Mathematics Paper II - Civil Services Main 2023
- Year 2023
- Conducted By Not specified in OCR
- 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.
Exam Details
| Detail | Information |
|---|---|
| Examination | MATHEMATICS (PAPER-II) |
| Year | 2023 |
| Conducting Body | Not specified in OCR |
| 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. |
| Maximum Marks | 250 |
| Number of Questions | 8 |
This document presents Mathematics Paper II for the Civil Services (Main) Examination 2023. It outlines an eight-question paper across two sections, with five questions to be attempted, including compulsory Q1 and Q5. Page 2 contains repaired OCR content covering problems in abstract algebra (groups and ideals), real/complex analysis (convergence tests, analyticity of log z), and an optimization task related to chemical procurement. The content is bilingual (Hindi and English) and includes the paper code SKYC-B-MTH/13. The OCR repair follows guidelines to separate merged stems, restore line breaks, and provide clearly delineated questions for SEO-ready metadata.
Major Topics Covered
- Abstract Algebra
- Group Theory
- Ring Theory
- Number Theory
- Real Analysis
- Sequences and Series
- Convergence
- Power Series
- Complex Analysis
- Analytic Functions
- Cauchy-Riemann Equations
- Analyticity
- Logarithmic Functions
- Multivariable Calculus
- Optimization
- Linear Programming
- Chemistry-Optimization Problem
- Mathematical Proof Techniques
- Homomorphisms
- Ideals
Why This Paper is Important
- Useful for MATHEMATICS (PAPER-II) 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
- Civil Services Main 2023 Zoology Paper II Question Paper PDF
- UPSC Civil Services Main 2023 Zoology Paper I Question Paper PDF
- UPSC Civil Services Mains 2023 Public Administration Paper II PDF
- UPSC Civil Services Main 2023 Political Science Paper I PDF
- UPSC Civil Services Main 2023 Philosophy Paper II PDF
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।
- प्राधिकृत 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 कोटि 10 का एक समूह G है तथा कोटि 6 का एक समूह G' है। जाँच कीजिए कि क्या G से G' पर एक आच्छादक समाकारिता का अस्तित्व है। Let G be a group of order 10 and G' be a group of order
Q1. (a) खण्ड—A / SECTION- 1. मान लीजिए कोटि 10 का एक समूह G है तथा कोटि 6 का एक समूह G' है। जाँच कीजिए कि क्या G से G' पर एक आच्छादक समाकारिता का अस्तित्व है। Let G be a group of order and G' be a group of order 6. Examine whether there exists a homomorphism of G onto G'. 10 गुणजावली 4Z +6Z को पूर्णांकीय प्रांत Z में एक मुख्य गुणजावली के रूप में व्यक्त कीजिए।
(a) खण्ड—A / SECTION- 1. मान लीजिए कोटि 10 का एक समूह G है तथा कोटि 6 का एक समूह G' है। जाँच कीजिए कि क्या G से G' पर एक आच्छादक समाकारिता का अस्तित्व है। Let G be a group of order and G' be a group of order 6. Examine whether there exists a homomorphism of G onto G'. 10 गुणजावली 4Z +6Z को पूर्णांकीय प्रांत Z में एक मुख्य गुणजावली के रूप में व्यक्त कीजिए।
(b) Express the ideal 4Z + 6Z as a principal ideal in the integral domain Z.
(c) श्रेणी ∑n=1∞ \frac1.3.5... (2n-1){
Q2. 6... (2n)} \cdot (x^2n + 1)/((2n + 1)), x > 0 के अभिसरण का परीक्षण कीजिए। Test the convergence of the series ∑n=1∞ (1 \cdot 3 \cdot 5 \cdot \ldots (2n-1))/(2 \cdot 4 \cdot 6 \cdot \ldots (2n)) \cdot (x^2n + 1)/((2n + 1)), \quad x > 0 10 एक फलन f(z) = f(x + iy) = u(x, y) + iv(x, y) के इसके प्रांत में विश्लेषिक होने के लिए पर्याप्त प्रतिबंध (d) लिखिए। तब दर्शाइए कि f(z) = log z अपने प्रांत में विश्लेषिक है तथा (df)/(dz) ज्ञात कीजिए। State the sufficient conditions for a function f(z) = f(x + iy) = u(x, y) + iv(x, y) to be analytic in its domain. Hence, show that f(z) = log z is analytic in its domain and find (df)/(dx). 10 एक व्यक्ति को अपने उद्यान के लिए रसायन A, B तथा C की क्रमशः 24, 24 तथा 20 इकाई की आवश्यकता (e) है। उत्पाद P के प्रत्येक मर्तबान में रसायन A, B तथा C की क्रमशः 2, 4 तथा 1 इकाई हैं तथा उत्पाद Q के प्रत्येक मर्तबान में रसायन A, B तथा C की क्रमशः 2, 1 तथा 5 इकाई हैं। यदि P के एक मर्तबान का मूल्य ₹ 30 है तथा Q के एक मर्तबान का मूल्य ₹ 50 है, तब न्यूनतम खर्च तथा आवश्यकताओं की पूर्ति के लिए प्रत्येक उत्पाद के कितने मर्तबान खरीदे जाएँ? A person requires 24, 24 and 20 units of chemicals A, B and C respectively for his garden. Product P contains 2, 4 and 1 units of chemicals A, B and C respectively per jar and product Q contains 2, 1 and 5 units of chemicals A, B and C respectively per jar. If a jar of P costs ₹ 30 and a jar of Q costs ₹ 50, then how many jars of each should be purchased in order to minimize the cost 10 and meet the requirements? (a) सिद्ध कीजिए कि 2p कोटि के एक अक्रमविनिमेय समूह, जहाँ p एक विषम अभाज्य संख्या है, में p कोटि का 2. एक उपसमूह होना आवश्यक है। Prove that a non-commutative group of order 2p, where p is an odd prime, 15 must have a subgroup of order p. SKYC-B-MTH/13 2
