## Applied Mathematics 4 - May 2015

### 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)** A continuous random variable with P.D.F. 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 (b)** $$ if \ A= \begin{bmatrix}
2 &2 &1 \\1
&3 &1 \\1
&2 &2
\end{bmatrix} $$ Find the characteristics roots of A and A^{3}+I.(5 marks)
**1 (c)** By using Green's theorem Show that the area bounded by a simple closed curve c is given by $$ \dfrac {1}{2} \int_c xdy-ydx $$(5 marks)
**1 (d)** If the tangent of the angle made by the line of regression of y on x is 0.6 and ?_{y}=2?_{x}. Find the correlation coefficient between x and y.(5 marks)
**2 (a)** The means of two random samples of size 9 and 7 are 196.42 and 198.82 respectively. The sum of the square of the deviation from the means is 26.94 and 18.73 respectively. Can the sample be considered to have been drawn from the same population?(6 marks)
**2 (b)** If the vector field F is irrotational, find the constants a,b,c where F=(x+2y+az)i+(bx-3y-z)j(4x+cy+2z)k show that F can be expressed as the gradient of a scalar function. Then find the work doen in moving a particle in this field from (1, 2, -4) to (3, 3, 2) along the straight line joining the points.(6 marks)
**2 (c)** Using the Kuhn Tucker conditions solve the following N.L.P.P Maximize [ Z=x^2_1 + x^2_2, subjected to x_1 + x_2 -4 le 0 and 2x_1-x_2 - 5 le 0, x_1, x_2 ge 0. ](8 marks)
**3 (a)** Seven dice are thrown 729 times. How many times do you expect at least four dice to show three or five?(6 marks)
**3 (b)** Evaluate by using Strokes theorem $$ \int_c xydx+xy^2dy, c $$ is the square in xy-plane with vertices (1,0), (0,1), (-1,0) and (0,-1).(6 marks)
**3 (c)** In a laboratory experiment two samples gave the following results. Test the equality of sample variance at 5% level of significance.

Sample |
Size |
mean |
Sum of square of thedeviations from mean |

1 | 10 | 15 | 90 |

2 | 13 | 14 | 108 |

**4 (a)**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 the standard deviation of 7.8 years.(6 marks)

**4 (b)**Use Gauss's divergence theorem to evaluate where [ iint_s overline{N}cdot overline{F}dS, overline{F}=(4xoverline {i} - 2y^2 overline {j} + z^2 overline {K}, ] and S is the region bounded by x

^{2}+y

^{2}=4, z=0, z=3.(6 marks)

**4 (c)**Using Lagrange's method of multipliers solve the NLPP, Optimize [ Z=4x_1 + 8x_2 - x^2_1 - x^2_2 subjected to x_1 + x_2 =4, x_1, x_2 ge 0 ](8 marks)

**5 (a)**Show that the matrix $$ A= \begin{bmatrix}8 &-6 &2 \\-6 &7 &-4 \\2 &-4 &3 \end{bmatrix} $$ is diagonalizable. Find the transforming matrix and the diagonal matrix.(6 marks)

**5 (b)**Calculate the Karl Pearson's coefficient of correlation for the following data.

x |
28 | 45 | 40 | 38 | 35 | 33 | 40 | 32 | 36 | 33 |

y |
23 | 34 | 33 | 34 | 30 | 26 | 28 | 31 | 36 | 35 |

**5 (c)**The following table gives the number of accidents in a city during a week. Find whether the accidents are uniformaly distributed over a week, using x

^{2}test.

day |
Sun | Mon | Tues | Wed | Thurs | Fri | Sat | Total |

No ofaccidents |
13 | 15 | 9 | 11 | 12 | 10 | 14 | 84 |

**6 (a)**$$ Find \ A^{50} \ if \ A = \begin{bmatrix} 2 &1 \\1 &2 \end{bmatrix} $$(6 marks)

**6 (b)**The monthly salary in a big organization is normally distributed with mean Rs. 3000, and standard deviation of Rs. 250. What should be the minimum salary of a worker in the organization so that the probability that he belongs to top 5% workers.(6 marks)

**6 (c)**Verify green's Theorem in the plane for where $$ \int_c (xy+y^2)dx+x^2 dy $$ c is the closed curve of the region bounded by y=x and y=x

^{2}.(8 marks)