What should be done with blank pages?

Any page or portion left blank must be clearly struck off.

What does the notation imply for symbols and equations?

Symbols and notations carry their usual standard meanings unless otherwise indicated.

What happens if you don’t strike off a page?

The page is counted as attempted; only struck-off pages are exempt from consideration.

Is the content translated in both languages?

Question text may include bilingual content; FAQ answers are in English for SEO clarity.

Where can I find the assignment-type question on page 2?

Page 2 contains a section describing a five-person, five-task assignment problem (OCR restored; content may include a matrix).

Mathematics Paper-II 2021 CS Main Exam

Central Government Jobs Other Jobs 2021

Year 2021
Conducted By UPSC
Questions 8
Maximum Marks 250
Duration Three Hours Maximum Marks: 250 Question Paper Specific Instructions (please Read Each Of The Following Instructions Carefully Before Attempting Questions) There Are Eight Questions Divided In Two Sections And Printed Both In Hindi And In English. Candidate Has To Attempt Five Questions In All. Question Nos. 1 And 5 Are Compulsory And Out Of The Remaining, Three Are To Be Attempted Choosing At Least One Question From Each Section. The Number Of Marks Carried By A Question/part Is Indicated Against It. Answers Must Be Written In The Medium Authorized In The Admission Certificate Which Must Be Stated Clearly On The Cover Of This Question-cum-answer (qca) Booklet In The Space Provided. No Marks Will Be Given For Answers Written In A Medium Other Than The Authorized One. Assume Suitable Data, If Considered Necessary, And Indicate The Same Clearly. Unless And Otherwise Indicated, Symbols And Notations Carry Their Usual Standard Meanings. Attempts Of Questions Shall Be Counted In Sequential Order. Unless Struck Off, Attempt Of A Question Shall Be Counted Even If Attempted Partly. Any Page Or Portion Of The Page Left Blank In The Question-cum-answer Booklet Must Be Clearly Struck Off. Hxs-b-mth/10 1

Exam Details

Detail	Information
Examination	Mathematics (PAPER-II)
Year	2021
Conducting Body	UPSC
Paper	Mathematics (PAPER-II)
Subject	Mathematics
Duration	Three Hours Maximum Marks: 250 Question Paper Specific Instructions (please Read Each Of The Following Instructions Carefully Before Attempting Questions) There Are Eight Questions Divided In Two Sections And Printed Both In Hindi And In English. Candidate Has To Attempt Five Questions In All. Question Nos. 1 And 5 Are Compulsory And Out Of The Remaining, Three Are To Be Attempted Choosing At Least One Question From Each Section. The Number Of Marks Carried By A Question/part Is Indicated Against It. Answers Must Be Written In The Medium Authorized In The Admission Certificate Which Must Be Stated Clearly On The Cover Of This Question-cum-answer (qca) Booklet In The Space Provided. No Marks Will Be Given For Answers Written In A Medium Other Than The Authorized One. Assume Suitable Data, If Considered Necessary, And Indicate The Same Clearly. Unless And Otherwise Indicated, Symbols And Notations Carry Their Usual Standard Meanings. Attempts Of Questions Shall Be Counted In Sequential Order. Unless Struck Off, Attempt Of A Question Shall Be Counted Even If Attempted Partly. Any Page Or Portion Of The Page Left Blank In The Question-cum-answer Booklet Must Be Clearly Struck Off. Hxs-b-mth/10 1
Maximum Marks	250
Number of Questions	8

This page contains Mathematics Paper-II (CS Main) for the 2021 examination. Page 1 provides the exam header, duration (three hours), maximum marks (250), bilingual (Hindi-English) printing, and the instruction context including compulsory questions and sectioning. Page 2 presents repaired OCR content for Section A with four subparts (a)–(d): (a) Bezout/GCD representation, (b) a uniform convergence check of a series on [0,1], (c) a statement about Riemann integrability for monotonic functions, and (d) a contour integration problem with a complex curve c(t) = e^{4πi t}. It also contains a second problem describing a five-person, five-task assignment scenario with a time-efficiency matrix (OCR garbled) to minimize total completion time. The OCR repair clarifies line breaks and separates merged stems for proper JSON extraction.

Major Topics Covered

GCD and linear combinations
Diophantine equations
Uniform convergence
Series convergence
Weierstrass M-test
Riemann integration
Monotonic functions
Riemann integrable
Contour integration
Complex analysis
Curve parameterization
e^{4πit}
Assignment problem
Hungarian algorithm
Optimization
Matrix time study
Makespan minimization
Two-language math exams
QCA instructions
Exam paper codes

