## Applied Mathematics 4 - May 2013

### Mechanical Engineering (Semester 4)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1 (a)** If X is a binomially distributed with E[X]=2 and Var[X]=4/3, find the probability distribution of X.(5 marks)
**1 (b) ** A discrete random variable has the probability function given below.

X : | -2 | -1 | 0 | 1 | 2 | 3 |

P(X=x) : | 0.2 | k | 0.1 | 2k | 0.1 | 2k |

**1 (c)**Find the Fourier series of f(x)=1-x

^{2}in (-1, 1)(5 marks)

**1 (d)**Obtain wave equation for vibration of a string.(5 marks)

**2 (a)**The marks obtained by a student in a college are normally distributed with mean 65 and variance 25. If 3 students are selected at random from this college, what is the probability that at least one of them would have scored more than 75 marks?(6 marks)

**2 (b)**Obtain the complex form of fourier series for f(x) = e

^{ax}in (0, a).(6 marks)

**2 (c)**A homogenous rod of conducting material of length l has ends kept at zero temperature and the temperature at centre is T and falls uniformly to zero at the two ends. Find the temperature u(x,t) at any time.(8 marks)

**3 (a)**Find a cosine series of period 2π to represent sin x in 0 ≤ x ≤ π(6 marks)

**3 (b)**In a study to investigate whether certain detonators used with explosives in coal mining meet the requirement that at least 90% will ignite the explosive, when charged it is found that 174 out of 200 detonators function properly. Test the null hypothesis that P = 0.90 against the alternative hypothesis that P < 0.9 at 0.05 level of significance.(6 marks)

**3 (c)**Calculate the correlation coefficient from the following data.

X : | 23 | 27 | 28 | 29 | 30 | 31 | 33 | 35 | 36 | 39 |

Y : | 18 | 22 | 23 | 24 | 25 | 26 | 28 | 29 | 30 | 32 |

**4 (a)**In a certain factory turning out blades, there is a small chance 1/500 for any blade to be defective. The blades are supplied in packets of 10. Use the poisson distribution to calculate the approximate number of packets containing no defective, one defective, two defective blades in a consignment of 10,000 packets.(6 marks)

**4 (b)**The regression lines of a sample are x+6y=6; 3x+2y+10. Find:

(i) Sample means ¯x, ¯y

(ii) Coefficient of correlation between x and y;

(iii) Also, verify that the sum of the coefficients of regression is greater than 2r.(6 marks)

**4 (c)**Expand f(x) = x sinx in the interval 0 ≤ x ≤ 2π. Deduce that:

(8 marks)

**5 (a)**Theory predicts that the proportion of beans in four groups A, B, C, D should be 9:3:3:1. In an experiment among 1600 beans the numbers in four groups were 882,313,287 and 118. Does experimental result support the theory?(6 marks)

**5 (b)**Find fourier integral represention of

f(x) = x for 0 < x < a

f(x) = 0 for x > a.

and f(-x) = f(x)(6 marks)

**5 (c)**

A tightly stretched string with fixed end points x=0 and x=l in the shape defined by y = kx(l-x) where k is a constant is released from this position of rest. Find y(x,t), the vertical displacement if:$\dfrac{\partial^2y }{\partial t^2}=c^{2}\dfrac{\partial^2y }{\partial x^2}$

(8 marks)**6 (a)**Nine items of a sample had the following values: 45, 47, 50, 52, 48, 47, 49, 53, 51. Does the mean of 9 items differ significantly from the assumed population mean 47.5?(6 marks)

**6 (b)**Find the Fourier expansion of f(x) = 2x - x

^{2}, 0 ≤ x ≤ 3 whose period is 3.(6 marks)

**6 (c)**A rectangular metal plate with insulated surfaces has width a and so long compared to its breadth that it may b considered infinite in length without introducing an appreciable error. If the temperature along one short edge y = 0 is given by u(x,0) = u

_{0}sin(πx/a) for 0 < x < a and the other long edges x = 0 and x = a and the other short edges are kept at zero degree temperature, find the temperature u(x,y) describing the steady state.(8 marks)

**7 (a)**Show that the functions f

_{1}(x)=1; f

_{2}(x)=x are orthogonal on (-1,1). Determine the constants a and b such that the function f

_{3}(x)=-1+ax+bx

^{2}is orthogonal to both f

_{1}and f

_{2}on that interval.(6 marks)

**7 (b)**Find the half range cosine series for f(x) = x in 0 < x < 2. Hence find the sum :

(6 marks)

**7 (c)**Fit a second degree parabolic curve to the following data:

X : | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Y : | 2 | 6 | 7 | 8 | 10 | 11 | 11 | 10 | 9 |