Pure Mathematics

100 marks · 3 hours · 8 questions

This paper

Pure Mathematics · all years Q. 1 · Show that the order and…Q. 2 · Show that the characteristic of…Q. 3 · Show that a one-to-one linear…Q. 4 · Solve ∫ 0 π /…Q. 5 · Show that in any conic…Q. 6 · Define Supremum and Infimum of…Q. 7 · Show that log ⁡ (…Q. 8 · Prove that the series z…With this paper← 2018 2020 →

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Not yet checked 6 of 8 questions have not yet been compared with the official paper.

Section A

Q. 1

(a)Show that the order and the index of a subgroup divides the order of a finite group. [10]
(b)Show that every finite integral domain is a field. [10]

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Q. 2

(a)Show that the characteristic of an integral domain is R is either zero or a prime. [10]
(b)Determine whether or not the set {(1, 2, −1), (0, 3, 1), (1, −5, 3)} of vectors is a basis for R³. [10]

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Q. 3

(a)Show that a one-to-one linear transformation preserves basis and dimension. [10]
(b)Solve the system of linear equations: 2x₁ + x₂ + 5x₃ = 4 3x₁-2x₂ + 2x₃ = 2 5x₁-8x₂ + 2x₃ = 1. [10]

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Section B

Q. 4

(a)Solve ∫ 0 π / 2 sin ⁡ 2 6 x cos ⁡ 3 x d x \int_{0}^{\pi/2} \sin^2 6x \cos 3x dx ∫ 0 π /2 sin 2 6 x cos 3 x d x . [10]
(b)Find the area enclosed by y = 6 2 − cos ⁡ θ y = \frac{6}{2-\cos \theta} y = 2 − c o s θ 6 [10]

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Q. 5

(a)Show that in any conic semi-latusrectum is the harmonic mean between the segments of focal chord. [10]
(b)Prove that the evolute of hyperbola 2 x y = $2xy = a^2$ is $(x+y)^2 - (x-y)^2 = 2a^2$ . [10]

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Q. 6

(a)Define Supremum and Infimum of a sequence. Find the supremum and infimum of the set{(-1)ⁿ(1-1/n), n = 1,2,3 ...}. [10]
(b)Evaluate lim ⁡ x → 0 ( 1 + x ) 1 / x − e x \lim_{x \to 0} \frac{(1+x)^{1/x}-e}{x} lim x → 0 x ( 1 + x ) 1/ x − e [10]

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Section C

Q. 7

(a)Show that log ⁡ ( 1 + cos ⁡ θ + i sin ⁡ θ ) = ln ⁡ ( 2 cos ⁡ θ ) + i θ 2 \log(1 + \cos \theta + i \sin \theta) = \ln(2 \cos \theta) + i \frac{\theta}{2} lo g ( 1 + cos θ + i sin θ ) = ln ( 2 cos θ ) + i 2 θ [10]
(b)Find v such that f(z) = u + iv is analytic. [10]

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Q. 8

(a)Prove that the series z ( 1 − z ) + z 2 ( 1 − z ) + z 3 ( 1 − z ) + … z(1 − z) + z²(1 − z) + z³(1 − z) + … z ( 1 − z ) + z 2 ( 1 − z ) + z 3 ( 1 − z ) + … converges for |z| < 1, and find its sum. [10]
(b)Find the residues of f ( z ) = $f(z) = \frac{z^2-2z}{(z+1)^2(z^2+4)}$ at all its poles in the finite plane. [10]

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About this paper

The 2019 CSS Pure Mathematics paper set by the FPSC. Question wording only; questions marked “Not yet checked” have not been compared with the official paper yet.

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