Calculus and Linear Algebra — Major (End Semester) 2020 question paper
MMMUT Electrical Engineering previous year question paper for Calculus and Linear Algebra (BSM-101), semester 1, Major (End Semester) 2020. All 28 questions are listed below, each with a written answer on Nexsus.
Paper details
Subject: Calculus and Linear Algebra (BSM-101)
Branch: Electrical Engineering
Semester: 1
Exam: Major (End Semester) 2020
Questions: 28
Questions asked in Calculus and Linear Algebra Major (End Semester) 2020
Q1(a). If x = r cos θ, y = r sin θ, then show that dr/dx = cos θ/r and (1/r) (∂x/∂θ) = r [2 marks]
Q1(b). If u = sin nx + cos nx, then show that u_r = n^r [1 + (−1)^n sin 2nx]^(1/2), where u_r is the rth differential coefficient of u w.r.t. x. [2 marks]
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]
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]
Q1(e). Evaluate ∫₀^π x sin^4 x cos^4 x dx. [2 marks]
Q1(f). Find the directional derivative of 1/r³ in direction of r^, where r = xi + yj + zk and r = |r| [2 marks]
Q1(g). Show that curl(grad φ) = 0. Here φ is the scalar point function. [2 marks]
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]
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) [5 marks]
Q2(c)(i). If u = log(x³ + y³ + z³ - 3xyz), show that (∂u/∂x + ∂u/∂y + ∂u/∂z)² = 9/(x+y+z)²
Q2(c)(ii). Given z = xⁿ f₁(y/x) + yⁿ f₂(y/x), prove that x² (∂²z/∂x²) + 2xy (∂²z/∂x∂y) + y² (∂²z/∂y²) = n²z.
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]
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]
Q4(a)(i). Using double integration show that β(m, n) = Γ(m)Γ(n)/Γ(m+n)
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.
Q4(b)(a). Evaluate ∫₀^π/2 cos⁴x dx.
Q4(b)(i). Evaluate ∫₀^∞ x e^(-x²/4) dx dy by changing the order of integration. [5 marks]
Q4(c)(i). Find the volume and mass contained in the solid region of the positive octant of the surface (x/a)³ + (y/b)³ + (z/c)³ = 1, where p, q & r > 0, given that density at any point p(x, y, z) = k√xyz. [5 marks]
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. [5 marks]
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 = Vφ. [5 marks]
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]
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]