Mathematics II

Daffodil International University Faculty of Science & Information Technology Department of Computer Science and Engineering Midterm Examination, Fall-2026 Course Code: MAT 102, Course Title: Mathematics II Level: 01 Term: 02 Batch: 72 Time: 1.5 Hours 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 value of ∫ (0 to π/2) 7√(64 sin^12 θ cos^2 θ dθ) using β-Γ function. [3] b) Illustrate β-Γ function to find the value of ∫_0^1 x^(3/2) (1 - √x)^5 dx [2] CO1 c) Demonstrate the value of ∫_0^∞ 5^10 e^(-5x^9) dx using Γ function. [2] 2. a) Identify the rate of volume changing of the given box [4] [Figure: Rectangular box with Height = 5 ft (increasing at 1 ft/sec.), Width = 8 ft (increasing at 3 ft/sec.), and Length = 10 ft (increasing at 4 ft/sec.)] b) Apply Euler's theorem to show that x (∂r/∂x) + y (∂r/∂y) + z (∂r/∂z) = (1/2) sin 2r, where r = tan^-1 ((x^2 + y^2 + z^2)/(x + y + z)) [4] CO2 c) Construct the Matrix form of, Z = (Z_1)^5 - 7Z_2 + 6Z_3 by following Z_1 = -8 + 3i, Z_2 = e^(2.5i), and Z_3 = (4.9, 231°) [5] 3. Solve the following triple integration of V = ∫_0^(2π) ∫_0^(π/4) ∫_0^4 (ρ cos φ) ρ^2 sin φ dρ dφ dθ [5] CO4