## Applied Mathematics 4 - Dec 2012

### 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)** Obtain half range sine series to represent

$$f(x) =\left\{\begin{matrix}
\frac{2x}{3} &0\leq x \leq \frac{\pi}{3} \\
\frac{\pi-x}{3}&\frac{\pi}{3}\leq x \leq \pi
\end{matrix}\right.$$(5 marks)
**1(b)** Let X be a continuous random variables with probability density function f(x)=k(x)(1-x) 0 ≤ x ≤1. Find k and determine a number 'b' such that p(x ≤ b)=P(X ≥ b).(5 marks)
**1(c)** A transmission channel has a per-digit error probability p=0.01.Calculate the probability of more than one error in 10 received Poission Distribution. Also find moment generation function in this case.(5 marks)
**1(d)** The coefficient of rank correlation of the marks obtained by 10 students in physics and chemistry was found to be 0.5. It was later discovered that the difference in ranks in the two subject obtained by one of the student was wrongly taken as 3 instead 0f 7. Find the correct coefficient of rank correlation.(5 marks)
**2(a)** Find the Fourier series for

$$f(x)=\dfrac{3x^{2}-6x\pi+2\pi^{2}}{12}\in (0,2\pi).$$(6 marks)
**2(b)** The income of group of 10,000 persons were found to be normally distributed with mean Rs.520/- and standard deviation Rs.60/- Find the

number of person having incomes between Rs.400/- and Rs.550

Lowest income of the richest 500.(6 marks)
**2(c)** The diameter of a semi-circular plate of radius 'a' is kept 0°C and the temperature at the semi circular boundary is T°C.Find the steady state temperature function u(r,θ).(8 marks)
**3(a)** Obtain the Fourier expansion of

$$f(x) =|cos x|in (-\lambda,\lambda)$$(6 marks)
**3(b)** If x and y are two independent normal random variates such that their means are 8,12 and standard deviation are 2 and 4√3 respectively. Find the values ? such that

$P[2X-Y]≤2]=P[(X+2Y)≥3]$(6 marks)
**3(c)** A panel of two judges A and B graded dramatic performance by independently awarding marks as follows:

Performance No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

8 | 36 | 32 | 34 | 31 | 31 | 32 | 35 |

Marks by B | 35 | 33 | 31 | 30 | 34 | 32 | 36 |

The eighth performance, however,which judge B could not attend,got 38 marks by judge A. If judge B had also been present, how many marks would he be expected to have awarded to the eighth performance?(8 marks)

**4(a)**Find Fourier sine integral of

$f(x) =\left\{\begin{matrix} x &0< x < 1 \\\\ {2-x}&1< x < 2 \\\\ 0 & x>2 \end{matrix}\right.$ (6 marks)

**4(b)**Investigation the association between the darkness of eye colour in father and son from the data:

**Colors of father's eyes**

Dark | Not dark | Total | |

Dark | 48 | 90 | 138 |

Not Dark | 80 | 782 | 862 |

Total | 128 | 872 | 1000 |

**4(c)**A string is stretched between x=0 and x=l and both ends given a displacement y=a sinpt perpendicular to the string. If the string the differential equation$\dfrac{\partial^{2}y}{\partial^{2}x}=\dfrac{1}{c^{}2}.\dfrac{\partial^{2}y}{\partial t^{2}}.$Show that the oscillation of string are given by $y= a sec (\dfrac{pl}{2c}) cos(\dfrac{Px}{c}-\dfrac{pl}{2c})sin pt $(8 marks)

**5(a)**The average of marks scored by 32 boys is 72 with standard deviation 8 while that of 36 girl is 70 with standard deviation 6. Test at 1% level of significance whether the boys perform better than the girls.(6 marks)

**5(b)**

Solve the partial differential equation by the method of separation of variables

$\dfrac{\partial u}{\partial t}=2\dfrac{\partial^{2}u}{\partial x^{2}} 00$

Given u(o,t)=u(3,t)=0

u(x,0)=5 sin 4π x-3 sin 8πx.

**5(c)**A biased coin is tossed 5 times and the whole experiment is repeated 200 times.The following frequencies of 0,1,2,...heads were obtained:

No.of heads | 0 | 1 | 2 | 3 | 4 | 5 |

Frequency | 12 | 56 | 74 | 39 | 18 | 1 |

Fit a Binominal distribution and find the theoretical frequencies.Also find mean and variance of the filled distribution.(8 marks)

**6(a)**A certain administered to 12 patient resultant in the following changes of blood pressure.

5,2,8,-1,3,0,6,-2,1,5,0,4

can it can concluded that the injection will be in general accomplished by an increase in blood pressure.(6 marks)

**6(b)**If $f(x) =\left\{\begin{matrix} 1 &0< x < 1 \\\\ 0&1< x <2 \end{matrix}\right.$ f(x+2)=f(x),find complex form of Fourier series. (6 marks)

**6(c)**

Obtain a solution of $$\dfrac{\partial^{2}u}{\partial x^{2}}+\dfrac{\partial^{2}u}{\partial y^{2}}=0$$ to satisfy the following equations.

u → 0 as y → ∞ for all x

u =0 if x=0 for all y

u -0 if x =l for all y

u =lx -x^{2} if y=o for all values of x between o and l.

**7(a)**Show that the set of functions sin(2n+1)x, n=0,1,2, ...is orthogonal over $$ \left[0,\dfrac{\pi}{2}\right ] $$Hence, construct orthnormal set of function.(6 marks)

**7(b)**can it be concluded that the average life-span of an Indian is more than 70 years, if a random sample of 100 Indians has an average life span of 71.8 years with standard deviation of 7.8 years?(6 marks)

**7(c)**A continuous random variables X has the density function $$f(x)=kx^{2}(1-x),0\leq x \leq 1.$$

Find k and p $$(|x-\mu|>2\sigma)$$where μ and σ

^{2}are the mean and variance of X.(8 marks)