CSS 2023 Pure Mathematics Past Paper

Paper details

Examination

CSS

Year

2023

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 2023.

1. Q2 · Group(a) For any group G , show that ≅ G and ≅ {e} . (10) {e} G (b) Suppose U and W are distinct four dimensional subspaces of a vector space V of (10) (20) dimension six. Find the possible dimension of U ∩ W .  − α α −1 α +1  
2. Q1 · Modern Algebra(a) Find centre of S3 . (10)  1 2 − 3   (b) Using the row operations, show that the matrix  1 − 2 1  has no inverse. (10) (20)  5 − 2 − 3  
3. Q6 · Modern Algebra(a) Evaluate ∫∫ ( x 2 + y 2 )dxdy , over the region bounded by xy=1, y=0, y=x and x=2. (10) x2 y2 (b) Find an equation of a normal to the hyperbola − = 1 in the form (10) (20) a 2 b2 ax cosθ + by cot θ = a 2 + b 2 . Prove that the normal is external bisector of the angle between the focal distances of its foot. \ \
4. Q7 · Conjugate elements, conjugate subgroups(a) Determine k such that U = e 2 x cos ky is harmonic and find a conjugate harmonic. (10) (b) Evaluate ∫ ( 5 + z 3 )dz from 1 to -1 along the upper arc of the unit circle. (10) (20) z
5. Q4 · Calculus & Analytic Geometry(a) Find the value of θ and the limit in order that lim be finite. (10) x →0 x3 x (b) Show that x < sin −1 x < , 0<x<1. (10) (20) 1 − x2
6. Q3 · Complex Variables(a) For what value of α is the matrix  1 2 3  is singular? (10)   2 − α α + 3 α + 7 (b) Define T : R3 → R3 by T ( x1 , x2 , x3 ) = (− x3 , x1 , x1 + x3 ) . Find N (T ) . Is T one-to- (10) (20) one? sin 2 x + θ sin x
7. Q8 · Cauchy’s integral formula, Taylor’s and Laurent’s series(a) Find the Laurent Series of in the region 0< z − 1 <2. (10) 1− z2 − Z 2 − 22 z + 8 (b) Find the residues at the singular points of which lie inside the (10) (20) Z 3 − 5z 2 + 4 z circle z =2.
8. Q5 · Integrated / Cross-Syllabus Questions(a) Given that U = . Verify that U xx + U yy + U zz = 0. x2 + y2 + z 2 (b) Evaluate ∫∫ ( x 2 + y 2 )dxdy , over the domain bounded by y = x 2 and x = y 2 . (10) (20)

Topic-wise Pure Mathematics past-paper analysis.