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}




There are 5 questions to complete.

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