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Engineering Mathematics-1 — Minor 2024 question paper

MMMUT Mechanical Engineering previous year question paper for Engineering Mathematics-1 (BSM-110), semester 1, Minor 2024. All 15 questions are listed below, each with a written answer on Nexsus.

Paper details

  • Subject: Engineering Mathematics-1 (BSM-110)
  • Branch: Mechanical Engineering
  • Semester: 1
  • Exam: Minor 2024
  • Questions: 15

Questions asked in Engineering Mathematics-1 Minor 2024

  1. Q1. Attempt any three parts of the following.
  2. Q1(a). If u, v, w are the roots of the cubic equation (λ - x)^3 + (λ - y)^3 + (λ - z)^3 = 0 in λ, then find ∂(u, v, w)/∂(x, y, z) [4 marks]
  3. Q1(b). If x^2/(a^2 + u) + y^2/(b^2 + u) + z^2/(c^2 + u) = 1, then prove that (∂u/∂x)^2 + (∂u/∂y)^2 + (∂u/∂z)^2 = 2(x ∂u/∂x + y ∂u/∂y + z ∂u/∂z) [4 marks]
  4. Q1(c). Find the inverse of the matrix A = [[0, 2, 1, 3], [1, 1, -1, -2], [1, 2, 0, 1], [-1, 1, 2, 6]] [4 marks]
  5. Q1(d). Let A = [[6, -2, 2], [-2, 3, -1], [2, -1, 3]]. Find matrix P such that P^-1AP is a diagonal matrix [4 marks]
  6. Q2. Attempt any three parts of the following.
  7. Q2(a). The Temperature T at any point (x, y, z) in space is T(x, y, z) = kxyz^2, k is positive constant. Find the highest temperature on the surface of sphere x^2 + y^2 + z^2 = a^2 [3 marks]
  8. Q2(b). If u be the homogeneous function of degree n in x and y, then show that x^2 ∂^2u/∂x^2 + 2xy ∂^2u/∂x∂y + y^2 ∂^2u/∂y^2 = n(n - 1)u [3 marks]
  9. Q2(c). Expand f(x, y) = tan^-1(xy) in powers of (x - 1) and (y - 1) up to the second-degree term. Hence compute f(0.9, 1.1) approximately [3 marks]
  10. Q2(d). If log y = tan^-1 x, show that (1 + x^2)y_n+2 + {2(n+1)x - 1}y_n+1 + n(n+1)y_n = 0 [3 marks]
  11. Q3. Attempt any three parts of the following.
  12. Q3(a). Find the rank of the matrix A by changing it into normal form. A = [[2, 3, -1, -1], [1, -1, -2, -4], [3, 1, 3, -2], [6, 3, 0, -7]] [3 marks]
  13. Q3(b). Find the value of a and b for which the system of equations 3x - 2y + z = b, 5x - 8y + 9z = 3, 2x + y + az = -1 has (i) unique solution (ii) no solution (iii) infinitely many solutions [3 marks]
  14. Q3(c). State the Cayley-Hamilton theorem. Verify this theorem for the matrix A = [[1, 2], [-1, 3]]. Hence express A^6 - 4A^5 + 8A^4 - 12A^3 + 14A^2 as linear polynomial in A [3 marks]
  15. Q3(d). Find the value of k such that the system of equations x + ky + 3z = 0, 4x + 3y + kz = 0, 2x + y + 2z = 0 has nontrivial solution [3 marks]