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 A³+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 x₁+x₂-4≤0 and 2x₁+x₂-5≤0, x₁,x₂≥ 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.