## Applied Mathematics 4 - Dec 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)** Find the fourier expansion of f(x) = 4 - x^{2} in the interval (0,2)(5 marks)
**1 (b)** Find the probability that at most 5 defective fuses will be found in a box of 200 fuses if experience shows that 2% of such fuses are defective.(5 marks)
**1 (c) ** Given 6y = 5x + 90, 15x = 8y + 130, (σ_{x})^{2} = 16. Find :

(i) ¯x and ¯y

(ii) r

(iii) (σ_{y})^{2}(5 marks)
**1 (d)**

Solve the two dimensional heat equation$\dfrac{d^{2}u}{dx^{2}}+\dfrac{d^{2}u}{dy^2}=0$ which satisfies the conditions u(0, y)=u(l, y)=u(x, 0) and u(x, a) = sin $\dfrac{n \pi x}{l}$

(5 marks)**2 (a)**Obtain fourier series for f(x) = x - x

^{2}, -π < x < π. Hence deduce that:

(7 marks)

**2 (b)**Seven dice are thrown 729 times. How many times do you expect atleast 4 dices to show three or five?(7 marks)

**2 (c)**A continuous random variable X had pdf f(x) = kx

^{2}e

^{-x}, x ≥ 0.Find k mean and variance.(6 marks)

**3 (a)**Using normal distribution find the probability that in a group of 100 persons there will be 55 males assuming that the probability of a person being male is 1/2(7 marks)

**3 (b)**Derive wave equation for vibration of string.(7 marks)

**3 (c)**Obtain fourier expansion of f(x)= sin ax in the interval (-l,l) where a is not an integer.(6 marks)

**4 (a)**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 (b)**A die was thrown 132 times and the following frequencies were observed

No. obtained : | 1 | 2 | 3 | 4 | 5 | 6 |

Frequency : | 15 | 20 | 25 | 15 | 29 | 28 |

**4 (c)**Obtain complex form of fourier series for f(x) = cosh 3x + sinh 3x in (-3,3)(6 marks)

**5 (a)**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.(7 marks)

**5 (b)**Obtain half range sine series for f(x) when

f(x) = x, for 0 < x < π/2

f(x) = π-x,for π/2 < x < π

Hence deduce

(7 marks)

**5 (c)**Two independent samples of sizes 8 and 7 gave the following results

Sample 1 : | 19 | 17 | 15 | 21 | 16 | 18 | 16 | 14 |

Sample 2 : | 15 | 14 | 15 | 19 | 15 | 18 | 16 |

**6 (a)**Find the expansion of f(x) = x(π-x), 0 < x < π as a half range cosine series. Hence show that

(i) (ii) (7 marks)

**6 (b)**The diameter of a semicircular plate of radius a is kept at 0°C and the temperature at the semicircular boundary T°C. Find the steady state temperature u(r,θ)(7 marks)

**6 (c)**The average of marks scored by 32 boys is 72 with standard deviation 8 while that of 36 girls is 70 with standard deviation 6. Test at 1% level of significance. Whether the boys perform better than the girls? (6 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.(7 marks)

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

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

f(x) = 0 for x > a.

and f(-x) = f(x)(7 marks)

**7 (c)**If u=x-y, v=x+y and if x,y are uncorrelated, prove that:

(6 marks)