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FPSC · CSS 2014

Pure Mathematics, Paper I

100 marks · 3 hours · 8 questions
This paper Pure Mathematics · all yearsQ. 1 · If G is a group…Q. 2 · If f : G be…Q. 3 · If in a ring R…Q. 4 · Prove that a non empty…Q. 5 · A company produces three products,…Q. 6 · Find the equation of the…Q. 7 · Find the equation of the…Q. 8 · Find the volume of the…With this paper← 20122014 →
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Section A

Q. 1
  1. (a)If G is a group in which (ab)2=a2b2(ab)^2 = a^2b^2 for three consecutive integers ii for all a , b a,b∈Ga,b \in G , show that G is abelian. [10]
  2. (b)The center Z of a group G is defined by } Z = \{z \in G / zx = xz \forall x \in G\} . Prove that Z is a subgroup of G. [10]
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Q. 2
  1. (a)If f : G f:G→G′f :G \to G' be a homomorphism. Prove that Ker ff is a normal subgroup of G. [10]
  2. (b)Prove that any group of order 15 is cyclic. [10]
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Q. 3
  1. (a)If in a ring R with unity, (xy)2=x2y2(xy)^2 = x^2y^2 for all x , y x,y∈Rx, y \in R , then show that R is commutative. [10]
  2. (b)Prove that the set } Z_7 = \{0,1,2,3,4,5,6\} forms a commutative ring with unit element under addition and multiplication module 7. [10]
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Q. 4
  1. (a)Prove that a non empty subset W of a vector space V(F) is a subspace of V if and only if αx+βy∈W\alpha x + \beta y \in W for α,β∈F,x,y∈W\alpha, \beta \in F, x, y \in W . [10]
  2. (b)Show that the vectors v 1 = ( 1 , – 1 , – 4 , 0 ) , v 2 = ( 1 , 1 , 2 , 4 ) , v 3 = ( 2 , − 1 , − 5 , 2 ) , v 4 = ( 2 , 1 , 1 , 6 ) v_1 = (1, – 1, – 4,0), v_2 = (1, 1, 2, 4), v_3 = (2,-1,-5,2), v_4 = (2, 1, 1,6) v 1 ​ = ( 1 , –1 , –4 , 0 ) , v 2 ​ = ( 1 , 1 , 2 , 4 ) , v 3 ​ = ( 2 , − 1 , − 5 , 2 ) , v 4 ​ = ( 2 , 1 , 1 , 6 ) are linearly dependent in R4(R)\mathbb{R}^4 (\mathbb{R}) . [10]
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Q. 5

A company produces three products, each of which must be processed through three different departments. Given table summarizes the hours required per unit of each product in each department. In addition, the weekly capacities are stated for each department in terms of work-hours available. What is desired is to determine whether there are any combinations of the three products which would exhaust the weekly capacities of the three departments.

  1. (a)Department Product Hours Available per Week A B C 1 2 3.5 3 1,200 2 3 2.5 2 1,150 3 4 3 2 1,400 [10]
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Section B

Q. 6
  1. (a)Find the equation of the straight line joining two points on the ellipse x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 whose eccentric angles are given. Hence find equations of the tangent and normal at any point on the ellipse. [10]
  2. (b)Find the angle of intersection of the cardioids r = a ( 1 + cos ⁡ θ ) r=a(1+\cos \theta) r = a ( 1 + cos θ ) and r = b ( 1 − cos ⁡ θ ) r=b(1-\cos \theta) r = b ( 1 − cos θ ) . [10]
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Q. 7
  1. (a)Find the equation of the line L through the point ( 5 , (5,72,5)(5, \frac{7}{2}, 5) and intersecting at right angles the line M with parametric equations x = 4 + 3 t , y = 1 + t , z = − 3 t x = 4 + 3t, y = 1 + t, z = −3t x = 4 + 3 t , y = 1 + t , z = − 3 t . [10]
  2. (b)Find the equation of the tangent plane at any point P(x1,y1,z1)P(x_1, y_1, z_1) of the elliptic paraboloid z = z=x2+4y2z = x^2 + 4y^2 . [10]
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Q. 8
  1. (a)Find the volume of the solid obtained by revolving the area enclosed by one arc of the cycloid x = a ( θ + sin ⁡ θ ) , y = a ( 1 + cos ⁡ θ ) x=a(\theta+\sin\theta), y=a(1+\cos\theta) x = a ( θ + sin θ ) , y = a ( 1 + cos θ ) about x – axis . [10]
  2. (b)Discuss the surface and make a sketch, x2–y2+z2=1x^2 – y^2 + z^2 = 1 . [10]
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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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