Applied Mathematics 4 : Question Paper May 2012 - Information Technology (Semester 4)

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Applied Mathematics 4 - May 2012

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) Probability of A, B and C hitting a target is 2/5, 1/5 and 4/5 respectively. If they fire the target what is the probability that at least two shots hit the target?(5 marks) 1 (b) Find the roots of the following equation using Newton-Raphson method correct upto 4 decimal places x³-5x+3=0 with x₀=2.(5 marks) 1 (c) A Company markets car tires. Their lives are normally distributed with mean 40,000 KM. and standard deviation 3,000 KM. A change in a production process is believed to result in a better product. A test sample of 64 new tires has mean life of 41,200 KM. Can you conclude that there is no significant difference between new product mean and current mean?(5 marks) 1 (d) A person wants to decide the constituents of a diet which will fulfil his daily requirements of proteins, carbohydrates at the minimum cost. The choice is to be made from four different foods. The yield per unit is given below:

Food Type	Proteins	Fats	Carbohydrates	Cost per units (Rs.)
1	3	2	6	45
2	4	2	4	40
3	8	7	7	85
4	6	5	4	65
Minimum Requirement	800	200	700

Formulate the L.P.P. (5 marks) 2 (a) Solve the following using Gauss-Seidel method:
20x + y - 2z = 17
3x + 20y - z = -18
2x - 3y + 20z = 25.(6 marks) 2 (b) An irregular 6 faced dice is thrown. The probability that in 10 throws it will give 5 even numbers is twice as likely that it will give four even numbers. How many times 10,000 sets of 10 throws, would you expect to no even number?(6 marks) 2 (c) Show that the second degree curve fitting the following data is given by
v = 10 + 0.85u - 0.27u²
where u = x-5, v = y-7.
Also find y when x=10.

X	1	2	3	4	5	6	7	8	9
Y	2	6	7	8	10	11	11	10	9

(8 marks) 3 (a) A fair coin is tossed till head appears. What is the expectation of the number of tosses required? (6 marks) 3 (b) Given a normal distribution with μ=50, σ=10. Find the value of x that has
(i) 13% of area to the left.
(ii) 14% of the area to the right.(6 marks) 3 (c) Solve by using Simplex method
Maximise z = 5x₁ + 4x₂
Subject to 6x₁ + 4x₂ ≤ 24
x₁ + 2x₂ ≤ 6
-x₁ + x₂ ≤ 1
x₂ ≤ 2
x₁ , x₂ ≥ 0(8 marks) 4 (a) Derive the pdf of Poisson's distribution as a limiting case of Binomial Distribution and hence find its mean.(6 marks) 4 (b) Calculate the Spearman's rank correlation for the following data

X	97.8	99.2	98.8	98.3	98.4	96.7	97.1
Y	73.2	85.8	78.9	75.8	77.2	81.2	83.8

(6 marks) 4 (c) Evaluate

by using
(i) Trapezoidal Rule.
(ii) Simpons 1/3^rd rule.
(iii) Simpons 3/8^th rule.(8 marks) 5 (a) For a random sample of 10 children fed on diet A the increase in weight was 10, 6, 16, 17, 13, 12, 8, 14, 15, 9.
For a random sample of 12 children fed on diet B the increase in weights was 7, 13, 22, 15, 12, 14, 18, 8, 21, 23, 10, 17.
Test whether the diets A & B differ significantly as regard effect in increase in weight. Use 5% LOC.(6 marks) 5 (b) From the following data find the equation of line of regression of y on x and estimate the most probable value of y when x=80.

x	89	86	74	65	64	64	66	67	72	79
y	92	91	84	75	73	72	71	75	78	94

(6 marks) 5 (c) While calculating correlation coefficient between x & y following constants are obtained.
N=25, ∑x=125, ∑y=100, ∑x²=650, ∑y²=460, ∑xy=508
It was later discovered that it had recorded two pairs x=6, y=14 and x=8, y=6 while the correct values were x=8, y=12 and x=6, y=8. Calculate correct correlation coefficient.(8 marks) 6 (a) Solve the following using Gauss elimination method-
2x + y + z = 10
3x + 2y + 3z = 18
x + 4y + 9z = 16(6 marks) 6 (b) The following table gives information regarding the colour of hair → and colour of eye ↓

Hair Colour ? Colour of Eye ?	Black	Fair	Brown	Total
Brown	10	22	32	64
Blue	15	28	29	72
Grey	25	20	19	64
Total	50	70	80	200

Use χ²test to determine whether there is any association between hair colour and eye colour.(6 marks) 6 (c) Apply Lagrange formuale to find f(5) & f(6) given that
f(1) = 2, f(2) = 4. f(3) = 8, f(7) = 128. Explain why the result differs obtained by f(x)=2^x.(8 marks) 7 (a) Solve graphically the following L.P.P.
Maximise, z = x - 2y
-x + y ≤ 1
Subject to, 6x + 4y ≥ 24
0 ≤ x ≤ 5, 2 ≤ y ≤ 4(6 marks) 7 (b) Explain:-
(i) Null Hypothesis.
(ii) Alternative Hypothesis.
(iii) Critical Region.
(iv) Level of Significance.
(v) Types of errors.
(vi) One Tailed Two Tailed Tests.(6 marks) 7 (c) Fit the binomial distribution for the following data and test the goodness of fit

No. of boys	0	1	2	3	4	5
No. of families	8	40	88	110	56	18

(8 marks)

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