Pure Mathematics, Paper II

100 marks · 3 hours · 8 questions

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Pure Mathematics · all years Q. 1 · Prove that every non-empty set…Q. 2 · Define continuity of a function…Q. 3 · Evaluate ∫ d x x…Q. 4 · Let be a metric space…Q. 5 · If converges absolutely then converges.Q. 6 · Let Z = ( cos…Q. 7 · Expand f ( x )…Q. 8 · Evaluate the integral by using…With this paper← 2009 2012 →

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Section A

Q. 1

(a)Prove that every non-empty set of real numbers that has an upper bound also has an supremum in R. [10]
(b)If x $x \in \mathbb{R}$ , set of real numbers, then there exists n $n \in \mathbb{N}$ such that x < n x < n x < n . [10]

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Q. 2

(a)Define continuity of a function at a point and also prove that if f and g be functions on A $A \subseteq \mathbb{R}$ then f + g f + g f + g and f $f \cdot g$ are continuous at c $c \in A$ . [10]
(b)If f : I $f: I \rightarrow \mathbb{R}$ is differentiable at c $c \in I$ , then $f$ is continuous at $c$ . [10]
(c)f'

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Q. 3

(a)Evaluate ∫ d x x − 2 \int \frac{dx}{\sqrt{x}-2} ∫ x − 2 d x [8]
(b)i) Define Complete metric space. ii) Prove that a sequence of real numbers is convergent iff it is a Cauchy sequence. This theorem is not in metric space, for justification give one example. [12]

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Q. 4

(a)Let $(X, d)$ be a metric space and $A$ a subset of $X$ . Then prove that: i) Interior $A$ of $A$ is an open subset of $X$ . ii) A ‾ \overline{A} A is the largest subset of $X$ contained in $A$ . [10]
(b)State and prove Mean value theorem. [10]
(c)= f ( b ) − f ( a ) b − a f'

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Q. 5

(a)If $\sum a_n$ converges absolutely then $\sum a_n$ converges. [10]
(b)Find the area enclosed by the parabola $y^2 + 16x - 71 = 0$ and the line $4x + y + 7 = 0$ . [10]

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SECTION–B

Q. 6

(a)Let Z = ( cos ⁡ θ + i sin ⁡ θ ) Z = (\cos\theta + i \sin\theta) Z = ( cos θ + i sin θ ) . Then prove that Z n = cos ⁡ n θ + i sin ⁡ n θ Z^n = \cos n\theta + i \sin n\theta Z n = cos n θ + i sin n θ for all $n$ . [10]
(b)Using De Moivre's Theorem evaluate ( 3 − i 3 + i ) 6 \left(\frac{\sqrt{3}-i}{\sqrt{3}+i}\right)^6 ( 3 + i 3 − i ) 6 . [10]

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Q. 7

(a)Expand f ( x ) = $f(x) = x^2$ , 0 < x < $0 < x < 2\pi$ in a Fourier series if period is $2\pi$ . [10]
(b)If $f(z)$ is analytic inside a circle $C$ with centre at $a$ , then for all $z$ inside $C$ f ( z ) = f ( a ) + f ′ ( a ) ( z − a ) + f ′ ′ ( a ) 2 ! ( z − a ) 2 + $.$ f(z) = f(a) + f'(a)(z-a) + \frac{f''(a)}{2!}(z-a)^2 + ... f ( z ) = f ( a ) + f ′ ( a ) ( z − a ) + 2 ! f ′′ ( a ) ( z − a ) 2 + ... [10]

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Q. 8

(a)Evaluate the integral by using Cauchy integral Formula $\oint_C \frac{(4-3z)dz}{z(z-1)(z-2)}$ where $C$ is a circle ∣ z ∣ = 3 2 |z| = \frac{3}{2} ∣ z ∣ = 2 3 . [10]
(b)Prove that ∫ 0 2 π d θ 1 − 2 p cos ⁡ θ + p 2 = 2 π 1 − p 2 \int_0^{2\pi} \frac{d\theta}{1-2p\cos\theta+p^2} = \frac{2\pi}{1-p^2} ∫ 0 2 π 1 − 2 p c o s θ + p 2 d θ = 1 − p 2 2 π [10]

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About this paper

The 2011 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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