Question Paper: Applied Mathematics 4 : Question Paper May 2013 - Information Technology (Semester 4) | Mumbai University (MU)
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Applied Mathematics 4 - May 2013

Information Technology (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 fair coin is tossed till head appears. What is the expectation of the number of tosses required?(5 marks) 1 (b) Solve using Bisection method
x - cosx = 0. Find the positive root.
(5 marks)
1 (c) Solve graphically the following L.P.P.
Maximize, z = x - 2y
Subject to, -x + y ≤ 1
6x + 4y ≥ 24
0 ≤ x ≤ 5, 2 ≤ y ≤ 4
(5 marks)
1 (d) The mean value of random sample of 60 items was found to be 145 with standard deviation of 40. Find the 95% confidence limits for the population mean.(5 marks) 2 (a) If p.d.f. of a random variable is given by,
f(x) = x for 0 ≤ x ≤ 1
f(x)=2 - x for 1 ≤ x ≤ 2
f(x)=0 Otherwise
Find the m.g.f. and hence find mean and variance.
(6 marks)
2 (b) If x1 and x2 are independent normal variates with mean 30 and 25 and variance 16 and 12 respectively and y = 3x1 -x2. Find P(60 ≤ y ≤ 80)(6 marks) 2 (c) Evaluate

by using
(i) Trapezoidal Rule.
(ii) Simpons 1/3rd rule.
(iii) Simpons 3/8th rule.
Take h = 0.25. Compare the results with exact value.(8 marks) 3 (a) Test significance of difference between the means of samples drawn from two normal populations with the following data:
Size Mean s.d.
Sample I 100 61 4
Sample II 200 63 6
(6 marks)
3 (b) If x = au + b, y = cv + d; a, b, c, d are constants then prove rxy=ruv
where rxy is coefficient of correlation between x and y.
(6 marks)
3 (c) Fit a second degree curve for the following data:
x 1 2 3 4 5
y 1250 1400 1650 1950 2300
(8 marks)
4 (a) Using Gauss-Seidel method, solve the equations
10x + y + z = 12;
2x + 10y + z = 13
2x + 2y + 10z = 14
(6 marks)
4 (b) According to theory of proportion of commodity in the four classes A, B, C, D should be 9 : 2 : 4 : 1. In a survey of 1600 items of this commodity the numbers in four classes were 882, 432, 168 and 118. Does this survey support the theory?(6 marks) 4 (c) Find the coefficient of correlation for the following data:
x 2 4 5 6 8 11
y 18 12 10 8 7 5
(8 marks)
5 (a) Let X be a random variable with the following p.d.f.
x -3 6 9
P(x) 1/6 1/2 1/3

Find the mean, variance and E(2X+1)2.
(6 marks)
5 (b) Explain:-
(i) Null Hypothesis.
(ii) Alternate Hypothesis.
(iii) Critical Region.
(iv) Level of Significance.
(v) Types of errors.
(vi) One Tailed Two Tailed Tests.
(6 marks)
5 (c) Find f(8) from the data -
x 5 7 11 13 17
f (x) 150 392 1452 2366 5202
(8 marks)
6 (a) Solve using Gauss-Jordan method
2x + y + 4z = 16;
3x + 2y + z = 10
1x + 3y + 3z = 16.
(6 marks)
6 (b) How many tosses of a coin are needed so that the probability of getting at least one head is 87.5%?(6 marks) 6 (c) Solve:
Maximize: z = 4x1 +x2 + 3x3 + 5x4;
Subject to, 4x1 - 6x2 - 5x3 - x4 ≤ 2
-3x1 - 2x2 + 4x3 + x4 ≤ 10
-8x1 - 3x2 + 3x3 +2x4 ≤ 20
x1 ,x2, x3, x4 ≥ 0.
(8 marks)
7 (a) Find mean and variance of Binomial Distribution.(6 marks) 7 (b) Two batches of 12 animals are taken for test of inoculation. One batch was inoculated and other was not from the data can it be regarded as effective against the disease?
Dead Survived Total
Inoculated 2 10 12
Non-Inoculated 8 4 12
Total 10 14 24
(6 marks)
7 (c) Show that R=r for the following data:
x 60 62 64 66 68 70 72 74
y 92 83 101 110 128 119 137 146
(8 marks)

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