Question 1 |
The minimum and the maximum eigen values of the matrix \begin{bmatrix} 1 & 1&3 \\ 1& 5 & 1\\ 3&1 & 1 \end{bmatrix} are -2 and 6, respectively. What is the other eigen value ?
5 | |
3 | |
1 | |
-1 |
Question 1 Explanation:
\begin{aligned} \sum \lambda _{i}&= \text{ Trace(A)}\\ \lambda _{1}+\lambda _{2}+\lambda _{3}&=1+5+1=7\\ \text{Now } \lambda _{1}&=-2,\; \lambda _{2}=6 \\ \therefore \;\; -2+6+\lambda _{3}&=7 \\ \lambda _{3}&=3\end{aligned}
Question 2 |
The degree of the differential equation \frac{d^{2}x}{dt^{2}}+2x^{3}=0 is
0 | |
1 | |
2 | |
3 |
Question 2 Explanation:
Degree of a differential equation is the power of its highest order derivative after the differential equation is made free of radicals and fractions if any, in derivative power.
Hence, here the degree is 1, which is power of \frac{d^{2}x}{dt^{2}}
Hence, here the degree is 1, which is power of \frac{d^{2}x}{dt^{2}}
Question 3 |
The solution for the differential equation \frac{dy}{dx}=x^{2}y with the condition that y = 1 at x = 0 is
y=e^{\frac{1}{2x}} | |
ln(y)=\frac{x^{3}}{3}+4 | |
ln(y)=\frac{x^{2}}{2} | |
y=e^{\frac{x^{3}}{3}} |
Question 3 Explanation:
\frac{dy}{dx}=x^{2}y
This is variable separable from,
\begin{aligned} \frac{dy}{y}&=x^{2}\, dx \\ \int \frac{dy}{y}&=\int x^{2}dx \\ \Rightarrow \;\; \log_{e}y&=\frac{x^{3}}{3}+C_{1} \\ \Rightarrow \;\; y&=^{\frac{x^{3}}{3}+C_{1}} \\ &=e^{C_{1}}\times e^{\frac{x^{3}}{3}} \\ y&=C\times e^{\frac{x^{3}}{3}} \\ \text{Now at }& x=0, \;y=1 \\ 1&=C\times e^\frac{0}{3} \\ \Rightarrow \;\; C&=1 \\ \therefore \;\; y&=e^{\frac{x^{3}}{3}} \end{aligned}
This is variable separable from,
\begin{aligned} \frac{dy}{y}&=x^{2}\, dx \\ \int \frac{dy}{y}&=\int x^{2}dx \\ \Rightarrow \;\; \log_{e}y&=\frac{x^{3}}{3}+C_{1} \\ \Rightarrow \;\; y&=^{\frac{x^{3}}{3}+C_{1}} \\ &=e^{C_{1}}\times e^{\frac{x^{3}}{3}} \\ y&=C\times e^{\frac{x^{3}}{3}} \\ \text{Now at }& x=0, \;y=1 \\ 1&=C\times e^\frac{0}{3} \\ \Rightarrow \;\; C&=1 \\ \therefore \;\; y&=e^{\frac{x^{3}}{3}} \end{aligned}
Question 4 |
An axially loaded bar is subjected to a normal stress of 173 MPa. The stress in the bar is
75MPa | |
86.5MPa | |
100MPa | |
122.3MPa |
Question 4 Explanation:
\begin{aligned} \text { Shear stress }&=\frac{\sigma_{1}-\sigma_{2}}{2} \\ \therefore \quad&=\frac{173-0}{2}=86.5 \mathrm{MPa} \end{aligned}
Question 5 |
A steel column, pinned at both end, has a buckling load of 200 kN. If the column is restrained against lateral movement at its mid-height, it buckling load will be
200kN | |
283kN | |
400kN | |
800kN |
Question 5 Explanation:
When both ends are hinged, the buckling load is given by,
P_{c r}=\frac{\pi^{2} E I}{L^{2}} \Rightarrow 200=\frac{\pi^{2} E l}{L^{2}}
When the lateral movement at the mid-height is not available, then buckling load
\begin{aligned} &=\frac{\pi^{2} E \mid}{\left(L_{1}\right)^{2}} \quad \text { where } L_{1}=\frac{L}{2} \\ &=\frac{4 \pi^{2} E \mid}{L^{2}}=4 \times 200=800 \mathrm{kN} \end{aligned}
P_{c r}=\frac{\pi^{2} E I}{L^{2}} \Rightarrow 200=\frac{\pi^{2} E l}{L^{2}}
When the lateral movement at the mid-height is not available, then buckling load
\begin{aligned} &=\frac{\pi^{2} E \mid}{\left(L_{1}\right)^{2}} \quad \text { where } L_{1}=\frac{L}{2} \\ &=\frac{4 \pi^{2} E \mid}{L^{2}}=4 \times 200=800 \mathrm{kN} \end{aligned}
There are 5 questions to complete.