Mathematics

Daffodil International University Department of Nutrition and Food Engineering Faculty of Health and Life Sciences B.Sc. in Nutrition and Food Engineering Midterm Examination Fall 2026 Course Code: 0541-111 Course Title: Mathematics Level and Term: L-1, T-2 Section: 262 All Course Teacher Initials: MRA/DMAI/SRM Time: 1 hour 30 minutes Total Marks: 25 "Maintaining the order of the questions is highly encouraged" 1 (a) Define Domain and Range of a function. Find the Domain and Range of the function f(x) = (3x - 7)/(8x + 5) [CLO1,PLO2,C1] 3 (b) Define Homogeneous Function. Determine whether the function f(x, y) = (x^2 y^2 + xy^3 - x^3 y)/(x^2 - 2xy + xy) is a homogeneous or not. [CLO1,PLO2,C1] 2 2 (a) State the conditions for the existence of the limit of a function. If f(x) = 1/(1 + e^(1/x)), does the limit exist at x = 0. [CLO1,PLO2,C2] 3 (b) If f(x) = { 3 - ax^2 when x < 1, x + 1 when x ≥ 1 for what value of a, f(x) is continuous at x = 1. [CLO1,PLO2,C3] 2 3 (a) Sketch the graph of the function y = √(2x - 4) + 3. [CLO1,PLO2,C4] 2 (b) Discuss the differentiability of the function at x = 1, where f(x) = { x + 1 when x ≤ 0, x^2 when 0 < x < 1, 1/x when x ≥ 1 . [CLO1,PLO2,C4] 3 4 (a) Find the differential coefficient dy/dx, d^2y/dx^2 at point x = 2 for the function y = 1 + x^2 + x^3 + x^4 + x^5 + x^6 [CLO2,PLO2,C3] 2 (b) Find the differential coefficient dy/dx for the following function: [CLO2,PLO2,C3] 3 (i) y = e^(sin^-1 x) + tan^-1 x (ii) y = cos x / (1 + sin x) 5 (a) If u = x^2 + y^2 + z^2 then show that x ∂u/∂x + y ∂u/∂y + z ∂u/∂z = 2u. [CLO3,PLO2,C2] 2 (b) State Rolle's theorem. Verify Rolle's theorem for the function f(x) = x^2 - 3x + 2 in the interval (1, 2). [CLO3,PLO2,C3] 3