Q2. 6... (2n)} \cdot (x^2n + 1)/((2n + 1)), x > 0 के अभिसरण का परीक्षण कीजिए। Test the convergence of the series ∑n=1∞ (1 \cdot 3 \cdot 5 \cdot \ldots (2n-1))/(2 \cdot 4 \cdot 6 \cdot \ldots (2n)) \cdot (x^2n + 1)/((2n + 1)), \quad x > 0 10 एक फलन f(z) = f(x + iy) = u(x, y) + iv(x, y) के इसके प्रांत में विश्लेषिक होने के लिए पर्याप्त प्रतिबंध (d) लिखिए। तब दर्शाइए कि f(z) = log z अपने प्रांत में विश्लेषिक है तथा (df)/(dz) ज्ञात कीजिए। State the sufficient conditions for a function f(z) = f(x + iy) = u(x, y) + iv(x, y) to be analytic in its domain. Hence, show that f(z) = log z is analytic in its domain and find (df)/(dx). 10 एक व्यक्ति को अपने उद्यान के लिए रसायन A, B तथा C की क्रमशः 24, 24 तथा 20 इकाई की आवश्यकता (e) है। उत्पाद P के प्रत्येक मर्तबान में रसायन A, B तथा C की क्रमशः 2, 4 तथा 1 इकाई हैं तथा उत्पाद Q के प्रत्येक मर्तबान में रसायन A, B तथा C की क्रमशः 2, 1 तथा 5 इकाई हैं। यदि P के एक मर्तबान का मूल्य ₹ 30 है तथा Q के एक मर्तबान का मूल्य ₹ 50 है, तब न्यूनतम खर्च तथा आवश्यकताओं की पूर्ति के लिए प्रत्येक उत्पाद के कितने मर्तबान खरीदे जाएँ? A person requires 24, 24 and 20 units of chemicals A, B and C respectively for his garden. Product P contains 2, 4 and 1 units of chemicals A, B and C respectively per jar and product Q contains 2, 1 and 5 units of chemicals A, B and C respectively per jar. If a jar of P costs ₹ 30 and a jar of Q costs ₹ 50, then how many jars of each should be purchased in order to minimize the cost 10 and meet the requirements? (a) सिद्ध कीजिए कि 2p कोटि के एक अक्रमविनिमेय समूह, जहाँ p एक विषम अभाज्य संख्या है, में p कोटि का 2. एक उपसमूह होना आवश्यक है। Prove that a non-commutative group of order 2p, where p is an odd prime, 15 must have a subgroup of order p. SKYC-B-MTH/13 2
Q3.
1.
(c) श्रेणी ∑_{n=1}^{∞} (1·3·5... (2n-1)) / (2·4·6... (2n)) · x^{2n+1} / (2n+1), x > 0 के अभिसरण का परीक्षण कीजिए। Test the convergence of the series ∑_{n=1}^{∞} (1 · 3 · 5 · ·· (2n-1)) / (2 · 4 · 6 · ·· (2n)) · x^{2n+1} / (2n+1), x > 0
Q4. 1. (d) एक फलन f(z) = f(x+iy) = u(x, y) + iv(x, y) के इसके प्रांत में विश्लेषिक होने के लिए पर्याप्त प्रतिबंध (d) लिखिए। तब दर्शाइए कि f(z) = log z अपने प्रांत में विश्लेषिक है तथा df/dz ज्ञात कीजिए। State the sufficient conditions for a function f(z) = f(x+iy) = u(x, y) + iv(x, y) to be analytic in its domain. Hence, show that f(z) = log z is analytic in its domain and find df/dz.
Q5. 1. (e) एक व्यक्ति को अपने उद्यान के लिए रसायन A, B तथा C की क्रमशः 24, 24 तथा 20 इकाई की आवश्यकता (e) है। उत्पाद P के प्रत्येक मर्तबान में रसायन A, B तथा C की क्रमशः 2, 4 और 1 इकाई हैं तथा उत्पाद Q के प्रत्येक मर्तबान में रसायन A, B तथा C की क्रमशः 2, 1 और 5 इकाई हैं। यदि P के एक मर्तबान का मूल्य ₹ 30 है तथा Q के एक मर्तबान का मूल्य ₹ 50 है, तब न्यूनतम खर्च तथा आवश्यकताओं की पूर्ति के लिए प्रत्येक उत्पाद के कितने मर्तबान खरीदे जाएँ? A person requires 24, 24 and 20 units of chemicals A, B and C respectively for his garden. Product P contains 2, 4 and 1 units of chemicals A, B and C respectively per jar and product Q contains 2, 1 and 5 units of chemicals A, B and C respectively per jar. If a jar of P costs ₹ 30 and a jar of Q costs ₹ 50, then how many jars of each should be purchased in order to minimize the cost
Q6. 2. Prove that a non-commutative group of order 2p, where p is an odd prime, must have a subgroup of order p.
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Frequently asked questions
What is the paper name and year?
Mathematics (Paper-II) for Civil Services Main Examination, 2023.
How many questions are in the paper?
There are eight questions in total, divided into two sections.
How many questions must be attempted?
Candidates must attempt five questions in all.
Which questions are compulsory?
Questions 1 and 5 are compulsory.
Are the questions bilingual?
Yes, the paper is printed in Hindi and English.
What determines the marks for each question?
The number of marks carried by a question/part is indicated against it on the paper.