Not yet checked 8 of 8 questions have not yet been compared with the official paper.
SECTION–A
Q. 1
(a)Let the function f = [ − 2 , 2 ] f=[−2,2]→R be defined by f ( x ) = ∣ x ∣ f(x) = |x| f ( x ) = ∣ x ∣ . Show that f is continuous at x = 0 x = 0 x = 0 but it is not differentiable at x = 0 x = 0 x = 0 . Will there exist a point c in ]–1,1[ such that f′(c)=0 or f(1)−f(−1)=2f′(c) ? [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]
Not yet checked
(10)
Q. 2
(a)Find the asymptotes of the curve defined by the equation ( x − y ) 2 ( x 2 + y 2 ) – 10 ( x − y ) x 2 + 12 y 2 + 2 x + y = 0 (x - y)² (x² + y²) – 10(x - y) x² + 12y² +2x + y = 0 ( x − y ) 2 ( x 2 + y 2 ) –10 ( x − y ) x 2 + 12 y 2 + 2 x + y = 0 [10]
(b)Test the convergence of the series ∑n=1∞nk1k>0 . How do we call this series? [10]
Not yet checked
(10)
Q. 3
(a)Find the area enclosed by the parabola y 2 + 16 x + 6 y – 71 = 0 y² + 16x + 6y – 71 = 0 y 2 + 16 x + 6 y –71 = 0 and the line 4x+y+7=0 [10]
(b)Find the volume of the solid generated by revolving about the y-axis the area of the triangle with vertices at (2,1), (6,1) and (4,5). [10]
Not yet checked
(10)
Q. 4
(a)If u = arcsin ( x 2 + y 2 x + y ) u = \arcsin \left( \frac{x^2 + y^2}{x + y} \right) u = arcsin ( x + y x 2 + y 2 ) , show that x ∂ u ∂ x + y ∂ u ∂ y = tan u x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = \tan u x ∂ x ∂ u + y ∂ y ∂ u = tan u . [10]
(b)Integrate F ( x , y ) = 1 F(x, y) = \frac{1}{y^4 + 1 1 over the region R : 0 ≤ x ≤ 8 , x ≤ y ≤ 2 R : 0 \le x \le 8, \sqrt{x} \le y \le 2 R : 0 ≤ x ≤ 8 , x ≤ y ≤ 2 [10]
Not yet checked
(10)
Q. 5
(a)Let X be the set of all (bounded or unbounded) sequences of complex numbers. If d : X d:X×X→R is defined as d ( x , y ) = ∑ j = 1 ∞ 1 2 j ∣ ζ j − η j ∣ 1 + ∣ ζ j − η j ∣ d(x, y) = \sum_{j=1}^{\infty} \frac{1}{2^j} \frac{| \zeta_j - \eta_j |}{1 + | \zeta_j - \eta_j |} d ( x , y ) = ∑ j = 1 ∞ 2 j 1 1 + ∣ ζ j − η j ∣ ∣ ζ j − η j ∣ , where x = ( ζ j ) x = (\zeta_j) x = ( ζ j ) and y = ( η j ) y = (\eta_j) y = ( η j ) , then show that d is a metric on X . [10]
(b)Prove that the mapping: T : T:(X,dy)→(Y,dx) is continuous at a point x ∈ X ⇔ x n → x x \in X \Leftrightarrow x_n \rightarrow x x ∈ X ⇔ x n → x implies Txn→Tx . [10]
Not yet checked
(10)
SECTION–B
Q. 6
(a)If Z = Z=1+it(1+i)+(3+2i)t , then show that the locus of Z is a circle. Also calculate the minimum and maximum distance of Z from the origin. [10]
(b)Find the complex number Z satisfying the equation Z 2 + ( 2 i – 3 ) Z + ( 5 – i ) = 0 Z² + (2i – 3) Z + (5 – i) = 0 Z 2 + ( 2 i –3 ) Z + ( 5– i ) = 0 [10]
Not yet checked
(10)
Q. 7
(a)Show that the function u(x,y)=4xy–3x+2 is harmonic. Construct the corresponding analytic function f(z)=u(x,y)+iv(x,y) [10]
(b)Find the Fourier Series of the function f ( x ) = { x 0 < x ≤ π 2 π − x π < x < 2 π f(x) = \begin{cases} x & 0 < x \le \pi \\ 2\pi - x & \pi < x < 2\pi \end{cases} f ( x ) = { x 2 π − x 0 < x ≤ π π < x < 2 π period 2π [10]
Not yet checked
(10)
Q. 8
(a)Evaluate the following integral by using Cauchy Integral Formula: ∮Cz(z−1)(z−2)4−3zdz where C is the 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 where 0 < p < 1 0 < p < 1 0 < p < 1 . [10]
The 2009 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.
Disclaimer Prep Right is independent and not affiliated with FPSC.