Mathematics I

Daffodil International University Faculty of Science & Information Technology Department of Computer Science & Engineering Midterm Examination, Fall 2026 Course Code: MAT101, Course Title: Mathematics I Level: L1 Term: T1 Batch: 73 Time: 01:30 Hrs Marks: 25 Answer ALL Questions [The figures in the right margin indicate the full marks and corresponding course outcomes. All portions of each question must be answered sequentially.] 1. a) Demonstrate the Continuity and Differentiability of the following function at the point x = 1 f(x) = { x^2 sin(1/x) ; x < 0 x^3 + x ; 0 ≤ x ≤ 1 3x^2 - 4x + 3 ; x > 1 [Marks: 4, CO1] 1. b) If a + b sqrt(6) = (75 sqrt(3) + 48 sqrt(2)) / (sqrt(432) + sqrt(162)), then demonstrate the value of (a, b). [Marks: 3] 2. Interpret the solution of the inequality (x - 5) / (x^3 + 24x^2 + 144x) ≤ 0 using sign table. [Marks: 3, CO1] 3. To determine the exact time complexity of the algorithm, the engineer derives the following characteristic polynomial equation: 4x^4 - 4x^3 - 61x^2 + 91x + 60 = 0 Where the largest real root x, represents the growth factor of the algorithm's execution time. Apply the Factor Theorem for Solving this equation to find the growth factor. [Marks: 5, CO2] 4. a) Inspect the rate of change of y with respect to x, or dy/dx of the function y = sin 4x + log_a x + sec x + 10^x + sec^-1 x + log x . [Marks: 3] 4. b) A motorcycle moves along a hill road. If the equation of the path is y = tan^-1 ((4 sqrt(x)) / (1 - 4x)) + (x^x)^x then, inspect the slope of the path at x = 0.2 [Marks: 4, CO3] 4. c) If y = ln(2θ + sqrt(θ^2 - 2)) and z = ln(θ e^θ ((1 + θ) / (1 - θ))^2), then, Inspect the rate of change of z with respect to y, or dz/dy . [Marks: 3]