RPSC Rajasthan State Services Mains 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. EXAM
Year 2018
Conducting Body RPSC
Paper Paper-1
Question Type Mixed

This is the question paper for the Raj. State and Sub. Services Comb. Comp. (MAINS) Exam 2018, Paper-1, conducted by RPSC. This paper is crucial for candidates aiming to secure a position in the Rajasthan state services. Aspirants can use this previous year's paper to understand the exam structure, question difficulty, and the types of questions asked in the Mains examination. Familiarizing oneself with Paper-1 content is essential for effective preparation and to gauge performance against the competitive standards set by RPSC.

Major Topics Covered

  • RAJ. STATE AND SUB. SERVICES COMB. COMP. 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

  • RPSC Rajasthan State Services Mains 2018 Paper-2
  • RPSC Rajasthan State Services Prelims 2018
  • RPSC Previous Year Question Papers
  • RPSC Rajasthan State Services Mains 2018 Paper-1 Answer Key
  • RPSC Rajasthan State Services Mains Syllabus
  • RPSC Paper-1 Syllabus
  • RPSC Rajasthan State Services Mains 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 \mathcalG be a group. Show that \mathcalG is abelian if and only if the map f: \mathcalG × \mathcalG → \mathcalG defined by f(x, y) = x y x-1 y-1 is the trivial map.

(b) Let V be a vector space over \mathbbR and let T: V → V be a linear transformation such that T2 = T. Show that V = ker(T) oplus extIm(T).

(c) Let G be a group and let H be a subgroup of G. Prove that if G is cyclic, then G/H is also cyclic.

(d) Let R be a commutative ring with unity. Show that an ideal I of R is a prime ideal if and only if for any a, b ∈ R, ab ∈ I implies a ∈ I or b ∈ I.

(e) Let F be a field. Show that the set of all polynomials in one variable over F, denoted by F[x], forms a Euclidean domain.

Q2.

(a) Let V be a finite-dimensional vector space over a field F and let T: V → V be a linear transformation. Prove that dim(ker(T)) + dim( extIm(T)) = dim(V).

(b) Let G be a group. Show that the set of all inner automorphisms of G, denoted by ext{Inn}(G), is a normal subgroup of the group of all automorphisms of G, denoted by ext{Aut}(G).

(c) Let R be a commutative ring with unity. Show that if R is an integral domain, then the polynomial ring R[x] is also an integral domain.

(d) Let V be a vector space over \mathbbR and let u, v ∈ V. Define an inner product \langle u, v \rangle on V. Show that the Cauchy-Schwarz inequality |langle u, v \rangle| ≤ |u| |v| holds.

(e) Let F be a field and let p(x) ∈ F[x] be a non-constant polynomial. Show that there exists an extension field K of F such that p(x) has a root in K.

Question paper preview

Scanned pages 1–2 for reference. Download the official PDF for the full paper.

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RPSC Rajasthan State Services Mains 2018 Paper-1 question paper page 1 scan PDF download, showing exam header and candidate details.
Page 2
RPSC Rajasthan State Services Mains 2018 Paper-1 question paper page 1 scan PDF download, showing exam header and candidate details.

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Frequently asked questions

What is the name of the exam for which this question paper is?

This is the question paper for the Raj. State and Sub. Services Comb. Comp. (MAINS) Exam 2018, Paper-1.

Which conducting body released this question paper?

This question paper was released by RPSC (Rajasthan Public Service Commission).

What is the year of this examination?

The examination year for this paper is 2018.

What is the paper code mentioned on the question 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 examination.

Can I download the RPSC Rajasthan State Services Mains 2018 Paper-1 question paper?

Yes, this page provides access to the RPSC Rajasthan State Services Mains 2018 Paper-1 question paper.

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