RAJ. STATE AND SUB. SERVICES COMB COMP (TSP) EXAM 2018 Paper-1 Question Paper PDF

Rajasthan Government Jobs Other Jobs 2018

  • Year 2018
  • Conducted By RPSC

Exam Details

Detail Information
Examination RAJ. STATE AND SUB. SERVICES COMB COMP (TSP) EXAM
Year 2018
Conducting Body RPSC
Paper Paper-1
Question Type Mixed

This document contains the question paper for the RAJ. STATE AND SUB. SERVICES COMB COMP (TSP) EXAM 2018, Paper-1. Conducted by RPSC, this exam is a crucial step for recruitment in Rajasthan state services. Aspirants can use this paper to understand the exam structure, question difficulty, and key topics covered in Paper-1 of the Mains examination. Analyzing previous year papers like this one is vital for effective preparation and to identify areas for improvement.

Major Topics Covered

  • RAJ. STATE AND SUB. SERVICES COMB COMP (TSP) EXAM
  • Paper-1
  • RPSC
  • 2018 question paper
  • previous year paper
  • PDF download

Why This Paper is Important

  • Useful for RAJ. STATE AND SUB. SERVICES COMB COMP ( preparation
  • Helps understand the latest exam pattern
  • Useful for practice and self-assessment
  • Covers frequently asked General Studies topics
  • Helpful for analysing question trends

Related Resources

  • RAJ. STATE AND SUB. SERVICES COMB COMP (TSP) EXAM 2018 Paper-2
  • RPSC RAS Mains 2018 Paper-1
  • RPSC RAS Mains 2018 Paper-2
  • RAJ. STATE AND SUB. SERVICES COMB COMP (TSP) EXAM 2018 Paper-1 Answer Key
  • RAJ. STATE AND SUB. SERVICES COMB COMP (TSP) EXAM Syllabus
  • RPSC Mains Syllabus
  • RAJ. STATE AND SUB. SERVICES COMB COMP (TSP) EXAM Pattern
  • RPSC Mains Exam Pattern

Instructions

  • PART - II 1 Paper Code P-1 Z lm i2 !տ Roll No.
  • PART - I Paper Code MARKET Date of Birth (DD/MM/YYYY) ш P-1 99703 5999 Father's Name HEAT SOUTH Signature of the candidate PERSONAL N m TO BE FILLED BY THE CANDIDATE lտ Z ۰ !เ∩ Roll No. \circledcirc \circledcirc \circledcirc \circledcirc \circledcirc \circledcirc \odot \odot \textcircled{\scriptsize 1} \odot \odot \odot ^{\circ} ^{\circ} ^{\circledR} ^{\circ} \circledcirc ^{\circledR} \circledcirc ^{\circledR} \circledS \circledcirc \circledcirc \circledcirc ш \circled{4} \circledcirc \circledcirc \circled{4} \circledcirc \circled{4} \circledcirc \circledcirc \circledcirc \circledS \circledS \circledS \circledcirc \circledcirc \circledcirc \circledcirc \circledcirc \circledcirc \circledcirc \circledcirc ^\circledR \circledcirc \circledcirc \circledcirc \circledcirc ^\circledR ^\circledR ^\circledR \circledcirc \circledcirc \circledcirc \circledcirc ^{\circ} \circledcirc ◎ ^{\circledR} Invigilator must check the Roll No. and Photo LD. of the candidate, them Sign. here: TOTAL COLLE TO BE FILLED BY INVIGILATOR SHOW If candidate found using unfair means them Invigilator should fill up this STATE bubble with black/blue ball pen & report to the Centre Superintendent: COMPANY دعو \circ STATE District STATE 1430_H2C-2 \blacksquare 58

Questions (page 2)

Q1.

(a) Let V be a vector space over a field F. Let W be a subspace of V. Prove that V/W is a vector space over F.

(b) Let T: V o W be a linear transformation. Prove that T is injective if and only if ext{ker}(T) = {0}.

(c) Let A be an n imes n matrix. Prove that A is invertible if and only if det(A) eq 0.

(d) Let V be a finite-dimensional vector space and let T: V o V be a linear operator. Prove that V = ext{Im}(T) oplus ext{ker}(T) if and only if ext{Im}(T) = ext{ker}(T).

(e) Let A be an n imes n matrix. Prove that A is diagonalizable if and only if the algebraic multiplicity of each eigenvalue equals its geometric multiplicity.

Q2.

(a) Let G be a group and let H be a subgroup of G. Prove that H is a normal subgroup of G if and only if gHg^{-1} = H for all g in G.

(b) Let G be a finite group and let p be a prime number dividing the order of G. Prove that G has an element of order p.

(c) Let R be a commutative ring with unity. Prove that an ideal I of R is a prime ideal if and only if R/I is an integral domain.

(d) Let F be a field. Prove that every ideal of F[x] is principal.

(e) Let G be a group and let H be a subgroup of G. Prove that the set of left cosets of H in G forms a group under the operation (aH)(bH) = (ab)H if and only if H is a normal subgroup of G.

Q3.

(a) Let f: mathbb{R} o mathbb{R} be a continuous function. Prove that the image of a connected set under f is connected.

(b) Let (X, d) be a metric space. Prove that the open ball B(x, r) = {y in X : d(x, y) < r} is an open set.

(c) Let f: X o Y be a function between topological spaces. Prove that f is continuous if and only if for every closed set C in Y, f^{-1}(C) is closed in X.

(d) Let X be a topological space. Prove that X is compact if and only if every open cover of X has a finite subcover.

(e) Let f: X o Y be a continuous bijection. Prove that if X is compact and Y is Hausdorff, then f is a homeomorphism.

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Frequently asked questions

What is the name of the exam?

The exam is the RAJ. STATE AND SUB. SERVICES COMB COMP (TSP) EXAM 2018.

Which paper is this question paper for?

This is Paper-1 of the RAJ. STATE AND SUB. SERVICES COMB COMP (TSP) EXAM 2018.

What is the conducting body for this exam?

The exam is conducted by RPSC (Rajasthan Public Service Commission).

What is the year of this examination?

The year of the examination is 2018.

What is the paper code mentioned on the paper?

The paper code mentioned is P-1.

What is the exam stage for this paper?

This paper is for the MAINS stage of the exam.

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