Pure Mathematics

100 marks · 3 hours · 7 questions

This paper

Pure Mathematics · all years Q. 1 · Let be subgroups of a…Q. 2 · If is a commutative ring…Q. 3 · Let be a finite-dimensional vector…Q. 5 · Evaluate the double integral ∬…Q. 6 · Find an equation of the…Q. 7 · Prove that a necessary and…Q. 8 · Let be analytic inside and…With this paper← 2016 2018 →

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Section A

Q. 1

(a)Let $H, K$ be subgroups of a group $G$ . Prove that H K HK H K is a subgroup of $G$ if and only if $HK=KH$ . [10]
(b)If $N, M$ are normal subgroups of a group $G$ , prove that $NM/M =N/NOM$ . [10]

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Q. 2

(a)If $R$ is a commutative ring with unit element and $M$ is an ideal of $R$ then show that $M$ is a maximal ideal of $R$ if and only if $R/M$ is a field. [10]
(b)If $F$ is a finite field and a ≠ 0 , β ≠ 0 a \neq 0, \beta \neq 0 a  = 0 , β  = 0 are two elements of $F$ then show that we can find elements $a$ and $b$ in $F$ such that 1 + a $1 + a a^2 + b^2 = 0$ . [10]

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Q. 3

(a)Let $V$ be a finite-dimensional vector space over a field $F$ and $W$ be a subspace of $V$ . Then show that $W$ is finite-dimensional, $\text{dim}W \leq \text{dim}V$ and $\text{dim} V/W = \text{dim} V - \text{dim} W$ . [10]
(b)Suppose $V$ is a finite-dimensional vector space over a field $F$ . Prove that a linear transformation T $T \in A(V)$ is invertible if and only if the constant term of the minimal polynomial for $T$ is not 0 0 0 . [10]

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Section B

Q. 5

(a)Evaluate the double integral ∬ R ( 3 x − 2 y ) d x d y \iint_{R} (3x - 2y) dx dy ∬ R ( 3 x − 2 y ) d x d y [10]
(b)Where $R$ is a region enclosed by the circle $x^2 + y^2 = 1$ . Find the area of the region enclosed by the curves y = sin ⁡ x , y = cos ⁡ x , x = 0 , x = 2 π y = \sin x, y = \cos x, x = 0, x = 2\pi y = sin x , y = cos x , x = 0 , x = 2 π . [10]

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Q. 6

(a)Find an equation of the ellipse traced by a point that moves so that the sum of its distance to (4,1) and (4,5) is 12. [10]
(b)Show that if $a, b$ and $c$ are nonzero, then the plane whose intercepts with the coordinate axes are x = a , y = b x = a, y = b x = a , y = b , and z = c z = c z = c is given by the equation $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ . [10]
(c)= D ⟹ C = D c A(0) + B(0) + C

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Section C

Q. 7

(a)Prove that a necessary and sufficient condition that w = f ( z ) = u ( x , y ) + i v ( x , y ) w = f(z) = u(x, y) + iv(x, y) w = f ( z ) = u ( x , y ) + i v ( x , y ) be analytic in a region $R$ is that the Cauchy-Riemann equations $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ are satisfied in $R$ where it is supposed that these partial derivatives are continuous in $R$ . [10]
(b)Show that the function f ( z ) = $ˉ f(z) = \bar{z}$ is not analytic anywhere in the complex plane $Z$ . [10]

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Q. 8

(a)Let $f(z)$ be analytic inside and on the boundary $C$ of a simply-connected region $R$ . Prove that f ′ ( a ) = $f'(a) = \frac{1}{2\pi i} \oint_{C} \frac{f(z)}{(z-a)^2} dz$ . [10]
(b)Show that ∫ 0 2 π d θ ( 5 − 3 sin ⁡ θ ) 2 = 5 π 32 \int_{0}^{2\pi} \frac{d\theta}{(5-3 \sin \theta)^2} = \frac{5\pi}{32} ∫ 0 2 π ( 5 − 3 s i n θ ) 2 d θ = 32 5 π . [10]

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About this paper

The 2017 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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Section A

Section B

Section C

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