Indian Forest Service 2019 Mathematics Paper-I Question Paper PDF

Central Government Jobs Other Jobs 2019

  • Year 2019
  • Conducted By UPSC
  • Questions 8
  • Maximum Marks 200
  • Duration Three Hours
  • Languages English

Exam Details

Detail Information
Examination Indian Forest Service (Main) Examination
Year 2019
Conducting Body UPSC
Paper Mathematics Paper-I
Subject Mathematics
Duration Three Hours
Maximum Marks 200
Number of Questions 8
Question Type Descriptive / Subjective

This is the Mathematics Paper-I from the Indian Forest Service (Main) Examination 2019, conducted by UPSC. The paper is designed to test candidates' in-depth knowledge of mathematics, with a duration of three hours and a maximum of 200 marks. It comprises eight questions, of which five are to be attempted, including compulsory questions 1 and 5. Candidates must select at least one question from each of the two sections, A and B. This paper is crucial for aspirants aiming for a career in the Indian Forest Service, providing a clear understanding of the expected difficulty and subject coverage.

Major Topics Covered

  • Linear Operators
  • Eigenvalues and Eigenvectors
  • Real Symmetric Matrices
  • Cylinders and Paraboloids
  • Rolle's Theorem
  • Mean Value Theorem
  • Coordinate Geometry
  • Extreme Values of Functions
  • Singular Matrices
  • Cube Diagonals

Why This Paper is Important

  • Useful for Indian Forest Service (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

  • Indian Forest Service (Main) Examination 2020 Mathematics Paper-I
  • Indian Forest Service (Main) Examination 2019 General Studies Paper-I
  • Indian Forest Service (Main) Examination 2019 Optional Subject Paper
  • Indian Forest Service 2019 Mathematics Paper-I Answer Key
  • Indian Forest Service Mathematics Syllabus
  • UPSC Main Exam Syllabus
  • Indian Forest Service Exam Pattern
  • UPSC Main Exam Pattern

Instructions

  • There are EIGHT questions in all, out of which FIVE are to be attempted.
  • Out of the remaining SIX questions, THREE are to be attempted selecting at least ONE question from each of the two Sections A and B.
  • The number of marks carried by a question/part is indicated against it.
  • Unless otherwise mentioned, symbols and notations have their usual standard meanings.
  • Assume suitable data, if necessary, and indicate the same clearly.
  • 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.
  • Answers must be written in ENGLISH only.

Questions (page 2)

Section A

Q1.

(a) Let T: \mathbbR3 → \mathbbR3 be a linear operator on \mathbbR3 defined by T(x, y, z) = (2y + z, x - 4y, 3x). Find the matrix of T in the basis (1, 1, 1), (1, 1, 0), (1, 0, 0).

(b) The eigenvalues of a real symmetric matrix A are -1, 1 and -2. The corresponding eigenvectors are (1)/(√(2))(-1\ 1\ 0)T, (0\ 0\ 1)T and (1)/(√(2))(-1\ -1\ 0)T respectively. Find the matrix A4.

(c) Find the volume lying inside the cylinder x2 + y2 - 2x = 0 and outside the paraboloid x2 + y2 = 2z, while bounded by xy-plane.

(d) Justify by using Rolle's theorem or mean value theorem that there is no number k for which the equation x3 - 3x + k = 0 has two distinct solutions in the interval [-1, 1].

(e) If the coordinates of the points A and B are respectively (bcosα, bsinα) and (acosβ, asinβ) and if the line joining A and B is produced to the point M(x, y) so that AM: MB = b: a, then show that x cos (α + β)/(2) + y sin (α + β)/(2) = 0.

Section A

Q2.

(a) Determine the extreme values of the function f(x, y) = 3x2 - 6x + 2y2 - 4y in the region {(x, y) in mathbb{R}^2 : 3x2 + 2y2 le 20}.

(b) Consider the singular matrix A = \beginbmatrix -1 & 3 & -1 & 1 \ -3 & 5 & 1 & -1 \ 10 & -10 & -10 & 14 \ 4 & -4 & -4 & 8 \endbmatrix. Given that one eigenvalue of A is 4 and one eigenvector that does not correspond to this eigenvalue 4 is (1\ 1\ 0\ 0)T. Find all the eigenvalues of A other than 4 and hence also find the real numbers p, q, r that satisfy the matrix equation A4 + pA3 + qA2 + rA = 0.

(c) A line makes angles α, β, γ, δ with the four diagonals of a cube. Show that cos2 α + cos2 β + cos2 γ + cos2 δ = (4/3).

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Indian Forest Service 2019 Mathematics Paper-I question paper page 1 instructions scan PDF download UPSC IFS Main Exam
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Frequently asked questions

What is the name of the examination?

The examination is the Indian Forest Service (Main) Examination.

What is the year of this question paper?

This question paper is from the year 2019.

Which board conducts the Indian Forest Service Examination?

The Union Public Service Commission (UPSC) conducts the Indian Forest Service Examination.

What is the subject of this paper?

This paper is for Mathematics, specifically Paper-I.

What is the maximum marks for Mathematics Paper-I?

The maximum marks for Mathematics Paper-I is 200.

What is the time allowed to complete the paper?

The time allowed to complete the paper is Three Hours.

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