Engineering B.E. (Computer Engineering) Sem 3 Mathematics for Computer Engineering May 2026 Question Paper - Mumbai University | munotes

Questions asked in this paper

• N.B. (1) Question No. 1 is Compulsory. (2) Answer any three questions from Q.2 to Q.6. (3) Figures to the right indicate full marks. (4) Statistical Table to be provided.

Q.1 Q.1 (Attempt any Five Questions from the following) 15 Marks

1. a) Find the sum and product of the eigenvalues of A, where A is the matrix with rows (-2, -9, 5), (-5, -10, 7), (-9, -21, 14).
   Read the model answer
2. b) Construct the dual of the following LPP: Maximise Z = 5x₁ + 4x₂ subject to constraint 2x₁ + 6x₂ <= 5, 7x₁ + 3x₂ <= 2, x₁, x₂ >= 0.
   Read the model answer
3. c) Find value of aₙ in the expansion of Fourier series for f(x) = x² in (-π, π).
   Read the model answer
4. d) Calculate first 2 raw moments for probability density function f(x) = xe⁻ˣ, x >= 0; f(x) = 0 otherwise.
   Read the model answer
5. e) What is the remainder when 5¹¹ is divided by 7?
   Read the model answer
6. f) Obtain Spearman's rank correlation coefficient for following data: X: 18, 17, 13, 17, 19, 21; Y: 21, 22, 14, 17, 18, 13.
   Read the model answer

Q.2 Q.2 15 Marks

1. a) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 7. Find the probability of observing exactly three accidents on this stretch of road next month.
   Read the model answer
2. b) Calculate the correlation coefficient for the following heights of fathers (X) and their sons (Y): X: 65, 66, 67, 67, 68, 69, 70, 72; Y: 67, 68, 65, 68, 72, 72, 69, 71.
   Read the model answer
3. c) Expand Fourier series for f(x) = x² in (0, 2).
   Read the model answer

Q.3 Q.3 15 Marks

1. a) Fit a straight line of the form y = a + bx to the following data: X: 1, 3, 5, 7, 8, 10; Y: 8, 4, 5, 7, 8, 4.
   Read the model answer
2. b) By using Chinese Remainder theorem, solve x = 3 (mod 4), x = 5 (mod 7).
   Read the model answer
3. c) Find eigen values and eigen vectors of A, where A is the matrix with rows (-9, 4, 4), (-8, 3, 4), (-16, 8, 7).
   Read the model answer

Q.4 Q.4 15 Marks

1. a) A random variable X has the following probability distribution: X: 1, 2, 3, 4; P(X = x): 0.2, 0.3, 0.3, 0.2. Find moment generating function about origin of x, hence find first two raw moments.
   Read the model answer
2. b) The distribution of wait times at a drive-through restaurant is approximately normal with mean 185 seconds and standard deviation 15 seconds. Find the percentage of wait times that are (i) between 155 and 215 seconds (ii) more than 170 seconds.
   Read the model answer
3. c) Check whether the matrix A with rows (8, -8, -2), (4, -3, -2), (3, -4, 1) is diagonalizable, if yes, find the diagonalizing matrix P and the diagonal matrix D.
   Read the model answer

Q.5 Q.5 15 Marks

1. a) Verify Cayley-Hamilton theorem for matrix A with rows (3, 2, 1), (1, 2, 2), (3, 2, 1).
   Read the model answer
2. b) Maximise Z = 7x₁ + 5x₂ subject to constraint -x₁ - 2x₂ >= -6, 4x₁ + 3x₂ <= 12, x₁, x₂ >= 0.
   Read the model answer
3. c) Perform key generation, encryption and decryption for RSA Algorithm with the following parameters: primes p = 3, q = 11, message M = 5.
   Read the model answer

Q.6 Q.6 15 Marks

1. a) Expand Fourier series for f(x) = x in (-π, π).
   Read the model answer
2. b) Obtain the equation of line of regression of X on Y: X: 21, 23, 25, 27, 28, 20; Y: 8, 12, 15, 17, 18, 20.
   Read the model answer
3. c) Using the method of Lagrange's multipliers, solve the following NLPP: Maximise Z = x₁² + x₂² + x₃² - 10x₁ - 6x₂ - 4x₃ subject to constraint x₁ + x₂ + x₃ = 7, x₁, x₂, x₃ >= 0.
   Read the model answer

Read from the scan above, so a character or two may differ. The scan is the original.