UPSC Mains Math Paper I 2017 – Questions & Topics

Central Government Jobs Other Jobs 2017

  • Year 2017
  • Conducted By Union Public Service Commission (UPSC)
  • Questions 8
  • Maximum Marks 250
  • Duration Three Hours Maximum Marks: 250 प्रश्न-पत्र सम्बन्धी विशिष्ट अनुदेश कृपया प्रश्नों के उत्तर देने से पूर्व निम्नलिखित प्रत्येक अनुदेश को ध्यानपूर्वक पढ़ें । इसमें आठ प्रश्न हैं जो दो खंडों में विभाजित हैं तथा हिन्दी और अंग्रेजी दोनों में छपे 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
  • Languages Hindi & English

Exam Details

Detail Information
Examination UPSC Civil Services Main Examination
Year 2017
Conducting Body Union Public Service Commission (UPSC)
Paper Mathematics (Paper I)
Subject Mathematics
Duration Three Hours Maximum Marks: 250 प्रश्न-पत्र सम्बन्धी विशिष्ट अनुदेश कृपया प्रश्नों के उत्तर देने से पूर्व निम्नलिखित प्रत्येक अनुदेश को ध्यानपूर्वक पढ़ें । इसमें आठ प्रश्न हैं जो दो खंडों में विभाजित हैं तथा हिन्दी और अंग्रेजी दोनों में छपे 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
Maximum Marks 250
Number of Questions 8

This JSON provides SEO-ready metadata for UPSC Civil Services Main Mathematics Paper I, 2017. Page 1 contains the exam header, bilingual instructions, and paper structure. Page 2 ( repaired OCR ) lists eight main questions distributed as 1.(a)–1.(e) and 2.(a)–2.(c), spanning linear algebra (diagonalization, similarity, characteristic polynomial), multivariable calculus (integration over regions), analytic geometry (tangent plane of conicoid, distance between skew lines), and 3D geometry (volume under elliptic paraboloid, locus of sphere center, and plane-sphere tangency). The SEO data below aggregating topics, keywords, FAQs, and image metadata is designed for strong search visibility.

Major Topics Covered

  • Linear Algebra
  • Diagonalization
  • Similarity of matrices
  • Characteristic polynomial
  • Eigenvalues
  • Eigenvectors
  • Tangent plane
  • Conicoid
  • Skew lines
  • Distance between lines
  • Implicit differentiation
  • Double integrals
  • Volume calculations
  • Elliptic paraboloid
  • Planes and spheres
  • Locus problems
  • Coordinate geometry
  • Three-dimensional calculus
  • Matrix equations
  • Eigen decomposition

Why This Paper is Important

  • Useful for UPSC Civil Services Main Examination 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

  • पुस्तिका के मुखपृष्ठ पर अंकित 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 ।
  • प्रश्नॅ-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 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. SECTION 'A' खण्ड 'A' मान लीजिए
Let A = [[2, 2], [1, 3]]. Find a non-singular matrix P such that P^{-1}AP is a diagonal matrix.

Q2. 1.(b) Show that similar matrices have the same characteristic polynomial.
समरूप आव्यूहों के समान अभिलक्षणिक बहुपद होते हैं।

Q3. 1.
(c) Integrate the function f(x, y) = xy(x2 + y2) over the domain R: {-3 ≤ x2 - y2 ≤ 3,
1 ≤ xy ≤ 4}.

Q4. 1.(d) Find the equation of the tangent plane at point (1, 1, 1) to the conicoid 3x2 - y2 = 2z.
स्पर्श-तल का समीकरण निकालिए।

Q5. 1.(e) Find the shortest distance between the skew lines: (x-3)/3 = (8 - y)/1 = (z-3)/1 and (x+3)/-3 = (y+7)/2 = (z-6)/4.
विषमतलीय रेखाओं के बीच न्यूनतम-दूरी ज्ञात।

Q6. 2.(a) Find the volume of the solid above the xy-plane and directly below the portion of the elliptic paraboloid x2 + y2/4 = z which is cut off by the plane z = 9.
कटा हुआ भाग आयतन का निर्धारण करें।

Q7. 2.(b) एक plane passes through a fixed point (a,b,c) and cuts the axes at the points A, B, C respectively. Find the locus of the centre of the sphere which passes through the origin O and A, B, C.
उच्चारण: अक्षों पर कटे A, B, C और origin O के द्वारा निर्मित गोले के केंद्र का लोकेशन्स।

Q8. 2.
(c) स्पर्श करता है। Show that the plane 2x-2y+z+12=0 touches the sphere x2+y2+z2-2x-4y+2z-3=0. Find the point of contact.

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Frequently asked questions

What is the aim of 1.(a) in Page 2?

To find a non-singular matrix P such that P^{-1}AP is diagonal, i.e., diagonalize A using P's eigenvectors.

Do similar matrices share the characteristic polynomial?

Yes. Similar matrices have the same characteristic polynomial.

What region is used for the integration in 1.(c)?

The region R is defined by -3 ≤ x^2 - y^2 ≤ 3 and 1 ≤ xy ≤ 4; the integral is ∬_R xy(x^2 + y^2) dA.

How is the tangent plane determined for the conicoid in 1.(d)?

Compute the gradient of the surface equation at (1,1,1) to obtain the normal vector and write the tangent plane accordingly.

What is the geometric meaning of the distance in 1.(e)?

It is the shortest distance between two skew lines in space, found via a cross-product approach.

How is the volume under the elliptic paraboloid computed in 2.(a)?

V = ∬_D (x^2 + y^2/4) dA over D defined by x^2 + y^2/4 ≤ 9, using the plane z=9 as the cut-off.

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