BTech 1 Sem Engineering Mathematics 1 Exam 2024 PDF

# 1E3101 B.Tech. I sem(Main/Back) Exam 2024 ## 1FY2-01 / Engineering Mathematics-I ### Total No. of Questions : 22 ### Total No. of Pages : 04 ### Time: 3 Hours ### Maximum Marks : 70 ## Instructions to Candidates - Attempt all ten questions from Part-A, five questions out of seven questions from...

# 1E3101 B.Tech. I sem(Main/Back) Exam 2024 ## 1FY2-01 / Engineering Mathematics-I ### Total No. of Questions : 22 ### Total No. of Pages : 04 ### Time: 3 Hours ### Maximum Marks : 70 ## Instructions to Candidates - Attempt all ten questions from Part-A, five questions out of seven questions from Part-B and three questions out of five questions from Part-C. - Schematic diagrams must be shown wherever necessary. - Any data you feel missing suitably be assumed and stated clearly. - Units of quantities used / calculated must be stated clearly. - Use of following supporting material is permitted during examination. (Mentioned in Form No. 205) ## PART-A (Answer should be given up to 25 words only) All questions are compulsory 1. What is the value of integral ∫ e⁻ˣ² dx? 2. Write the formula of surface area of solid of revolution when the revolution is about x-axis. 3. What do you mean by convergence of a sequence? 4. Find whether the following series is convergent or not? 1/2.3 + 1/3.4 + 1/4.5 + ... 5. State Parseval's theorem. 6. Find the value of a₀ for the function f(x) = |x| in the interval (-π,π). 7. State the necessary and sufficient conditions for the minimum of a function f(x,y). 8. Find the gradient of f(x,y,z)=x²y +xy² -z² at (3, 1, 1). 9. Evaluate ∫∫ xydxdy. 10. State the Gauss Divergence theorem. ## PART-B (Analytical/Problem solving questions) Attempt any five questions 1. Use beta and gamma functions, to evaluate : ∫ ˣ²(1+x) / (x+x⁶)¹⁰ dx 2. Expand sin x in the powers of (x-π/2) using Taylor's series. 3. Find Fourier series of x² in (-π,π), and use Parseval's identity to prove : π²/90 = 1 + 1/2⁴ + 1/3⁴ + ... 4. If u = eˣʸᶻ, then show that: ∂³u / ∂x∂y∂z = (1+3xyz + x²yz²) eˣʸᶻ 5. Whether the fluid motion given by V=(y+z)i+(z+x)j+(x+y)k is incompressible or not? 6. Change the order of integration and hence evaluate : ∫∫ 1 / logy dxdy 7. Evaluate ∫∫∫ (xyz)dxdydz ## PART-C (Descriptive/Analytical/Problem Solving/Design question) Attempt any three questions 1. Use beta and gamma functions, to evaluate: (a) ∫ x / (1+x⁶) dx (b) ∫ 1 / (x√(1-x)) dx 2. Find the Fourier series expansion of the following periodic function with period 2π. f(x) = -1, -π<x<0 0, x=0 1, 0<x<π Hence, show that 1 - 1/3 + 1/5 - 1/7 + 1/9 - ... = π/4 3. Use Lagrange's method to find the maximum and minimum distance of the point (3, 4, 12) from the sphere x² + y² + z² = 1. 4. If u=f(r), where r² = x² + y², then prove that : ∂²u/∂x² + ∂²u/∂y² = f''(r) + 1/r f'(r) 5. Verify Green's theorem for ∫ [(xy+y²)dx+x²dy] , where C is the closed curve of the region bounded by y = x and y = x².