CSS past papers · 2019
CSS 2019 Pure Mathematics Past Paper
Download and review the CSS 2019 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
- 2019
- 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 2019.
- Q1 · Group(a) Show that the order and the index of a subgroup divides the order of a finite (10) group. (b) Show that every finite integral domain is a field. (10) (20)
- Q2 · Modern Algebra(a) Show that the characteristic of an integral domain is R is either zero or a (10) prime. (b) Determine whether or not the set {(1, 2, −1), (0, 3, 1), (1, −5, 3)} of vectors (10) (20) is a basis for R3 .
- Q3 · Rank and nullity of a linear transformation(a) Show that a one-to-one linear transformation preserves basis and dimension. (10) (b) Solve the system of linear equations: (10) (20) 2 x1 + x2 + 5 x3 = 3 x1 − 2 x2 + 2 x3 = 5 x1 − 8 x2 + 2 x3 = 1. π
- Q5 · Calculus & Analytic Geometry(a) Show that in any conic semi-latusrectum is the harmonic mean between the (10) segments of focal chord. (b) Prove that the evolute of hyperbola (10) (20) 2xy = a is (x + y) − (x − y) = 2a .
- Q7 · Calculus & Analytic Geometry(a) Show that Log(1 + cos θ + i sin θ) = ln(2 cos ) + i . (b) Find v such that f (z) = u + iv is analytic. (10) (20)
- Q8 · The plane(a) Prove that the series z(1 − z) + z 2 (1 − z) + z 3 (1 − z) + ⋯ converges (10) for |z| < 1, and find its sum. z 2 −2z (b) Find the residues of f(z) = (z+1)2 at all its poles in the finite plane. (z 2 +4) (10) (20)
- Q4 · Integrated / Cross-Syllabus Questions(a) Solve ∫0 sin 6x cos4 3x dx. (b) Find the area enclosed by y = . (10) (20) 2−cos θ
- Q6 · Integrated / Cross-Syllabus Questions(a) Define Supremum and Infimum of a sequence. Find the supremum and infimum (10) of the set �(−1)n �1 − n� , n = 1,2,3 ... �. (b) Evaluate (10) (20) (1+x)x −e limx→0 . x θ θ (10)