CSS 2021 Pure Mathematics Past Paper

Paper details

Examination

CSS

Year

2021

Subject

Pure Mathematics

Paper

Single Paper

Subject type

Optional

Mode

Subjective

Questions recorded from this paper

These 8 question wordings are transcribed from CSS Vista’s structured Pure Mathematics past-paper record for CSS 2021.

1. Q2 · Group(a) Find elements of the cyclic group generated by the permutation. (10) 1 2 3 4 5 6    3 4 5 2 6 1 (b) (10) (20) Verify that the polynomials 2-x , x -x, 2-3x and 3-x form a basis for the set P3 (x); the set of all polynomials of degree three. Also express the vectors 1+x2 and x+x3 as a linear combination of these basis vectors. 
2. Q1 · subgroups(a) Let Ψ be a homomorphism of group G into group G with kernel K, prove that K is (10) a normal subgroup of G. (b) Prove that if H and K are two subgroups of a group G, then HK is a subgroup of G (10) (20) if and only if HK=KH.
3. Q3 · Linear independence(a) Let V be the real vector space of all function from R to R. Show that {cos 2 x, sin2 (10) x, cos 2x}is linearly dependent while { cosx, sinx, coshx, sinhx} are linearly independent. (b) Solve the system of linear equations: (10) (20) x1 – 2x2 – 7x3 + 7x4 = 5 – x1 + 2x2 + 8x3 – 5x4 = – 7 3x1 – 4x2 – 17x3 + 13x4 = 14 2x1 – 2x2 + 11x3 + 8x4 = 7  y 1  x  If f  x, y   x tan  1
4. Q7 · Calculus & Analytic Geometry(a) Sum the series1  cos  cos 2  cos3  ... (10) 2 2.4 2.4.6  2 3 1 (b) Prove that co s  co s  co s  (10) (20) Construct the analytic function f whose real part is U  x  3xy  3x  1
5. Q6 · Curve tracing(a) Evaluate  dydx and  dxdy where D is the region bounded by the y-axis, the x lines x=2 and the curve e . x3  x (b) Investigate the curve y  2 for points of inflexion. (10) (20) 3x  1 \ 1 1.3 1.3.5
6. Q5 · Cartesian and spherical polar coordinates in three dimensions(a)  Let I n  x n e  x dx where n is an integer. Prove that In = n In−1 Hence show that In = n! (b) i. Write r  in rectangular coordinates. (10) (20) 2  cos Write x  2 x y  y  6 x y  2 y  0 ii. in polar coordinates.
7. Q4 · Integrated / Cross-Syllabus Questions(a)   y 2 tan  . (10) x  y 2 f  x 2  y2  Show that    2 2 x, y  yx  x +y   x 2 whenx  2 (b) Evaluate  f  x dx where f ( x)   3x  2whenx  2 
8. Q8 · Integrated / Cross-Syllabus Questions(a) (10) dz (b) Evaluate  z  2 z  2 Where C is a square with corners (0,0),(-2,0),(-2,-2) and (0,-2).

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