# GATE CE 2017 SET-1

 Question 1
The matrix P is the inverse of a matrix Q. If $I$ denotes the identity matrix, which one of the following options is correct?
 A $PQ = I \text{ but }QP \neq I$ B $QP = I \text{ but }PQ \neq I$ C $PQ = I \text{ and } QP = I$ D $PQ - QP = I$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Given that P is inverse of Q.
\begin{aligned} P&=Q^{-1} & P&=Q^{-1} \\ PQ&=Q^{-1}Q & QP&=QQ^{-1} \\ PQ&=I & QP&=I \\ \therefore PQ&=QP=I \end{aligned}
 Question 2
The number of parameters in the univariate exponential and Gaussian distributions, respectively, are
 A 2 and 2 B 1 and 2 C 2 and 1 D 1 and 1
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
In exponential,
$f\left ( x \right )=\lambda e^{-\lambda x}$; $x=0$
The parameter is $\lambda$
In Gaussian, $f(x)=\frac{1}{\sigma \sqrt{2\pi }}e^{-\frac{1}{2}\left ( \frac{x-\mu }{\sigma } \right )^{2}}; \;\; -\infty \lt x\lt \infty$
The parameters are $\mu$ and $\sigma$.

 Question 3
Let x be a continuous variable defined over the interval ($-\infty ,\infty$), and $f(x)=e^{-x-e^{-x}}$. The integral $g(x)=\int f(x)dx$ is equal to
 A $e^{e^{-x}}$ B $e^{-e^{-x}}$ C $e^{-e^{x}}$ D $e^{-x}$
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} f\left ( x \right )&=e^{-x-e^{-x}}= e^{-x}.e^{-e^{-x}} \\ y\left ( x \right )&=\int f\left ( x \right )dx=\int e^{-x}.e^{-e^{-x}}dx\\ \text{Let } e^{-x}&=t \\ -e^{-x}dx&=dt \\ \int f\left ( x \right )dx&=\int e^{-t}.\left ( -dt \right ) \\ &=\frac{e^{-t}}{-1}.\left ( -d \right ) \\ &=e^{-t} \\ &=e^{-\left ( e^{-x} \right )} \\ &=e^{-e^{-x}} \end{aligned}
 Question 4
An elastic bar of length L, uniform cross sectional area A, coefficient of thermal expansion $\alpha$, and Young's modulus E is fixed at the two ends. The temperature of the bar is increased by T, resulting in an axial stress $\sigma$. Keeping all other parameters unchanged, if the length of the bar is doubled, the axial stress would be
 A $\sigma$ B 2$\sigma$ C 0.5$\sigma$ D 0.25$\alpha \sigma$
Solid Mechanics   Properties of Metals, Stress and Strain
Question 4 Explanation: $\sigma=\alpha T E$
$\therefore \;$ Length have no effect on thermal stress.
$\therefore \;$ Axial stress is only $\sigma$.
 Question 5
A simply supported beam is subjected to uniformly distributed load. Which one of the following statements is true?
 A Maximum or minimum shear force occurs where the curvature is zero. B Maximum or minimum bending moment occurs where the shear force is zero. C Maximum or minimum bending moment occurs where the curvature is zero. D Maximum bending moment and maximum shear force occur at the same section.
Solid Mechanics   Shear Force and Bending Moment

There are 5 questions to complete.