## Applied Mathematics - 4 - May 2015

### Civil 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 characteristic roots of A and A^{3}+l.(5 marks)
**1 (c)** By using Green's theorem show that the area bounded by a simple closed curve c is given by $$\dfrac{1}{2}\int_{c}\limits xdy-ydx$$.(5 marks)
**1 (d)** If the tangent of the angle made by the line of regression of y on x is 0.6 and σ_{y}=2σ_{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 squares 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)J+(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 done 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 $$Z=x_{1}^{2}+x_{2}^{2}$$ subjected to x_{1}+x_{2}-4≤0 and 2x_{1}+x_{2}-5≤0, x_{1},x_{2}≥ 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 stokes theorem $$\int_{c}\limits xydx+xy^{2}dy,c$$ is the square in xy-plane with vertices(1,0),(0,1),(-1,0) and (0,-1).(6 marks)
**3 (c)** In laboratory experiment two samples gave the following result. Test the equality of sample variances at 5% level of significance.

Sample | Size | mean | Sum of squares of the deviations 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 $$\iint_{s}\limits \bar{N}\bar{F}ds, \bar{F}=(4x\bar{l}-2y^{2}\bar{J})+z^{2}\bar{k}$$ and s is 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 $$Z=4x_{1}+8x_{2}-x_{1}^{2}-x_{2}^{2}$$ subjected to x

_{1}+x

_{2}=4,x

_{1},x

_{2}≥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 matriz and the diagonal matrix."(6 marks)

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

<colgroup span="11" width="85"> </colgroup>

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 uniformly distributed over a week, using x

^{2}test.

<colgroup span="9" style="text-align: center;" width="85"> </colgroup>

Day | Sun | Mon | Tues | Wed | Thrus | Fri | Sat | Total |

No of accidents | 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 $$\int_{c}\limits (xy+y^{2})dx+x^{2}dy$$ c is the closed curve of the region bounded by y=x and y=x

^{2}(8 marks)