B. Tech Mathematics III Past Paper PDF Dec 2023

## December 2023 ### B. Tech (IT/CE (Hindi Medium)/CE/CSE/CSE (AIML)) – III SEMESTER ### Mathematics III (Calculus and Ordinary Differential Equations) (BSC-301) **Time:** 3 Hours **Max. Marks:** 75 **Instructions:** 1. It is compulsory to answer all the questions (1.5 mark each) of Part -A in shor...

## December 2023 ### B. Tech (IT/CE (Hindi Medium)/CE/CSE/CSE (AIML)) – III SEMESTER ### Mathematics III (Calculus and Ordinary Differential Equations) (BSC-301) **Time:** 3 Hours **Max. Marks:** 75 **Instructions:** 1. It is compulsory to answer all the questions (1.5 mark each) of Part -A in short. 2. Answer any four questions from Part -B in detail. 3. Different sub-parts of a question are to be attempted adjacent to each other. ### **PART-A** **Q1** (a) Write the type of the sequence {-1, 1, -1, 1,...}. Is it convergent? (1.5) (b) What is positive term series? (1.5) (c) Test lim(x,y)→(0,0) (x2+y2)/(2xy) exists or not. (1.5) (d) If u = (x - y)(y - z)(z - x), then find (∂u)/(∂y). (1.5) (e) Evaluate ∫0^1∫x^2 e^y dy dx. (1.5) (f) State Green's theorem. (1.5) (g) Find the integrating factor for the differential equation: dy/dx + 4 sin x y = 2 cos x (1.5) (h) Check if the following differential equation is exact: (y²+2x²y)dx + (2x³-xy)dy = 0 (1.5) (i) What is Clairaut's type equation? Give an example. (1.5) (j) Identify the nature of the singular points of the differential equation: x²(x - 2)y" + (x - 1)y' + 2xy = 0 (1.5) ### **PART-B** **Q2** (a) Test the convergence of Σ(n=1 to ∞) (√(n+1) - √(n-1)). (8) (b) Using Taylor's series expansion, prove that loge(1 + e^x) = loge(2) + (x/2) + (x^2/8) + (x^4/192) + ... (8) **Q3** (a) If z = f(x, y) where x = u² - v², y = 2uv, prove that (∂²z/∂u²) + (∂²z/∂v²) = 4(u² + v²)[(∂²z/∂x²) + (∂²z/∂y²)] (8) (b) Find the minimum value of the function x² + y² + z² subject to the condition xy + yz + zx = 3a². (7) **Q4** (a) Using Gauss divergence theorem, evaluate ∫∫S F. ds where F = 4xzī- y²j + yzk and S is the surface of the cube bounded by the planes x = 0, x = 2, y = 0, y = 2, z = 0, z = 2. (8) (b) Change the order of integration ∫0^2∫x^2 xy dy dx and hence evaluate. (7) **Q5** (a) Solve the differential equation p² - p(e^x + e^-x) + 1 = 0 where p has usual meaning. (8) (b) Solve (2x + y + 1)dy = (x + y + 1)dx. (7) **Q6** (a) Solve the following differential equation by using variation of parameter d²y/dx² + 2 dy/dx + y = e^-x/x² (8) (b) Find the power series solution of (1-x²)(d²y/dx²) - 2x dy/dx + 2y = 0 in powers of x. (7) **Q7** (a) Solve (D⁴ - 1)y = e^x cos x. (8) (b) Find the directional derivative of 2yz + z²in the direction of the vector ī + 2j + 2k at the point (1, -1, 3). (7)