Calculus Final Exam 2022-23 (Re-exam 2021-22) PDF

# SVKM'S NMIMS ## School of Technology Management & Engineering / School of Technology Management & Engineering ### Academic Year: 2022-23 | Programme: | B.Tech (All Streams) / MBA Tech (All Streams) | |---|---| | Year: | I | | Semester: | I | | Subject: | Calculus/ | | Date: | 03 January 2023 | |...

# SVKM'S NMIMS ## School of Technology Management & Engineering / School of Technology Management & Engineering ### Academic Year: 2022-23 | Programme: | B.Tech (All Streams) / MBA Tech (All Streams) | |---|---| | Year: | I | | Semester: | I | | Subject: | Calculus/ | | Date: | 03 January 2023 | | Marks: | 100 | | Time: | 11.00 am - 2.00 pm | | Durations: | 3 (Hrs) | | No. of Pages: | 3 | ## Final Examination (2022-23)/Re-Examination (2021-22) **Instructions:** Candidates should read carefully the instructions printed on the question paper and on the cover of the Answer Book, which is provided for their use. 1) Question No. 1 is compulsory. 2) Out of remaining questions, attempt any 4 questions. 3) In all 5 questions to be attempted. 4) All questions carry equal marks. 5) Answer to each new question to be started on a fresh page. 6) Figures in brackets on the right hand side indicate full marks. 7) Assume Suitable data if necessary. ## Q1 **Answer briefly:** | (a) | Test the convergence of ∑ _n=1_^∞ 2ⁿ/nⁿ using ratio test. | [4] | |---|---|---| | (b) | Evaluate lim_(x,y)→(0,0) x - y + 2√x - 2√y | [4] | | (c) | Evaluate ∫_0^∞ e^x dx | [4] | | (d) | Evaluate ∫_0^a ∫_0^(√(a²-x²)) dy dx / √(x² + y² + a²) | [4] | | (e) | If v = 3x²y + 6xy - y³, show that v is harmonic function. | [4] | ## Q2 | (a) | Verify Rolle's mean value theorem for f(x) = (x² - 4x)/(x + 2) in [0, 4]. Also find 'c'. | [6] | |---|---|---| | (b) | If u = f(x - y / xy, z - x / xz) then prove that x (∂u / ∂x) + y (∂u / ∂y) + z (∂u / ∂z) = 0. | [6] | | (c) | Change the order of integration and evaluate ∫_0^1 ∫_0^(√(1-x²)) (1 + y²) / √(1 - x² -y²) dydx. | [8] | ## Q3 | (a) | Expand x³ - 3x² + 4x + 3 in powers of (x - 2). | [6] | |---|---|---| | (b) | Find the maximum and minimum values of the function x³ + 3xy² - 15x² - 15y² + 72x. | [6] | | (c) | Evaluate ∫_0^∞ ∫_0^∞ ∫_0^∞ (log₂(x) / (x + y + z)) dzdydx. | [8] | ## Q4 | (a) | Find a, b, c if lim_(x→0) (aex - be^-cx / x - sin x) = 4. | [6] | |---|---|---| | (b) | Expand e^x cos y near the point (1, π/4) by Taylor's theorem up to second degree terms. | [6] | | (c) | (i) Find Vf and Vf| if f = 2xz - x²y at (2, -2, -1). (ii) Find the rate of change of φ = xyz in the direction normal to the surface x²y + y²x + yz² = 3 at the point (1, 1, 1). | [8] | ## Q5 | (a) | Using Lagrange's mean value theorem prove that ((b - a) / b) < log(b/a) < ((b - a) / a) , 0 < a < b. Hence deduce that 1 / 4 < log(4 / 3) < 1 / 3. | [6] | |---|---|---| | (b) | If u = (a³ / x²) + (b³ / y²) +( c³ / z²) where x + y + z = 1, prove that stationary point of u is given by x = (a³ / (a³ + b³ + c³)), y = (b³ / (a³ + b³ + c³)), z = (c³ / (a³ + b³ + c³)) using Lagrange's Multipliers Method. | [6] | | (c) | Evaluate: (i) ∫_1^∞ (1 / √(1 - x)) dx (ii) ∫_0^1 (x / √(1 - x²)) dx | [8] | ## Q6 | (a) | Evaluate ∫_0^1 (x log x) dx. | [6] | |---|---|---| | (b) | Using double integration, find the area bounded by the lines y = 2 + x, y = 2 - x, x = 5. | [6] | | (c) | Prove that F = (x + 2y + az)i + (bx - 3y - z)j + (4x + cy + 2z) k is solenoidal and determine a, b and c such that F is irrotational. | [8] | ## Q7 | (a) | Find the area of the surface of the solid of revolution generated by revolving the parabola y² = 4ax, 0 ≤ x ≤ 3a about the X-axis. | [6] | |---|---|---| | (b) | Evaluate ∫∫∫ (x + y + z) dx dy dz over the volume of the sphere x² + y² + z² = a². | [6] | | (c) | Using Stoke's theorem, calculate the circulation of the field F = yi + xz j + x²k around the curve C: the boundary of triangle cut from the plane x + y + z = 1 by the first octant counter clockwise when viewed from above. | [8] |