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}

Question 6 |

The stiffness coefficient k_{ij} indicates

force at i due to a unit deformation at j | |

deformation at j due to a unit force at i | |

deformation at i due to a unit force at j | |

force at j due to a unit deformation at i |

Question 6 Explanation:

Stiffness(k) is the force required to produce unit deformation.

Thus k_{ij} denotes force required in direction i due to unit deformation (displacement) in direction j.

Thus k_{ij} denotes force required in direction i due to unit deformation (displacement) in direction j.

Question 7 |

For an isotropic material, the relationship between the Young's modulus (E), shear modulus (G) and Poisson's ratio (\mu) is given by

G=\frac{E}{2(1+\mu )} | |

E=\frac{G}{2(1+\mu )} | |

G=\frac{E}{(1+2\mu )} | |

G=\frac{E}{2(1-\mu )} |

Question 8 |

A clay soil sample is tested in a triaxial apparatus in consolidated-drained conditions at a cell pressure of 100 kN/m^{2}. What will be the pore water pressure
at a deviator stress of 40 kN/m^{2} ?

0kN/m^{2} | |

20kN/m^{2} | |

40kN/m^{2} | |

60kN/m^{2} |

Question 8 Explanation:

In the consolidated drained test, the soil sample is first consolidated under an appropriate cell pressure. With the cell pressure kept at the same value, the soil sample is then sheared by applying the deviator stress so slowly that excess pore Water pressure does not develop during the test. Thus, at any stage of the test, the total stresses are the effective stresses.

Question 9 |

The number of blows observed in a Standard Penetration Test (SPT) for different
penetration depths are given as follows :

The observed N value is

The observed N value is

8 | |

14 | |

18 | |

24 |

Question 9 Explanation:

The number of blows for the first 150mm
penetration of the sampler are disregarded. The
number of blows for the next 300 mm penetration
are recorded as the observed N value.

Observed N value =8+10=18

Observed N value =8+10=18

Question 10 |

The vertical stress at some depth below the corner of a 2m x 3m rectangular footing due to a certain load intensity is 100 kN/m^{2}. What will be the vertical
stress in kN/m^{2} below the centre of a 4m x 6m rectangular footing at the same depth and same load intensity ?

25 | |

100 | |

200 | |

400 |

Question 10 Explanation:

For same load intensity and same depth

\begin{aligned} \Rightarrow \quad \frac{\sigma_{1}}{\sigma_{2}}&=\frac{A_{1}}{A_{2}} \Rightarrow \frac{100}{\sigma_{2}}=\frac{2 \times 3}{4 \times 6} \\ \Rightarrow \quad \sigma_{2}&=400 \mathrm{kN} / \mathrm{m}^{2} \end{aligned}

\begin{aligned} \Rightarrow \quad \frac{\sigma_{1}}{\sigma_{2}}&=\frac{A_{1}}{A_{2}} \Rightarrow \frac{100}{\sigma_{2}}=\frac{2 \times 3}{4 \times 6} \\ \Rightarrow \quad \sigma_{2}&=400 \mathrm{kN} / \mathrm{m}^{2} \end{aligned}

There are 10 questions to complete.