CSS past papers · 2022
CSS 2022 Pure Mathematics Past Paper
Download and review the CSS 2022 Pure Mathematics Past Paper on CSS Vista, with archive details, related papers and a direct PDF link for examination preparation.
Paper details
- Examination
- CSS
- Year
- 2022
- Subject
- Pure Mathematics
- Paper
- Single Paper
- Subject type
- Optional
- Mode
- Subjective
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Questions recorded from this paper
These 8 question wordings are transcribed from CSS Vista’s structured Pure Mathematics past-paper record for CSS 2022.
- Q1 · Group(a) Let G be a group and H be a subgroup of index 2 in G. Show that H is normal in (10) G. (b) Let G be any group, g a fixed element in G. Define : → by (10) (20) = , ∀ ∈ . Prove that is an automorphism of G onto G.
- Q4 · Integral domains(a) Find lim 1 + 2 . (10) (→) (b) Evaluate the integral - . ( cos 2 3 . (10) (20)
- Q2 · Field extension and finite fields(a) Prove that a finite integral domain is a field. (10) (b) Let W be the subspace of R spanned by = 1,2, −1,3,4 , = 2,4, −2,6,8 , (10) (20) = 1,3,2,2,6 , = 1,4,5,1,8 , = 2,7,3,3,9 . Find a subset of the vectors that form a basis of W. Also extend the basis of W to a basis of R .
- Q3 · Solution of a system of homogeneous and non-homogeneous linear equations(a) Let : R → R be defined by (10) , , !, " = − + ! + ", 2 − 2 + 3! + 4", 3 − 3 + 4! + 5" Find the rank and nullity of T. (b) Find all possible solutions of the following homogeneous system of equations. (10) (20) + + − =0 +2 − 2 + = 0 2 +4 −3 + =0 4 +7 −4 + =0 ⁄ *+ (
- Q6 · Calculus & Analytic Geometry(a) Find the area of the region bounded above by = + 6, bounded below by (10) = , and bounded on the sides by the lines = 0 and = 2. (b) Find the foci, vertices and center of the ellipse: (10) (20) 9 + 16 − 72 − 96 + 144 = 0 \ \
- Q7 · Implicit function theorem(a) Prove that the function , = . ( sin − cos is harmonic. Also find (10) a function B , such that 4 ! = , + C B , is analytic. (b) Evaluate ∮F !̅ 3! around the circle |!| = 1. (10) (20)
- Q8 · Fourier series and Fourier transforms(a) Use residues to prove that -M ( I J = √ (b) Find the Fourier series of the following function 4 which is assumed to have (10) (20) the period 2N. 4 = | |, −N < < N
- Q5 · Integrated / Cross-Syllabus Questions(a) If =4 , and = 5 cos 6 , = 5 sin 6, then show that (10) 9: 9: 9: 9: 89( ; + 89<; = 8 9= ; + = > 89?; (b) Evaluate ∫∫A 3 3 over the region bounded by = and = . (10) (20)