# 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.