Engineering B.E. (Computer Engineering) Sem 3 Mathematics for Computer Engineering Dec 2025 Question Paper - Mumbai University | munotes

Questions asked in this paper

Que. 1 Que. 1 (Attempt any Five questions of the following) 15 Marks

1. a. If A is the matrix with rows (3, 2, 1), (0, -1, 4), (0, 0, -2), find the Eigen values of A⁻¹ + 6I.
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2. b. Transform the LPP to standard form: Maximise Z = 2x₁ + x₂ + 4x₃ subject to x₁ - x₂ + 2x₃ <= 5, 3x₁ + x₂ + 5x₃ >= 14, x₁, x₂ >= 0, x₃ is unrestricted.
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3. c. Find the value of aₙ in the expansion of Fourier Series for f(x) = 4 - x² in (0, 2π).
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4. d. A random variable x defines the possible outcomes of tossing of a fair die, find moment generating function of x about origin.
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5. e. What is the remainder when 13⁷⁰ is divided by 21?
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6. f. Obtain Spearman's rank correlation coefficient for following data: X: 24, 24, 24, 55, 25; Y: 24, 31, 51, 40, 38.
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Que. 2 Que. 2 15 Marks

1. a. It is known that the probability of an item produced by a certain machine will be defective is 0.001. If the produced items are sent to the market in packets of 100, find the number of packets containing at least 2 are defective in the consignment of 2000 Packets.
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2. b. Ten students got the following percentage of marks in mathematics and statistics. Maths: 28, 34, 18, 25, 75, 23, 24, 33, 37, 43; Stats: 24, 31, 11, 34, 65, 24, 23, 53, 53, 42. Calculate the coefficient of correlation.
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3. c. Expand Fourier series for f(x) = x² in (0, 2).
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Que. 3 Que. 3 15 Marks

1. a. Obtain equations of line of regression of x on y for following data: X: 31, 44, 65, 49, 41; Y: 48, 32, 21, 66, 43.
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2. b. By using Chinese Remainder theorem, Solve x = 1 (mod 3), x = 2 (mod 4), x = 3 (mod 5).
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3. c. Find Eigen values and Eigen vectors of the matrix A with rows (3, -1, 1), (-1, 5, -1), (1, -1, 3).
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Que. 4 Que. 4 15 Marks

1. a. A random variable X has probability density function f(x) = x²e⁻ˣ, x >= 0. Find 2 moments about origin and mean.
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2. b. Show that the matrix A with rows (2, 2, 1), (1, 3, 1), (1, 2, 2) is diagonalisable, hence find it's diagonalisable matrix.
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3. c. In RSA System the public key (E, N) of user A is defined as (7, 187). Calculate φ(N) and private key 'D'. What is the cipher text for M = 10 using the public key.
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Que. 5 Que. 5 15 Marks

1. a. Verify Cayley-Hamilton theorem for matrix A with rows (2, -1, 1), (-1, 2, -1), (1, -1, 2).
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2. b. The height of school children of one school is normally distributed with mean of 54 inches and standard deviation 12 inches. What proportion of students have height between (i) 46 and 56 inches (ii) More than 56.
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3. c. Applying Simplex method, Solve the following LPP: Maximise Z = 3x₁ + 2x₂ subject to constraint x₁ + x₂ <= 4, x₁ - x₂ <= 2, x₁, x₂ >= 0.
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Que. 6 Que. 6 15 Marks

1. a. Fit a straight line of the form y = a + bx to the following data: X: 1, 4, 7, 9, 8, 10; Y: 18, 14, 19, 27, 18, 28.
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2. b. Find Fourier series for the function f(x) = x, 0 < x < 2π.
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3. c. Using the method of Lagrange's multipliers solve the following NLPP: Optimize z = -x₁² - x₂² + 4x₁ + 8x₂ subject to x₁ + x₂ = 4, x₁, x₂ >= 0.
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