Why This Paper is Important

Useful for CS (Main) Exam 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

Instructions

इसमें आठ प्रश्न हैं जो दो खण्डों में विभाजित हैं तथा हिन्दी और अंग्रेजी दोनों में छपे हुए हैं।
परीक्षार्थी को कुल पाँच प्रश्नों के उत्तर देने हैं।
प्रश्न संख्या 1 और 5 अनिवार्य हैं तथा बाकी प्रश्नों में से प्रत्येक खण्ड से कम-से-कम एक प्रश्न चुनकर तीन प्रश्नों के उत्तर दीजिए।
प्रत्येक प्रश्न/भाग के अंक उसके सामने दिए गए हैं।
प्रश्नों के उत्तर उसी प्राधिकृत माध्यम में लिखे जाने चाहिए, जिसका उल्लेख आपके प्रवेश-पत्र में किया गया है, और इस माध्यम का स्पष्ट उल्लेख प्रश्न-सह-उत्तर (क्यू॰ सी॰ ए॰) पुस्तिका के मुखपृष्ठ पर निर्दिष्ट स्थान पर किया जाना चाहिए।
उल्लिखित माध्यम के अतिरिक्त अन्य किसी माध्यम में लिखे गए उत्तर पर कोई अंक नहीं मिलेंगे।
यदि आवश्यक हो, तो उपयुक्त आँकड़ों का चयन कीजिए, तथा उनको निर्दिष्ट कीजिए।
जब तक उल्लिखित न हो, संकेत तथा शब्दावली प्रचलित मानक अर्थों में प्रयुक्त हैं।
प्रश्नों के उत्तरों की गणना क्रमानुसार की जाएगी।
यदि काटा नहीं हो, तो प्रश्न के उत्तर की गणना की जाएगी चाहे वह उत्तर अंशतः दिया गया हो।
प्रश्न-सह-उत्तर पुस्तिका में खाली छोड़ा हुआ पृष्ठ या उसके अंश को स्पष्ट रूप से काटा जाना चाहिए।

There are EIGHT questions divided in two Sections and printed both in HINDI and in ENGLISH.
Candidate has to attempt FIVE questions in all.
Question Nos. 1 and 5 are compulsory and out of the remaining, THREE are to be attempted choosing at least ONE question from each Section.
The number of marks carried by a question/part is indicated against it.
Answers must be written in the medium authorized in the Admission Certificate which must be stated clearly on the cover of this Question-cum-Answer (QCA) Booklet in the space provided.
No marks will be given for answers written in a medium other than the authorized one.
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless and otherwise indicated, symbols and notations carry their usual standard meanings.
Attempts of questions shall be counted in sequential order.
Unless struck off, attempt of a question shall be counted even if attempted partly.
Any page or portion of the page left blank in the Question-cum-Answer Booklet must be clearly struck off.

Questions (page 2)

Q1. 1. (a) मान लीजिए कि m1, m2, ..., mk धनात्मक पूर्णांक हैं तथा d > 0, m1, m2, ..., mk का महत्तम समापवर्तक है। दर्शाइए कि ऐसे पूर्णांक x1, x2, ..., xk अस्तित्व में हैं ताकि d = x1 m1 + x2 m2 + ⋯ + xk mk.
Let m1, m2, ..., mk be positive integers and d > 0 the greatest common divisor of m1, m2, ..., mk. Show that there exist integers x1, x2, ..., xk such that d = x1 m1 + x2 m2 + ⋯ + xk mk.

Q2. 1. (b) x⁴ + x⁴/(1+x⁴) + x⁴/(1+x⁴)^2 + x⁴/(1+x⁴)^3 + ... के [0, 1] पर एकसमान अभिसरण की जाँच कीजिए। Test the uniform convergence of the series x⁴ + x⁴/(1+x⁴) + x⁴/(1+x⁴)^2 + x⁴/(1+x⁴)^3 + ⋯ on [0, 1].

Q3.

Q4.

(d) मान लीजिए कि c: [0, 1] → C, c(t) = e^{4π it}, 0 ≤ t ≤ 1 के द्वारा परिभाषित एक वक्र है। कन्टूर समाकल ∮ dz/(2z²-5z + 2) का मान निकालिए। Let c:[0, 1] → C be the curve, where c(t) = e^{4π i t}, 0 ≤ t ≤ 1. Evaluate the contour integral ∮ dz/(2z²-5z + 2).

Q5. 2. 10 एक कम्पनी के एक विभाग के पाँच कर्मचारियों को पाँच कार्य सम्पन्न करने हैं। जितना समय (घंटों में) एक व्यक्ति एक कार्य को सम्पन्न करने के लिए लेता है, वह प्रभाविता आव्यूह में दिया गया है। इन पाँच कर्मचारियों को इन सभी कार्यों को इस तरह निर्धारित कीजिए जिससे कि समस्त कार्य सम्पन्न करने का समय न्यूनतम हो : कर्मचारी Ш IV V ... [OCR matrix garbled; contains a times-matrix and labeling; original formatting partially garbled in OCR].

Question paper preview

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

Mathematics Paper-II 2021 CS Main Exam — page 1 instructions scan PDF download — Page 1

Free question paper download

Download question paper PDF

2.7 MB
8 pages
PDF format

Frequently asked questions

What is the official name of the paper and its year?

Mathematics Paper-II for CS Main Examination, 2021.

How many questions are there and how should they be attempted?

There are eight questions in two sections. Candidates must attempt five questions in total, with questions 1 and 5 compulsory, choosing at least one question from each section.

What are the maximum marks and duration?

Maximum marks are 250 and the duration is three hours.

In what languages is the paper printed?

The paper is printed in Hindi and English.

What is the medium of answers?

Answers must be written in the medium authorized in the Admission Certificate, clearly stated on the cover of the QCA Booklet.

Are there any constraints on the use of marks or sections?

Yes. Q1 and Q5 are compulsory. From the remaining questions, pick at least one from each section; marks are indicated against each question.

← Back to Other Jobs papers