CSS past papers · 2024
CSS 2024 Pure Mathematics Past Paper
Download and review the CSS 2024 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
- 2024
- 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 2024.
- Q1 · Group(a) Let N be a normal subgroup of a group G. If H is a subgroup of G, then prove (10) that ɐa = {ɑh : ɑ∈ɐɐɑɑh∈a} is a subgroup of G. (b) If ᵲ is an epimorphism from a group a onto a group a then prove that a/a is (10) (20) isomorphic to a, where a=aɑɑᵲ.
- Q2 · Ring(a) Let ɐ be a ring. If every ᴂ∈ɐ satisfies ᴂ2 =ᴂ then prove that ɐ is a (10) commutative. (b) For which value(s) of ɐ will the following system have no solution? Exactly (10) (20) one solution? Infinitely many solutions? ᴂ + 2ᴂ- 3ᴂ=4 3ᴂ -ᴂ + 5ᴂ=2 4ᴂ+ᴂ+ (ɐ2 -14) z=ɐ+2.
- Q3 · Rank and nullity of a linear transformation(a) Determine a basis for and the dimension of the solution space of the system (10) ᴂ-2ᴂ+ᴂ=0 3ᴂ+ᴂ-5ᴂ=0 ᴂ+2ᴂ-5ᴂ=0. (b) Let ᴂ1 = (1, 1, 1) , ᴂ2 = (1, 1, 0) and ᴂ3 = (1, 0, 0) be a basis forR3. Find (10) (20) a linear transformation ɐ: R3 → R2 such that ɐ (ᴂ1) = (1, 0) , ɐ (ᴂ2) = (2,-1) and ɐ (ᴂ3) = (4, 3) .
- Q6 · Calculus & Analytic Geometry(a) Evaluate the integral ∞ and show that the result is 9ᵰ , (10) ∫ᴂ ((1+2ᴂ)) ɑᴂ using Beta function. (b) Find the vertices and foci of the hyperbola (10) (20) 25 ᴂ2 -16 ᴂ2 +250ᴂ+32ᴂ+109=0.
- Q4 · Mean value theorems(a) Evaluate the limit: (10) (i) lim (1+ cos ᴂ )tan ᴂ ᵰ ᴂ→ (ii) 1 1 lim ( - ) ᴂ→0 sin ᴂ ᴂ (b) State and prove the Mean Value Theorem. (10) (20)
- Q5 · Maxima and minima(a) If ᴂ=ɑ( ᴂ2 + ᴂ2 ) then show that ᴂ ∂ ᴂ -ᴂ ∂ ᴂ =0. ∂ᴂ ∂ᴂ (b) Find all the local maxima, local minima and saddle points of the given (10) (20) function 2 ᴂ3 + ᴂ2 -9 ᴂ2 -4ᴂ+12ᴂ-2.
- Q7 · Implicit function theorem(a) Verify that ᴂ (ᴂ,ᴂ) =ɑɑᴂᴂɑɑᴂhᴂ is harmonic function and find a (10) corresponding analytic function ɑ (ᴂ) =ᴂ (ᴂ,ᴂ) +ɑᴂ (ᴂ,ᴂ). is (b) Use Residue theorem to evaluate the integral ∫ a ( 3 ᴂ2 +ᴂ-1 ᴂ 2 -1)(ᴂ-3) ɑᴂ , where a the circle |ᴂ| =4.
- Q8 · Cauchy’s integral formula, Taylor’s and Laurent’s series(a) Use the Cauchy’s integral formula to evaluate the integral (10) (10) ᴂ+4 ∫ 2 aᴂ +2ᴂ+5 ɑᴂ , where a is the circle |ᴂ+1-ɑ| =2. (b) Find the three cube roots of 3 +ɑ. (10) (20)