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Engineering Mathematics-1 — Major (End Semester) 2024 question paper

MMMUT Mechanical Engineering previous year question paper for Engineering Mathematics-1 (BSM-110), semester 1, Major (End Semester) 2024. All 24 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: Major (End Semester) 2024
  • Questions: 24

Questions asked in Engineering Mathematics-1 Major (End Semester) 2024

  1. Q1. Attempt any five parts of the following:
  2. Q1(a). If u = sin(nx) + cos(nx) then show that u_r = n^r [1 + (-1)^r sin 2nx]^(1/2), where u_r is the rth differential coefficient of u w.r.t x. [2 marks]
  3. Q1(b). Find the n^(th) derivative of e^(ax) sin(bx + c). [2 marks]
  4. Q1(c). Find the extreme values of the function f(x,y) = x^3 + y^3 - 3x - 12y + 20. [2 marks]
  5. Q1(d). If u = f(x^2 - y^2, y^2 - z^2, z^2 - x^2) then prove that 1/x * ∂u/∂x + 1/y * ∂u/∂y + 1/z * ∂u/∂z = 0 [2 marks]
  6. Q1(e). The eigen values of a matrix [[a, 4], [1, b]] are -2 and 3, then find the value of a and b. [2 marks]
  7. Q1(f). Check the consistency of the system of equations x + 2y - z = 6, 3x - y - 2z = 3 and 4x + 3y + z = 9. [2 marks]
  8. Q1(g). Find inverse of the following matrix using Gauss Jordan Method. [2 marks]
  9. Q2. Attempt any two parts of the following.
  10. Q2(a). Show that Γ(1/n) * Γ(2/n) * Γ(3/n) * ... * Γ((n-1)/n) = (2π)((n-1)/2) / (n^1/2). [5 marks]
  11. Q2(b). Show that Γ(m) * Γ(m + 1/2) = (√π) / 2^(2m-1)) * Γ(2m). [5 marks]
  12. Q2(c). Evaluate (i) ∫_0^∞ (x^c / c^x) dx, (ii) ∫_0^1 (1/ √(1-x^4))dx. [5 marks]
  13. Q3. Attempt any two parts of the following.
  14. Q3(a). Evaluate ∫∫_R (x - y)^4 * e^(x+y) dx dy, where R is the square with vertices at (1,0), (2,1), (1,2) and (0,1). [5 marks]
  15. Q3(b). Change the order of integration in ∫_0^1 ∫_(x^2)^(2-x) xy dy dx, and hence evaluate it. [5 marks]
  16. Q3(c). (i)Using double integration, find the area enclosed by curves y = 2x^2 and y^2 = 4x. (ii) Find the volume of the solid region contained in the first octant of the ellipsoid x^2 /a^2+ y^2/b^2 + z^2/c^2 = 1. [5 marks]
  17. Q4. Attempt any two parts of the following.
  18. Q4(a). Evaluate ∫∫_S F · n dS, where F = 4y*i + 18z*j - x*k and S is the surface of the plane 3x + 2y + 6z = 6, contained in the first octant. [5 marks]
  19. Q4(b). State Stoke's theorem. Verify Stoke's theorem for the vector field F = (2x - y)*i - y*z^2*j - y^2*z*k, over the upper half surface of the sphere x^2 + y^2 + z^2 = 1, bounded by its projection on the xy-plane. [5 marks]
  20. Q4(c). Verify Green's theorem in a plane for the integral ∫_C (x - 2y)dx + xdy taken around the circle x^2 + y^2 = 4. [5 marks]
  21. Q5. Attempt any two parts of the following.
  22. Q5(a). (i) Show that r^n * r is solenoidal for n = -3 and irrotational for all values of n. (ii) Find the directional derivative of xyz^2 + xz at (1,1,1) in the direction of the normal to the surface 3xy^2 + y + z at (0,1,1). [5 marks]
  23. Q5(b). Verify Gauss divergence theorem for F = (x^2 - yz)*i + (y^2 - zx)*j + (z^2 - xy)*k taken over the rectangular parallelepiped bounded by 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c. [5 marks]
  24. Q5(c). (i) Show that div{grad(x/r^3)} = 0, where r is the magnitude of position vector r = x*i + y*j + z*k. (ii) Find the work done in moving a particle by the force field F = 3x^2*i + (2xz - y)*j - z*k, from t = 0 to t = 1 along the curve x = 2t^2, y = t, z = 4t^3. [5 marks]