JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year MATHEMATICAL AND STATISTICAL FOUNDATIONS Previous Question Paper (March - 2022 - R18)
Answer any five questions
All questions carry equal marks
Main
Find x if possible such that x ≡ 2 (mod 6), x ≡ 5 (mod 8) and x ≡ 5 (mod 7), x ≡ 4 (mod 8)
Answer
The problem consists of two separate systems of linear congruences. For the first system: x ≡ 2 (mod 6) and x ≡ 5 (mod 8). Since gcd(6, 8) = 2, a solution exists only if 5 - 2 is divisible by 2. Because 3 is not divisible by 2, this system has no solution. For the second system: x ≡ 5 (mod 7) and x ≡ 4 (mod 8). Since gcd(7, 8) = 1, a unique solution exists modulo 56. Using the Chinese Remainder Theorem, x = 7k + 5. Substituting into the second congruence: 7k + 5 ≡ 4 (mod 8) implies -k ≡ -1 (mod 8), so k ≡ 1 (mod 8). Thus, k = 8m + 1. Substituting back: x = 7(8m + 1) + 5 = 56m + 12. The solution is x ≡ 12 (mod 56).
Solve 8x ≡ 13 (mod 29)
Answer
To solve 8x ≡ 13 (mod 29), we find the modular multiplicative inverse of 8 modulo 29 using the Extended Euclidean Algorithm. We solve 8a + 29b = 1: 29 = 3(8) + 5; 8 = 1(5) + 3; 5 = 1(3) + 2; 3 = 1(2) + 1. Working backwards: 1 = 3 - 1(2) = 3 - 1(5 - 1(3)) = 2(3) - 5 = 2(8 - 5) - 5 = 2(8) - 3(5) = 2(8) - 3(29 - 3(8)) = 11(8) - 3(29). Thus, 8⁻¹ ≡ 11 (mod 29). Multiplying both sides of the congruence by 11: x ≡ 13 × 11 (mod 29) ≡ 143 (mod 29). Dividing 143 by 29 gives 143 = 4 × 29 + 27. Therefore, x ≡ 27 (mod 29).
A continuous Random variable has the p.d.f f(x) = Kx² for 0 < x < 3, and 0 otherwise. Determine: i) K ii) The mean iii) Variance.
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Ten coins are thrown simultaneously. Find the probability of getting i) At least 1 head ii) At least seven heads.
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In a distribution exactly normal 7% of the items are under 35 and 89% are under 63. What are the mean and standard deviation of the distribution.
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Five measurements of the tar content of a certain kind of cigarette yielded 14.5, 14.2, 14.4, 14.3 and 14.6 mg per cigarette. Show that the difference between the mean of this sample x̄ = 14.4 and the average tar claimed by the manufacture μ = 14.0 is significant α = 0.05.
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Among 64 offsprings of a certain cross between guinea pigs 34 were red, 10 were black and 20 were white. According to the genetic model these numbers should be in the ratio 9 : 3 : 4. Are the data consistent with model at 5% level?
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A professor has three pet questions, one of which occurs on every test he gives. He never uses the same question twice in successive examinations. If he used the question no. 1, he tosses a coin and uses the question no. 2. If he uses the question no. 2, he tosses two coins and use the question no. 3, if both are heads. If he uses the question no. 3, he tosses three coins and use the question no. 1, if all are heads. In long run which question does he use most often and with how much frequency is it used.
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Solve x ≡ 2 (mod 6), x ≡ 5 (mod 8)
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Solve 9x ≡ 12 (mod 16)
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A continuous Random variable has the p.d.f f(x) = 1/2(1+|x|) for -1 ≤ x ≤ 1, and 0 otherwise. Determine: i) P(2 ≤ x ≤ 4) ii) The mean iii) Variance.
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If the variance of a Poisson variate is 3. Find the probability that i) P(x=0) ii) P(1 ≤ x < 4)
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A study of passage of English text to find a vowel followed by a vowel or a consonant followed by a consonant or a vowel reveal the following transition probability matrix [[0.12, 0.88], [0.54, 0.46]]. Find the percentage of letters in the text book which are expected to be vowels.
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