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PPSC · PMS Punjab 2023

Mathematics, Paper II

100 marks · 3 hours · 8 questions
This paper Mathematics · all yearsQ. 1(a) · Show that if every element…Q. 2(a) · Show that normalizer of a…Q. 3(a) · Define an integral domain. If…Q. 4(a) · If G is a non-Abelian…Q. 5(a) · Let X be any infinite…Q. 6(a) · Let (X, <.,.>) be an…Q. 7(a) · Show that the following can…Q. 8 · Find the eigen values and…With this paper← 2023
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Q. 1(a)
  1. (a)Show that if every element of a group G is its own inverse, then G is an Abelian group. [10]
  2. (b)In a symmetric group of degree 3, give an example of two elements x, y such that (x. y)² = x².y² [10]
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Q. 2(a)
  1. (a)Show that normalizer of a subset H of a group G is a subgroup of G. [10]
  2. (b)Show that the centralizer of a subset H in a group G is a subgroup of G. [10]
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Q. 3(a)
  1. (a)Define an integral domain. If p is a prime number, then show that ring of integers mod p is an integral domain. [10]
  2. (b)Let F be a field of real numbers. Then set of all real valued functions whose nth derivative exist for n = 1,2,..., is a subspace of all real valued continuous function on [0,1]. [10]
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Q. 4(a)
  1. (a)If G is a non-Abelian group of order 6, then show that G is isomorphic on to S₃. [10]
  2. (b)Let G₁ be the group of positive real numbers under multiplication and G₂ the group of all real numbers under addition. Define : G₁ → G₂ by ϕ ( x ) = log ⁡ 10 x \phi(x) = \log_{10} x ϕ ( x ) = lo g 10 ​ x . Show that ϕ\phi is a bijective homomorphism. [10]
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Q. 5(a)
  1. (a)Let X be any infinite set, and let the set / consist of the empty set ϕ\phi together with all the subsets of X whose complements are finite. Show that / is a topology on X. [10]
  2. (b)Define a metrizable topological space. Give an example of a topological space which is not metrizable. [10]
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Q. 6(a)
  1. (a)Let (X, <.,.>) be an inner product space. Show that | < x,y > | < ||x||||y|| for all x, y ∈ X, where the equality sign holds if and only if {x,y} is a linearly dependent set. [10]
  2. (b)Show that the space C[a, b] is not an inner product space [10]
Not yet checked
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Q. 7(a)
  1. (a)Show that the following can happen for 2x2 matrices A and B. (1) A² = 0 even though A ≠ 0. (2) AB + BA. [10]
  2. (b)Suppose that T: R⁴ → R³ is a linear transformation such that T([1,0,-1,2]) = [2,1,0] and T([1,1,-2,0]) = [-1,2,4]. Compute T'([-2, -5,7,6]). [10]
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Q. 8
  1. (a)Find the eigen values and the corresponding eigen vectors of A = [ 1 2 ] [2 3 ] [10]
  2. (b)Any two eigen vectors corresponding to two distinct eigen values of orthogonal matrix are orthogonal. [10]
Not yet checked
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About this paper

The 2023 PMS Punjab Mathematics paper set by the PPSC. Question wording only; questions marked “Not yet checked” have not been compared with the official paper yet.

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