CSS past papers · 2025
CSS 2025 Pure Mathematics Past Paper
Download and review the CSS 2025 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
- 2025
- 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 2025.
- Q1 · subgroups(a) Let Q + be the set of positive rational numbers and define ∗ by a ∗ b = then (10) prove that (Q + , ∗) is a group. (b) Find all the cyclic subgroups of Z18 . (10)
- Q2 · Linear independence(a) A homomorphism φ: Z6 → Z6 is a one to one mapping then calculate Ker(φ). (10) (b) Check whether the vectors a = (1, 2, 3), b = (2, 5, 7)and c = (1, 3, 5) are linearly (10) dependent or independent.
- Q3 · Linear transformations, Matrices and their algebra(a) Let V be a vector space of all 2× 2 matrices. W be a subspace of V which consists (10) of all symmetric matrices then find two different basis of W. (b) Consider the transformation T: R 3 → R 2 given by T(x, y, z) = (|x|, y+z). (10) Check whether T is linear or not.
- Q4 · Real Numbers(a) Find all the real numbers x ∈ R such that x 2 + x > 2. (10) (b) x 3 −4 4 (10) Use the definition of limit to establish that lim x 2 +1 = 5 . x→2
- Q6 · Indefinite integration(a) 2 √x (10) Change the order of integration in double integral ∫0 ∫0 f (x, y)dydx. (b) Draw the graph of the conic r = 2Cosθ. (10)
- Q5 · Implicit function theorem(a) Use Mean Value Theorem to show that ex ≥ 1 + x ∀ xεR . (10) (b) Find the absolute extrema of the function f(x, y) = xy − 2x on the region R given (10) by vertices (0, 4), (4, 0) and (0, 0).
- Q8 · Implicit function theorem(a) 5Z−2 Use Cauchy Residue Theorem to evaluate the integral ∫C Z(Z−1) dZ (10) where C is the circle |Z| = 2. (b) Find the Maclaurin series for the function f(Z) = Z 2 e3Z . (10)
- Q7 · Cauchy residue theorem and contour integration(a) Check that the Cauchy-Riemann Equations are satisfied in polar coordinates for (10) f(Z) = Z. (b) Evaluate the contour integral ∮C f(Z)dZ where f(Z) = y − x − 3ix 2 and C is (10) simple closed contour OABO with O = 0+0i, A = 0+i and B = 1+i.