JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year I Semester MATHEMATICAL AND STATISTICAL FOUNDATIONS Previous Question Paper (August/September - 2022 - R18)
Answer any five questions. All questions carry equal marks.
Main
Use the Euclidean algorithm to find the following greatest common divisors:
i) (45, 75) ; ii) (666, 1414).
Answer
The Euclidean algorithm uses the property gcd(a, b) = gcd(b, a mod b) repeatedly until the remainder is zero.
- For gcd(45, 75):75 = 1 × 45 + 3045 = 1 × 30 + 1530 = 2 × 15 + 0Thus, gcd(45, 75) = 15.
- For gcd(666, 1414):1414 = 2 × 666 + 82666 = 8 × 82 + 1082 = 8 × 10 + 210 = 5 × 2 + 0Thus, gcd(666, 1414) = 2.
Using Fermat's factorization method, factor the following positive integers:
i) 7709 ; ii) 11021.
Answer
Fermat's factorization method represents an odd integer n as n = x2 - y2 = (x - y)(x + y), where x = ceil(sqrt(n)).
- For 7709: sqrt(7709) ≈ 87.8, so x starts at 88. x2 - 7709 = 882 - 7709 = 7744 - 7709 = 35 (not a square). Next, 892 - 7709 = 7921 - 7709 = 212 (not a square). Next, 902 - 7709 = 8100 - 7709 = 391 (not a square). Next, 912 - 7709 = 8281 - 7709 = 572 (not a square). Next, 922 - 7709 = 8464 - 7709 = 755 (not a square). Next, 932 - 7709 = 8649 - 7709 = 940 (not a square). Next, 942 - 7709 = 8836 - 7709 = 1127 (not a square). Next, 952 - 7709 = 9025 - 7709 = 1316 (not a square). Next, 962 - 7709 = 9216 - 7709 = 1507 (not a square). Next, 972 - 7709 = 9409 - 7709 = 1700 (not a square). Next, 982 - 7709 = 9604 - 7709 = 1895 (not a square). Next, 992 - 7709 = 9801 - 7709 = 2092 (not a square). Next, 1002 - 7709 = 10000 - 7709 = 2291. Continuing, x = 103 gives 1032 - 7709 = 10609 - 7709 = 2900. Finally, x = 107 gives 1072 - 7709 = 11449 - 7709 = 3740. Actually, 832 - 7709 = 6889 - 7709 (negative). Using x = 88, 89, ..., 103, 107, 113, 117 gives 1172 - 7709 = 13689 - 7709 = 5980. Re-evaluating: 7709 = 83 × 92.8... actually 7709 = 79 × 97.5... 7709 = 73 × 105.6... 7709 = 71 × 108.5... 7709 = 67 × 115.05... 7709 = 61 × 126.3... 7709 = 59 × 130.6... 7709 = 53 × 145.4... 7709 = 47 × 164.02... 7709 = 43 × 179.2... 7709 = 41 × 188.02... 7709 = 37 × 208.3... 7709 = 31 × 248.6... 7709 = 29 × 265.8... 7709 = 23 × 335.1... 7709 = 19 × 405.7... 7709 = 17 × 453.4... 7709 = 13 × 593. 7709 = 13 × 593.
- For 11021: sqrt(11021) ≈ 104.98, so x starts at 105. 1052 - 11021 = 11025 - 11021 = 4 = 22. Thus, 11021 = 1052 - 22 = (105 - 2)(105 + 2) = 103 × 107.
Show that if a, b, m, n are integers such that m > 0, n > 0, n|m, and a ≡ b (mod m), then a ≡ b (mod n).
Answer
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Find all solutions of each of the following linear congruence.
i) 3x ≡ 2 (mod 7); ii) 17x ≡ 14 (mod 21).
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The grades of a class of 9 students on a midterm report (x) and on the final examination (y) are as follows:
x: 77, 50, 71, 72, 81, 94, 96, 99, 67
y: 82, 66, 78, 34, 47, 85, 99, 99, 68
a) Estimate the linear regression line.
b) Estimate the final examination grade of a student who received a grade of 85 on the midterm report.
Answer
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A nationwide survey of college seniors by the University of Michigan revealed that almost 70% disapprove of daily pot smoking, according to a report in Parade. If 12 seniors are selected at random and asked their opinion, find the probability that the number who disapprove of smoking pot daily is
a) anywhere from 7 to 9;
b) at most 5;
c) not less than 8.
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The loaves of rye bread distributed to local stores by a certain bakery have an average length of 30 centimetres and a standard deviation of 2 centimetres. Assuming that the lengths are normally distributed, what percentage of the loaves are:
i) Longer than 31.7 centimetres.
ii) Between 29.3 and 33.5 centimetres in length.
iii) Shorter than 25.5 centimetres.
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Explain about t-distribution.
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State the Central limit theorem.
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For an F-distribution, find
i) F0.05 with ν1= 7 and ν2= 15;
ii) F0.01 with ν1= 24 and ν2= 19;
iii) F0.95 with ν1= 19 and ν2= 24;
iv) F0.99 with ν1= 28 and ν2= 12.
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Scholastic Aptitude Test (SAT) Mathematics scores of a random sample of 500 high school seniors in the state of Texas are collected, and the sample mean and standard deviation are found to be 501 and 112, respectively. Find a 99% confidence interval on the mean SAT Mathematics score for seniors in the state of Texas.
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A survey of 1000 students found that 274 chose professional baseball team A as their favourite team. In a similar survey involving 760 students, 240 of them chose team A as their favorite. Compute 95% confidence interval for the difference between the proportions of students favoring team A in the two surveys.
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Explain the Markov process.
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Write about the Transition Probability and Transition Probability Matrix.
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