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?
PQ = I \text{ but }QP \neq I | |
QP = I \text{ but }PQ \neq I | |
PQ = I \text{ and } QP = I | |
PQ - QP = I |
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}
\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
2 and 2 | |
1 and 2 | |
2 and 1 | |
1 and 1 |
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.
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
e^{e^{-x}} | |
e^{-e^{-x}} | |
e^{-e^{x}} | |
e^{-x} |
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
\sigma | |
2\sigma | |
0.5\sigma | |
0.25\alpha \sigma |
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?
Maximum or minimum shear force occurs where the curvature is zero. | |
Maximum or minimum bending moment occurs where the shear force is zero. | |
Maximum or minimum bending moment occurs where the curvature is zero. | |
Maximum bending moment and maximum shear force occur at the same section. |
There are 5 questions to complete.