Question Paper: Applied Mathematics - 4 : Question Paper May 2014 - Civil Engineering (Semester 4) | Mumbai University (MU)
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## Applied Mathematics - 4 - May 2014

### 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)

Find a,b,c if $$\bar{F}=(axy+bz^{3})\bar{i}+(3x^{2}-cz)\bar{j}+(3xz^{2}-y)\bar{k}\$$is irrational.

(5 marks) 1(b)

Find $$A^{5}-4A^{4}-7A^{3}+11A^{2}-A=10 I$$  in terms of A using Cayley-Hamilton theorem for $$A=\begin{bmatrix} 1 &4 \\ 2& 3 \end{bmatrix}\$$

(5 marks)
1(c)

A continuous random variable x has the p.d.f defined by f(x)=A+Bx, 0≤ x ≤1 if the mean of the distribution is $\dfrac{1}{3}$,Find A and B.

(5 marks)
1(d) A sample of 50 pieces of certain type of string was tested.he mean breaking strength turned out to be 14.5 pounds.Test whether the sample is from a batch of a string having a mean breaking strength of 15.6 pounds and S.D of 2.2 pounds.(5 marks) 2(a) Obtain the rank correlation coefficient from the following data:
 X 10 12 18 18 15 40 Y 12 18 25 25 50 25
(6 marks)
2(b) The marks of 1000students of university are found to be normally distributed with mean 70 & SD 5.Estimate the number of student whose marks will be
between 60 & 75 (ii)more than 75.
(6 marks)
2(c)

Show that the matrix $$A=\begin{bmatrix} 8 & -6 & 2\\ -6& 7 &-4 \\ 2&-4 & 3 \end{bmatrix}\$$Find the diagonal form and transforming matrix.

(8 marks)
3(a) A certain injection administered to 12 patients resultant in the following changes of blood pressure:
5,2,8,-1,3,0,6,-2,1,5,0,4.can it be concluded that the injection will be in general accompanied by an increase in blood pressure?
(6 marks)
3(b)

Optimize $Z=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-6x_{1}-8x_{2}-10x_{3}$

(6 marks)
3(c) Verify Green's theorem in the plane for
$$\displaystyle \oint (x^{2}-y)dx+(2y^{2}+x)dy$$ around the boundary of the region defined by $$y =x^{2}$$  and y=4.
(8 marks)
4(a) A car hire firm has two cars which it hire out day by day.The number of demands for a car each day is distributed as Poission variates with mean 1.5. Calculate the proportion of days on which (i)neither car is used (ii)some demand is refused.(6 marks) 4(b) Evaluate $$\bar{F}=(2x-y+z))i+(x+y-z^{2})j+(3x-2y+4z)k$$ and S is the surface of the cylinder $$x^{2}+y^{2}=4$$ bounded by the plane z=9 and open at the other end. (6 marks) 4(c) Table below shows the performances of students in Mathematics and physics.test the hypothesis that the performance in mathematics is independent of performance in physics.
 Grades in Physics Grade in Maths High Medium Low High 56 71 12 Medium 47 163 38 Low 14 42 81
(8 marks)
5(a) The ratio of the probability of 3 successes in 5 independent trials to the probability of 2 successes In 5 independent trials is 1/4.What is the probability of 4 successes in 6 independent trials?(6 marks) 5(b) Evaluate$$\displaystyle \iint \bar{F}.\bar{ds} \ where \ \bar{F}=4xi-2y^{2}j+z^{2}k$$ and S is the region bounded by $$y^{2}=4,x=1,z=0,z=3.$$(6 marks) 5(c) Find (i)the lines of regression (ii)coefficient of correlation for the following data.
 X 65,66,67,67,68,69,70,72 Y 67.68,65,66,72,72,69,71
(8 marks)
6(a) A group of 10 rats fed on diet A and another group of 8 rats fed on different diet B,recorded the following increase in weight
 Diet A 5,6,8,1,12,4,3,9,6,10gms Diet B 2,3,6,8,1,10,2,8gms

Find if the variances are significantly different?
(6 marks)
6(b)

If $A=\begin{bmatrix} -1 & 4\\\\ 2& 1 \end{bmatrix}$then prove that 3tan A=Atan3.

(6 marks)
6(c)

Using the Kuhn-Tucker condition solve the following N.L.P.P.
Maximize$Z=x_{1}^{2}-x_{2}^{2}-x_{3}^{2}+4x_{1}+6x_{2}$
Subject to    $x_{1}+x_{2}\leq 2$
$2x_{1}+3x_{2} \leq12$
$x_{1},x_{2} \geq 0.$

(8 marks)