CSS past papers · 2016
CSS 2016 Pure Mathematics Past Paper
Download and review the CSS 2016 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
- 2016
- 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 2016.
- Q1 · GroupProve that the normaliser of a subset of a group G is a Subgroup of G. (b) Let A be a normal subgroup and B a subgroup of a group G. Then prove that (10) (20) < A,B > = AB
- Q2 · Group(a) Let a be a fixed point of a group G and consider the mapping Ia : G→G defined (10) by Ia(g)= aga where g∈G. Show that Ia is an automorphism of G. Also show that for a, b ∈ G, Ia .Ib=Iab (10) (20) a b (b) Let M2 (R) = c : a , b , c , d R be the set of all 2×2 matrices with d real entries. Show that ( M2(R), +, ∙ ) forms a ring with identity. Is ( M2(R), +, ∙ ) a field?
- Q3 · Solution of a system of homogeneous and non-homogeneous linear equations(a) Let T: X→Y be a linear transformation from a vector space X into a Vector (10) space Y. Prove that Kernal of T is a subspace. (b) Find the value of λ such that the system of equations (10) (20) x + λy + 3z = 0 4x + 3y + λz = 0 2x + y + 2 z = 0 has non-trivial solution.
- Q5 · Calculus & Analytic Geometry( 1x ) x (10) (a) Prove that the maximum value of is e1/e .
- Q4 · Asymptotes(a) Using - ∈ definition of continuity, prove that the function Sin x is continuous (10) for all x ∈ R. (b) Find the asymptotes of the curve (x -y )(x+2y) + 5 (x +y ) + x+y =0
- Q6 · the sphere(a) Find the area enclosed between the curves y=x and y=x. (10) (b) A plane passes through a fixed point (a, b, c) and cuts the coordinate axes in (10) (20) A,B,C. Find the locus of the centre of the sphere OABC for different positions of the plane, O is the origin.
- Q8 · Cauchy residue theorem and contour integration(a) Use Cauchy Integral Formula to evaluate ∫ dz along the simple z closed counter C: | z |=3 described in the positive direction. (b) State and prove Cauchy Residue Theorem. (10) (20)
- Q7 · Integrated / Cross-Syllabus Questions(a) P(z) (10) Determine where i P(z) = (z z1 )(z z2 )(z z3 )(z z4 ) with z1 e 4 , z2 z1 , z3 z1 and z4 z1 . z z d z, (n any integer) along the circle C (b) n Find value of the integral 0 (10) (20) c ..........with centre and z 0 radius r, described in the counter clock wise direction. z z (10)