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PPSC · PMS Punjab 2016

Mathematics, Paper II

100 marks · 3 hours · 8 questions
This paper Mathematics · all yearsQ. 1 · Show that the set G…Q. 2 · Define a convergent sequence {X…Q. 3 · Show that the vectors (1-i,i)…Q. 4 · Suppose U and W are…Q. 5 · Show that a subspace of…Q. 6 · Let (X,d) be a metric…Q. 7 · Using the row operation, show…Q. 8 · Solve the differential equations (D…With this paper2021 →
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Not yet checked 2 of 8 questions have not yet been compared with the official paper.

Q. 1
  1. (a)Show that the set G of all non-singular matrices of order 2 is a non abelian group under matrix multiplication. [10]
  2. (b)If n is the order of an element of a group G, then a n =e iff n divides m. [10]
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Q. 2
  1. (a)Define a convergent sequence {X n } in a metric space (X,d). show that the limit of a convergent sequence in a metric space is unique. [10]
  2. (b)Show that {(1, 2, 2), (-1,0,2), (0,0,1)} is a basis of R 3 . Using Gram-schmidt orthonormalization process, transform this basis into an orthonormal basis. [10]
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Q. 3
  1. (a)Show that the vectors (1-i,i) and (2,-1+i) in C x C are linearly dependent over C but linearly independent over R. [10]
  2. (b)Determine whether or not the given set of vectors is a basis for R 2 : (i) {(1,1),(3,1)} (ii) {(1,2,-1), (0,3,1), (1,-5,3)} [10]
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Q. 4

Suppose U and W are distinct four dimensional subspaces of a vector space V of dimension six. Find the possible dimension of U intersection W.

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Q. 5
  1. (a)Show that a subspace of topological space is itself topological space. [10]
  2. (b)Prove that cofinite topology is discrete if X is finite. [10]
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Q. 6
  1. (a)Let (X,d) be a metric space and T be a collection of all open subsets of X. Show that T is a topology on X. [10]
  2. (b)A topological space is normal iff for any closed set A and an open set U containing A, there is atleast one open set V containing A such that A ⊆ V ⊆ U. [10]
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Q. 7
  1. (a)Using the row operation, show that the matrix [[2, -3, 1], [1, -2, 1], [5, -2, -3]] has no inverse. [10]
  2. (b)Solve the system of equations having the given matrices as their augmented matrices. [10]
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Q. 8
  1. (a)Solve the differential equations (D 3 -6D 2 +3D+10)y=0 [10]
  2. (b)Find the general solution of each of the following:- (D 2 +3D-4)y=15e x [10]
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The 2016 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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