Question 1 |

The estimate of \int_{0.5}^{0.5}\frac{dx}{x} obtained using Simpson's rule with three-point function evaluation exceeds the exact value by

0.235 | |

0.068 | |

0.024 | |

0.012 |

Question 1 Explanation:

Exact value of \int_{0.5}^{1.5}\frac{dx}{x}=\left [ \log x \right ]_{0.5}^{1.5} = \log (1.5)- \log(0.5) =1.0986

Approximate value by Simpson's rule with 3 point is,

\begin{aligned} I&=\frac{h}{3}\left ( f\left ( 0 \right )+4f\left ( 1 \right )+f\left ( 2 \right ) \right ) \\ n_{i}&=n_{pt}-1=3-1=2 \end{aligned}

(n_{pt} is the number of pts and n_{i} is the number of intervals)

Here h=\frac{b-a}{n_{i}}=\frac{1.5-0.5}{2}=0.5

The table is

I=\frac{0.5}{3}\left ( \frac{1}{0.5}+4 \times 1+\frac{1}{1.5}\right ) = 1.1111

So the estimate exceeds the exact value by,

Approximate value-Exact value

=0.012499 \approx 0.012499

Approximate value by Simpson's rule with 3 point is,

\begin{aligned} I&=\frac{h}{3}\left ( f\left ( 0 \right )+4f\left ( 1 \right )+f\left ( 2 \right ) \right ) \\ n_{i}&=n_{pt}-1=3-1=2 \end{aligned}

(n_{pt} is the number of pts and n_{i} is the number of intervals)

Here h=\frac{b-a}{n_{i}}=\frac{1.5-0.5}{2}=0.5

The table is

I=\frac{0.5}{3}\left ( \frac{1}{0.5}+4 \times 1+\frac{1}{1.5}\right ) = 1.1111

So the estimate exceeds the exact value by,

Approximate value-Exact value

=0.012499 \approx 0.012499

Question 2 |

The annual precipitation data of a city is normally distributed with mean and standard deviation as
1000 mm and 200 mm, respectively. The probability that the annual precipitation will be more than
1200 mm is

\lt50% | |

50% | |

75% | |

100% |

Question 2 Explanation:

The annual precipitation is normally distributed with \mu =1000\: mm and \sigma =200\: mm

\begin{aligned} p\left ( x\gt 1200 \right ) &=p\left ( z\gt \frac{1200-1000}{200} \right ) \\ &=p\left ( z\gt 1 \right ) \end{aligned}

Where z is the standard normal variate.

In normal distribution

Now, since

p\left ( -1\lt z\lt 1 \right )\approx 0.68

(\simeq 68% of data is within one standard deviation of mean)

\begin{aligned} p\left ( 0\lt z\lt 1 \right ) &=\frac{0.68}{2}&=0.34 \\ \text{So} p\left ( z\gt 1 \right ) &=0.5-0.34 \\ &=0.16\simeq 16\% \end{aligned}

which is \lt 50\%

\begin{aligned} p\left ( x\gt 1200 \right ) &=p\left ( z\gt \frac{1200-1000}{200} \right ) \\ &=p\left ( z\gt 1 \right ) \end{aligned}

Where z is the standard normal variate.

In normal distribution

Now, since

p\left ( -1\lt z\lt 1 \right )\approx 0.68

(\simeq 68% of data is within one standard deviation of mean)

\begin{aligned} p\left ( 0\lt z\lt 1 \right ) &=\frac{0.68}{2}&=0.34 \\ \text{So} p\left ( z\gt 1 \right ) &=0.5-0.34 \\ &=0.16\simeq 16\% \end{aligned}

which is \lt 50\%

Question 3 |

The infinite series 1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+... corresponds to

sec x | |

e^{x} | |

cos x | |

1+sin^{2}x |

Question 3 Explanation:

e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}...

(By McLaurin's series expansion)

(By McLaurin's series expansion)

Question 4 |

The Poisson's ratio is defined as

|\frac{\text{Axial tress}}{\text{Lateral stress}}| | |

|\frac{\text{Lateral strain}}{\text{Axial strain}}| | |

|\frac{\text{Lateral stress}}{\text{Axial stress}}| | |

|\frac{\text{Axial strain}}{\text{Lateral strain}}| |

Question 4 Explanation:

\mu =\left | \frac{\text{Lateral strain}}{\text{Axial strain}} \right |

Question 5 |

The following statements are related to bending of beams:

I The slope of the bending moment diagram is equal to the shear force.

II The slope of the shear force diagram is equal to the load intensity.

III The slope of the curvature is equal to the flexural rotation.

IV The second derivative of the deflection is equal to the curvature.

The only FALSE statement is

I The slope of the bending moment diagram is equal to the shear force.

II The slope of the shear force diagram is equal to the load intensity.

III The slope of the curvature is equal to the flexural rotation.

IV The second derivative of the deflection is equal to the curvature.

The only FALSE statement is

I | |

II | |

III | |

IV |

Question 5 Explanation:

We know that

\begin{aligned} \frac{d S}{d x} &=W ; \frac{d M}{d X}=S_{x} \\ E 1 \cdot \frac{d^{2} y}{d x^{2}} &=M \\ \therefore\quad \frac{d^{2} y}{d x^{2}} &=\frac{M}{E I} \\ \text{Also}\quad \frac{M}{I} &=\frac{f}{y}=\frac{E}{R} \quad \therefore \frac{M}{E I}=\frac{1}{R} \\ \therefore\quad \frac{d^{2} y}{d x^{2}} &=\frac{1}{R} \end{aligned}

\begin{aligned} \frac{d S}{d x} &=W ; \frac{d M}{d X}=S_{x} \\ E 1 \cdot \frac{d^{2} y}{d x^{2}} &=M \\ \therefore\quad \frac{d^{2} y}{d x^{2}} &=\frac{M}{E I} \\ \text{Also}\quad \frac{M}{I} &=\frac{f}{y}=\frac{E}{R} \quad \therefore \frac{M}{E I}=\frac{1}{R} \\ \therefore\quad \frac{d^{2} y}{d x^{2}} &=\frac{1}{R} \end{aligned}

There are 5 questions to complete.