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## DOON UNIVERSITY, DEHRADUN Mid Semester Examination,(1) Semester, 2023 School of Physical Sciences B.Sc. Hons. (Mathematics) Course: MAC-101: Calculus ### **Time Allowed:** 1.5 Hours ### **Maximum Marks:** 30 ### **Note:** Attempt any four questions from section A, B and any two questions Sectio...

## DOON UNIVERSITY, DEHRADUN Mid Semester Examination,(1) Semester, 2023 School of Physical Sciences B.Sc. Hons. (Mathematics) Course: MAC-101: Calculus ### **Time Allowed:** 1.5 Hours ### **Maximum Marks:** 30 ### **Note:** Attempt any four questions from section A, B and any two questions Section C. ### **Section: A** (1.5 x 4 = 6 Marks) 1. Find the odd and even parts of the function f(x), where f(x) = x³ + 5sin x + 10x². 2. Compute y₇ if y = x² sin 5x + e²ˣ. 3. Find n-th derivative of the function e¹⁰ˣ sin (4x + 5). 4. If Iₙ = Dⁿ (xⁿ log x), then prove that Iₙ = Iₙ₋₁ + (n - 1)! 5. Expand the function x⁴ - 12 in powers of (x - 2). ### **Section: B** (3 x 4 = 12 Marks) 6. If y¹/² + y⁻¹/² = 2x, then prove that (x² - 1) yₙ₊₂ + (2n + 1) x yₙ₊₁ + (n² - 4) yₙ = 0. 7. Without calculating the derivative, find the tangent at (1, 2) to the curve 3x² + 2xy + y² - 4 - y = 0. 8. Find all the asymptotes to the curve (y - 1)(x - 2)² + (y + 3x)(y - 2x) + 2x + 2y = 1. 9. Compute the limit lim_(x → 0) (1/x² - 1/sin² x) 10. Prove that a curve r = f(θ) is symmetric about the line θ = α if f(2α - θ) = f(θ). ### **Section: C** (6 x 2 = 12 Marks) 11. Prove the Leibnitz rule, Dⁿ(uv) = Σ(k=0)ⁿ C(n, k) uₖ vₙ₋ₖ. 12. Prove that a curve y = f(x) is concave or convex at a point x according as f''(x) < 0 or f''(x) > 0 13. Trace the curve y²(9 - x²) - x²(9 + x²) = 4.