Calculus and Linear Algebra — Major (End Semester) 2021 question paper

Paper details

• Subject: Calculus and Linear Algebra (BSM-101)
• Branch: Computer Science and Engineering
• Semester: 1
• Exam: Major (End Semester) 2021
• Questions: 23

Questions asked in Calculus and Linear Algebra Major (End Semester) 2021

1. Q1(a). If x = r cos θ, y = r sin θ, then show that ∂r/∂x = cos θ / r, ∂r/∂y = sin θ / r, ∂θ/∂x = -sin θ / r, ∂θ/∂y = cos θ / r. [2 marks]
2. Q1(b). 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(c). Find the sum and product of all the Eigen values of the matrix [[-2, 2, -3], [1, 2, -6], [-1, -2, 0]]. [2 marks]
4. Q1(d). Compute the area bounded by the parabola y = x² + 2 and the straight lines x = 0, x = 1, x + y = 0. [2 marks]
5. Q1(e). Evaluate ∫₀^∞ x sin x cos x dx. [2 marks]
6. Q1(f). Find the directional derivative of 1/r³ in direction of r, where r = xî + yĵ + zk and r = |r|. [2 marks]
7. Q1(g). Show that curl(grad φ) = 0. Here φ is the scalar point function. [2 marks]
8. Q2(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). [5 marks]
9. Q2(b). If x²/(a²+u) + y²/(b²+u) + z²/(c²+u) = 1, then prove that (∂u/∂x)² + (∂u/∂y)² + (∂u/∂z)² = 2(x ∂u/∂x + y ∂u/∂y + z ∂u/∂z) / (x+y+z). [5 marks]
10. Q2(c)(i). If u = log(x³ + y³ + z³ - 3xyz), show that (∂u/∂x + ∂u/∂y + ∂u/∂z)² = 9 / (x+y+z)².
11. Q2(c)(ii). Given z = x^n f₁(y/x) + y^n f₂(y/x), prove that x² ∂²z/∂x² + 2xy ∂²z/∂x∂y + y² ∂²z/∂y² = n²z.
12. Q3(a). Find the inverse of following matrix by using elementary operations A = [[0, 2, 1, 3], [1, 1, -1, -2], [1, 2, 0, 1], [1, 1, 2, 6]]. [5 marks]
13. Q3(b). Discuss the consistency of the following system of equations for various values of λ: 2x₁ - 3x₂ + 6x₃ - 5x₄ = 3, x₂ + 4x₃ + x₄ = 1, 4x₁ - 5x₂ + 8x₃ - 9x₄ = λ, and if consistent, solve it. [5 marks]
14. Q3(c). Diagonalize the matrix [[3, 1, -1], [-2, 1, 2], [0, 1, 2]]. [5 marks]
15. Q4(a)(i). Using double integration show that β(m, n) = Γ(m)Γ(n) / Γ(m+n).
16. Q4(a)(ii). Evaluate ∫∫_R (x + y)²dxdy, where R is the parallelogram in the x - y plane with vertices (1,0), (3,1), (2,2), (0,1) using the transformation u = x + y and v = x - 2y.
17. Q4(b)(a). Evaluate ∫₀^π/2 cos⁴x dx.
18. Q4(b)(i). Evaluate ∫₀^∞ x e^(-x²/4) dx dy by changing the order of integration. [5 marks]
19. Q4(c)(i). Find the volume and mass contained in the solid region of the positive octant of the surface (x/a)^p + (y/b)^q + (z/c)^r = 1, where p, q & r > 0, given that density at any point ρ(x, y, z) = k√xyz. [5 marks]
20. Q5(a)(i). Find the work done in moving a particle by force field F = 3xyî – 5zî + 10xk along the curve x = t², y = 2t², z = t³ from t = 0 to t = 2.
21. Q5(a)(iii). Show that the vector field F = (6xy + z³)î – (3x² – z)ĵ + (3xz² - y)k is irrotational. Find the scalar potential φ such that F = ∇φ.
22. Q5(b)(i). Verify Stokes theorem for F = xyî + xy²ĵ taken around a square having vertices (1,1), (−1, −1), (1, −1) and (−1, 1) in x - y plane. [5 marks]
23. Q5(c)(ii). Verify Gauss's divergence theorem for F = (x² – yz)î + (y² – zx)ĵ + (z² – xy)k taken over the rectangular parallelepiped 0 ≤ x ≤ a, 0 ≤ y ≤ b and 0 ≤ z ≤ c. [5 marks]