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. |

Question 6 |

According to IS 456-2000, which one of the following statements about the depth of neutral axis \chi _{u,bal} for a balanced reinforced concrete section is correct?

\chi _{u,bal} depends on the grade of concrete only. | |

\chi _{u,bal} depends on the grade of steel only. | |

\chi _{u,bal} depends on both the grade of concrete and grade of steel. | |

\chi _{u,bal} bal does not depend on the grade of concrete and grade of steel. |

Question 6 Explanation:

x_{u, \text { bal }}=\frac{700}{1100+0.87 f_{y}} \times d

So it depends upon grade of steel only.

So it depends upon grade of steel only.

Question 7 |

The figure shows a two-hinged parabolic arch of span L subjected to a uniformly distributed load of intensity q per unit length

The maximum bending moment in the arch is equal to

The maximum bending moment in the arch is equal to

\frac{qL^{2}}{8} | |

\frac{qL^{2}}{12} | |

Zero | |

\frac{qL^{2}}{10} |

Question 7 Explanation:

If a two hinged or three hinged parabolic arch is
subjected to UDL throughout its length, bending
moment is zero everywhere.

Question 8 |

Group I lists the type of gain or loss of strength in soils. Group II lists the property or process responsible for the loss or gain of strength in soils.

The correct match between Group I and Group II is

The correct match between Group I and Group II is

P-4, Q-1, R-2, S-3 | |

P-3, Q-1, R-2, S-4 | |

P-3, Q-2, R-1, S-4 | |

P-4, Q-2, R-1, S-3 |

Question 8 Explanation:

Loss in strength of soil due to remoulding
at same water content is termed as
sensitivity.

Over a period of time soil regain a part of its lost strength is termed as thixotropy. When seepage takes place in upward direction, seepage pressure acts in upward direction and effective stress is reduced consequently shear strength is reduced.

In liquefaction, due to dynamic/cyclic loading in loose saturated sand, effective stress decreases and decrease in shear strength is recorded.

Over a period of time soil regain a part of its lost strength is termed as thixotropy. When seepage takes place in upward direction, seepage pressure acts in upward direction and effective stress is reduced consequently shear strength is reduced.

In liquefaction, due to dynamic/cyclic loading in loose saturated sand, effective stress decreases and decrease in shear strength is recorded.

Question 9 |

A soil sample is subjected to a hydrostatic pressure, \sigma. The Mohr circle for any point in the soil sample would be

a circle of radius \sigma and center at the origin | |

a circle of radius \sigma and center at a distance \sigma from the origin | |

a point at a distance \sigma from the origin | |

a circle of diameter \sigma and center at the origin |

Question 9 Explanation:

Hydrostatic pressure acts equally in all directions.

Question 10 |

A strip footing is resting on the ground surface of a pure clay bed having an undrainedcohesion C_{u}. The ultimate bearing capacity of the footing is equal to

2\pi C_{u} | |

\pi C_{u} | |

(\pi+1) C_{u} | |

(\pi+2) C_{u} |

Question 10 Explanation:

Footing is at surface Hence,

\begin{aligned} \text{Hence},\quad D_{t}&=0 \\ q_{u}&=C N_{c}+\gamma D_{f} N_{q}+0.5 \mathrm{B} \gamma N_{\gamma} \end{aligned}

\Rightarrow \quad For clay

\begin{aligned} N_{Y} &=0, N_{q}=1 \\ \therefore \quad q_{U}&=C N_{c} \end{aligned}

As per Terzaghi

N_{c}=5.7

and as per Meyerhoff and Prandtl.

\begin{aligned} N_{c} &=5.14 \\ \therefore \quad q_{u} &=(\pi+2) C_{u} \end{aligned}

\begin{aligned} \text{Hence},\quad D_{t}&=0 \\ q_{u}&=C N_{c}+\gamma D_{f} N_{q}+0.5 \mathrm{B} \gamma N_{\gamma} \end{aligned}

\Rightarrow \quad For clay

\begin{aligned} N_{Y} &=0, N_{q}=1 \\ \therefore \quad q_{U}&=C N_{c} \end{aligned}

As per Terzaghi

N_{c}=5.7

and as per Meyerhoff and Prandtl.

\begin{aligned} N_{c} &=5.14 \\ \therefore \quad q_{u} &=(\pi+2) C_{u} \end{aligned}

There are 10 questions to complete.