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

Paper details

• Subject: Engineering Mathematics-I (BSM-110)
• Branch: Computer Science and Engineering
• Semester: 1
• Exam: Major (End Semester) 2024
• Questions: 23

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

1. 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]
2. Q1(b). Find the nth derivative of e^ax sin(bx + c). [2 marks]
3. Q1(c). Find the extreme values of the function f(x, y) = x^3 + y^3 - 3x - 12y + 20. [2 marks]
4. 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]
5. 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]
6. 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]
7. Q1(g). Find inverse of the following matrix using Gauss Jorden Method. A = [[2, 1, 1], [3, 2, 3], [1, 4, 9]] [2 marks]
8. 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]
9. Q2(b). Show that Γ(m) Γ(m + 1/2) = (√π / 2^(2m-1)) Γ(2m). [5 marks]
10. Q2(c)(i). Evaluate ∫₀^∞ (x^c / c^x) dx
11. Q2(c)(ii). Evaluate ∫₀¹ (dx / √(1 - x^4)) [5 marks]
12. Q3(a). Evaluate ∫∫_R (x - y)^4 e^(x+y) dxdy, where R is the square with vertices at (1,0), (2,1), (1,2) and (0,1). [5 marks]
13. Q3(b). Change the order of integration in ∫₀¹ ∫_{x^2}^{2-x} xy dy dx and hence evaluate it. [5 marks]
14. Q3(c)(i). Using double integration, find the area enclosed by curves y = 2x^2 and y^2 = 4x.
15. Q3(c)(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]
16. Q4(a). Evaluate ∫∫_S F . n dS. Here F = 4yi + 18zj - xk and S is the surface of the plane 3x + 2y + 6z = 6, contained in the first octant. [5 marks]
17. Q4(b). State Stoke's theorem. Verify Stoke's theorem for the vector field F = (2x - y)i - yz^2j - y^2zk, 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]
18. 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]
19. Q5(a)(i). Show that r^n r_vector is solenoidal for n = -3 and irrotational for all values of n.
20. Q5(a)(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]
21. 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]
22. Q5(c)(i). Show that div{grad (x/r^3)} = 0, where r is the magnitude of position vector r_vector = xi + yj + zk.
23. Q5(c)(ii). Find the work done in moving a particle by the force field F = 3x^2i + (2xz - y)j - zk, from t = 0 to t = 1 along the curve x = 2t^2, y = t, z = 4t^3. [5 marks]