m t} \\ T &=2\tau A_mt \end{aligned}
 Question 6
A steel column is restrained against both translation and rotation at one end and is restrained only against rotation but free to translate at the other end. Theoretical and design (IS:800-2007) values, respectively, of effective length factor of the column are
 A 1.0 and 1.0 B 1.2 and 1.0 C 1.2 and 1.2 D 1.0 and 1.2
Design of Steel Structures   Compression Member
Question 6 Explanation:
The given support conditions indicates the following support/ end conditions of column

$l_{eff}$ as per theoretical conditions = 1.0$l_{o}$
$l_{eff}$ as per IS 800 : 2007 = 1.2 $l_{o}$
Considering the errors that may occur due to construction of supports on site.
 Question 7
If the fineness modulus of a sample of fine aggregates is 4.3, the mean size of the particles in the sample is between
 A $150\mu m \; and \; 300\mu m$ B $300\mu m \; and \; 600\mu m$ C $1.18\mu m \; and \; 2.36\mu m$ D $2.36\mu m \; and \; 4.75\mu m$
Geotechnical Engineering   Properties of Soils
Question 7 Explanation:
F.M of mean size 2.36 = 5
F.M of mean size 1.18 = 4
Hence for F.M of 4.3, the mean size of aggregate will be between 1.18 mm and 2.36 mm.
 Question 8
For a channel section subjected to a downward vertical shear force at its centroid, which one of the following represents the correct distribution of shear stress in flange and web?
 A A B B C C D D
Design of Steel Structures   Structural Fasteners
Question 8 Explanation:
Shear flow in horizontal member (flange) is linear with zero at free end and in vertical member (web) it is parabolic.
 Question 9
Which one of the following options contains ONLY primary air pollutants?
 A Hydrocarbons and nitrogen oxides B Hydrocarbons and ozone C Ozone and peroxyacetyl nitrate D Nitrogen oxides and peroxyacetyl nitrate
Environmental Engineering   Air and Noise Pollution
Question 9 Explanation:
Hydrocarbons and nitrogen oxides are considered primary air pollutants.
 Question 10
Analysis of a water sample revealed that the sample contains the following species.
$CO_3^{2-}, Na^{+}, H^{+}$, $PO_4^{3-}, AI^{3+}, H_2CO_3$, $CI^{-}, Ca^{2+},Mg^{2+}$, $HCO_3^{-}, Fe^{2+}, OH^-$
Concentrationsof which of the species will be required to compute alkalinity?
 A $CO_3^{2-},H^+,HCO_3^{-},OH^-$ B $CO_3^{2-},H^+,H_2CO_3,HCO_3^-$ C $CO_3^{2-},H_2CO_3,HCO_3^-,OH^-$ D $H^+,H_2CO_3,HCO_3^-,OH^-$
Environmental Engineering   Quality Characteristics of Water
Question 10 Explanation:
Alkalinity is defined as ability of water to neutralize the acid or hydronium ion
$\text{Alkalinity} (A_T) \text{of water} = [HCO_3^-] + [CO_3^{2-}] + [B(OH)_4^-] + [H_3(SiO_4)^-] + [HS^-] + [\text{organic anions}] + [OH^-] - [H^+]$
From given options of ions in problem answer is $CO_3^{2-}, H^+, HCO_3^-, OH^-$
There are 10 questions to complete.