Pure Mathematics, Paper II

100 marks · 3 hours · 8 questions

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Pure Mathematics · all years Q. 1 · Prove that if is a…Q. 2 · For what value of ,…Q. 3 · Find the area of the…Q. 4 · Find the sum of the…Q. 5 · Let be a non-empty set…Q. 6 · Using De Moivre's Theorem evaluate…Q. 7 · Evaluate , where is the…Q. 8 · Find the Fourier transform of…With this paper← 2014 2015 →

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Section A

Q. 1

(a)Prove that if $n$ is a positive integer which is not a perfect square, then n \sqrt{n} n is an irrational number. [10]
(b)Show that every non-empty set of real numbers which has a lower bound has the infimum. [10]

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Q. 2

(a)For what value of $a$ , $m$ , and $b$ does the function $$ f(x) = \begin{cases} 3 & x=0 \\ x^2+3x+a & 0 [10]
(b)For what value of $a$ is f ( x ) = { x 2 − 1 x < 3 2 a x x ≥ 3 f(x) = \begin{cases} x^2-1 & x < 3 \\ 2ax & x \ge 3 \end{cases} f ( x ) = { x 2 − 1 2 a x x < 3 x ≥ 3 Continuous at every $x$ ? [10]
(c)f

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Q. 3

(a)Find the area of the surface generated by revolving r = 2 a sin ⁡ θ r = 2a\sin\theta r = 2 a sin θ about the polar axis. [6]
(b)Find the area enclosed by the graph of the cardioid r = a ( 1 − sin ⁡ θ ) r = a(1-\sin\theta) r = a ( 1 − sin θ ) . [7]
(c)Evaluate the integral $\int_{2}^{\infty} \frac{1}{(x-2)^{2/3}} dx$ . [7]

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Q. 4

(a)Find the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ . [6]
(b)For what value of $x$ does the series $\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n^2+3}$ converges absolutely, converges conditionally and diverges? [7]

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Q. 5

(a)Let $x$ be a non-empty set and define d ( a , b ) = { 1 if a ≠ b 0 if a = b d(a,b) = \begin{cases} 1 & \text{if } a \neq b \\ 0 & \text{if } a = b \end{cases} d ( a , b ) = { 1 0 if a  = b if a = b Show that $d$ is a metric on $x$ . Also describe open and closed balls in this metric space. [10]
(b)Prove that a function $f$ from a metric space $(x, d)$ into a metric space ( Y , $(Y, d^1)$ is continuous if and only if $f^{-1}(A)$ is a closed subset of $X$ for every closed subject $A$ of $Y$ . [10]

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Section B

Q. 6

(a)Using De Moivre's Theorem evaluate ( 1 + i 3 + i ) 6 \left(\frac{1+i}{\sqrt{3}+i}\right)^6 ( 3 + i 1 + i ) 6 . [10]
(b)Find real constants $a$ , $b$ , $c$ and $d$ so that the given function is analytic f ( z ) = $f(z) = x^2 + axy + by^2 + i (cx^2 + dxy + y^2)$ [10]

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Q. 7

(a)Evaluate $\oint_{c} \frac{dz}{z^2+1}$ , where $c$ is the circle ∣ z ∣ = 4 |z|=4 ∣ z ∣ = 4 . [10]
(b)Expand f ( z ) = $f(z) = \frac{1}{z(z-1)}$ in a Laurent series valid for 1 < ∣ z − 2 ∣ < 2 1 < |z-2| < 2 1 < ∣ z − 2∣ < 2 . [10]

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Q. 8

(a)Find the Fourier transform of f ( z ) = e − ∣ z ∣ f(z) = e^{-|z|} f ( z ) = e − ∣ z ∣ . [10]
(b)Evaluate $\oint_{c} \frac{1}{(z-1)^2(z-3)} dz$ , where the contour $C$ is the rectangle defined by $x=0, x=4, y=-1, y=1$ . [10]

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About this paper

The 2014 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

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