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FPSC · CSS 2016

Pure Mathematics

100 marks · 3 hours · 8 questions
This paper Pure Mathematics · all yearsQ. 1 · Prove that the normaliser of…Q. 2 · Let a be a fixed…Q. 3 · Let T : X be…Q. 4 · Using definition of continuity, prove…Q. 5 · Prove that the maximum value…Q. 6 · Find the area enclosed between…Q. 7 · Determine where P ( z…Q. 8 · Use Cauchy Integral Formula to…With this paper← 20152017 →
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Section A

Q. 1
  1. (a)Prove that the normaliser of a subset of a group G is a Subgroup of G. [10]
  2. (b)Let A be a normal subgroup and B a subgroup of a group G. Then prove that < A , B > = A B < A,B > = AB < A , B >= A B . [10]
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Q. 2
  1. (a)Let a be a fixed point of a group G and consider the mapping Ia:G→GI_a : G \rightarrow G defined by Ia(g)=aga−1I_a(g) = aga^{-1} where g g∈Gg \in G . Show that IaI_a is an automorphism of G. Also show that for a , b a,b∈G,Ia⋅Ib=Iaba, b \in G, I_a \cdot I_b = I_{ab} . [10]
  2. (b)Let M 2 ( R ) = { ( a b c d ) : a , b , c , d ∈ R } M_2 (R) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in R \right\} M 2 ​ ( R ) = { ( a c ​ b d ​ ) : a , b , c , d ∈ R } be the set of all 2x2 matrices with real entries. Show that (M2(R),+,⋅)(M_2(R), +, \cdot) forms a ring with identity. Is (M2(R),+,⋅)(M_2(R), +, \cdot) a field? [10]
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Q. 3
  1. (a)Let T : X T:X→YT: X \rightarrow Y be a linear transformation from a vector space X into a Vector space Y. Prove that Kernal of T is a subspace. [10]
  2. (b)Find the value of λ\lambda such that the system of equations x + x+λy+3z=0x + \lambda y + 3z = 0 , 4 x + 3 y + 4x+3y+λz=04x + 3y + \lambda z = 0 , 2x+y+2z=02x + y + 2z = 0 has non-trivial solution. [10]
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Section B

Q. 4
  1. (a)Using δ−ϵ\delta - \epsilon definition of continuity, prove that the function Sin2xSin^2 x is continuous for all x x∈Rx \in R . [10]
  2. (b)Find the asymptotes of the curve (x2−y2)(x+2y)+5(x2+y2)+x+y=0(x^2-y^2)(x+2y) + 5(x^2+y^2) + x+y = 0 . [10]
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Q. 5
  1. (a)Prove that the maximum value of (1/x)x(1/x)^x is e1/ee^{1/e} . [10]
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Q. 6
  1. (a)Find the area enclosed between the curves y=x3y=x^3 and y=xy=x . [10]
  2. (b)A plane passes through a fixed point (a,b,c)(a, b, c) and cuts the coordinate axes in A, B, C. Find the locus of the centre of the sphere OABC for different positions of the plane, O is the origin. [10]
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Section C

Q. 7
  1. (a)Determine P(z)P(z) where P ( z ) = P(z)=(z−z1)(z−z2)(z−z3)(z−z4)P(z) = (z-z_1)(z-z_2)(z-z_3)(z-z_4) with z 1 = e i π / 4 , z 2 = z ˉ 1 , z 3 = − z 1 z_1 = e^{i\pi/4}, z_2 = \bar{z}_1, z_3 = -z_1 z 1 ​ = e iπ /4 , z 2 ​ = z ˉ 1 ​ , z 3 ​ = − z 1 ​ , and z 4 = − z ˉ 1 z_4 = -\bar{z}_1 z 4 ​ = − z ˉ 1 ​ . [10]
  2. (b)Find value of the integral ∫c(z−z0)ndz\int_c (z-z_0)^n dz ( nn any integer) along the circle C with centre z0z_0 and radius rr , described in the counter clock wise direction. [10]
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Q. 8
  1. (a)Use Cauchy Integral Formula to evaluate ∫ccosz+isinzz−π/2dz\int_c \frac{cos z + i sin z}{z - \pi/2} dz along the simple closed counter C: ∣ z ∣ = 3 |z|=3 ∣ z ∣ = 3 described in the positive direction. [10]
  2. (b)State and prove Cauchy Residue Theorem. [10]
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The 2016 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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