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Questions › Mathematics › Real analysis

Real analysis: previous year questions

Asked 38 times in 5 exams across 15 years. Newest first.

Practise Real analysis, easy to hard →

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  • Topology of the real line: open, closed and compact sets 29
  • Metric spaces and norms 5
  • Supremum, infimum and epsilon arguments 4

Topology of the real line: open, closed and compact sets

Which sets are open, closed or compact and how unions and intersections behave.

  1. CUET-PG 2025 Q67Match List-I (Types of Interval for variable x) with List-II (Mathematical Representation, given a,b ∈ ℝ with b>a): List-I: (A) Closed Interval; (B)…
  2. UGC-NET 2023 Q62Consider the following statements: (A) The union of a finite number of closed sets is a closed set. (B) The empty set is an open set but not a…
  3. UoH 2021 Q17Consider an economy with 2-goods. The set of all possible positive relative prices is given as: P = {(p₁, p₂) | p₁, p₂ > 0, and p₁ + p₂ = 1}.…
  4. DSE 2020 Q33The interval [0,∞) can be expressed as
  5. DSE 2020 Q36Let B be the collection of sets E⊆ℝ such that for every x∈ E there are real numbers a<b with x∈(a,b)⊆ E. Let C be the collection of sets E⊆ℝ such…
  6. DSE 2020 Q38Suppose the distance between x,y∈ℝ is |x-y| and f:ℝ→ℝ is continuous. If E is an open subset of ℝ, then {x∈ℝ| f(x)∈ E} is
  7. UoH 2020 Q75Consider set X = {1/n | n is a natural number}. Then X is a,
  8. DSE 2019 Q48The set (0,∞) can be expressed as:
  9. DSE 2018 Q7The set [0,1]
  10. DSE 2018 Q48Let ℝ be the set of real numbers. A subset E of ℝ is said to be open if, for every x ∈ E, there exists r > 0 such that (x - r, x + r) is a subset of…
  11. DSE 2017 Q17The interval (0,1) can be expressed as
  12. DSE 2017 Q18The interval [0,1] can be expressed as
  13. DSE 2016 Q13If f:ℝ→ℝ is a continuous function (ℝ is the set of real numbers) and E is an open subset of ℝ, then the set {x∈ℝ| f(x)∈ E} is necessarily
  14. DSE 2015 Q33The set (0,1) can be expressed as
  15. DSE 2015 Q34The set [0,1] can be expressed as
  16. DSE 2011 Q27Consider a Society of individuals who may belong to sets called Clubs and Tribes, satisfying: the entire Society is a Tribe; the empty subset is a…
  17. DSE 2011 Q28(Same Clubs-and-Tribes setup as the previous question.) The union of a collection of Clubs is necessarily
  18. DSE 2011 Q29(Same Clubs-and-Tribes setup.) Which of the following statements is necessarily true?
  19. DSE 2011 Q30(Same Clubs-and-Tribes setup.) Given a Club and a Tribe, the set of individuals who belong to the given Tribe but not to the given Club necessarily…
  20. DSE 2007 Q44The set ⋂ₙ₌₁^∞(-1-n⁻¹, 1+n⁻¹) is identical to
  21. DSE 2007 Q45The set ⋃ₙ₌₁^∞[-1+n⁻¹, 1-n⁻¹] is identical to
  22. DSE 2006 Q54Consider a Society consisting of individuals who may belong to various sets called Families and/or Gangs, with the rules: the entire Society is a…
  23. DSE 2006 Q55For the Society in which Families are closed under arbitrary intersection and finite union (with the whole Society and the empty set as Families)…
  24. DSE 2006 Q56For the Society in which Families are closed under arbitrary intersection and finite union (with the whole Society and the empty set as Families)…
  25. DSE 2006 Q57For the Society in which Families are closed under arbitrary intersection and finite union and a Gang is the complement of a Family: suppose we are…
  26. DSE 2005 Q5Consider a Society consisting of individuals who may belong to sets called Clubs and/or Tribes, with the rules: the entire Society is a Club; the…
  27. DSE 2005 Q6For the Society in which Clubs are closed under arbitrary union and finite intersection (with the whole Society and empty set as Clubs) and a Tribe…
  28. DSE 2005 Q7For the Society in which Clubs are closed under arbitrary union and finite intersection and a Tribe is the complement of a Club: which of the…
  29. DSE 2005 Q8For the Society in which Clubs are closed under arbitrary union and finite intersection and a Tribe is the complement of a Club: suppose we are…

Metric spaces and norms

Distance functions and the properties that make them a metric.

  1. DSE 2018 Q4Let ℝ be the set of real numbers and let D be the set of functions d: ℝ × ℝ → ℝ that satisfy, for all x, y, z ∈ ℝ: (i) d(x,y) ≥ 0; (ii) d(x,y) = 0…
  2. DSE 2014 Q25Let d((x₁,x₂),(y₁,y₂))=max {|x₁-y₁|,|x₂-y₂|} be the distance between two points on the plane. Then the circle centred at the origin with radius 1 is
  3. JNU 2011 Q7Suppose we know that |a| < 3. Which of the following conditions is enough to imply that |b| < 5?
  4. DSE 2010 Q33Consider the statements for x,y∈ℜ: (i) |x|-|y|≤ |x-y|; (ii) | x|-|y | =|x-y|. In general,
  5. DSE 2007 Q46For x,y∈ℜⁿ, let d(x,y)=max {|xᵢ-yᵢ | i=1,…,n}. Which of the following relations holds for all x,y,z∈ℜⁿ?

Supremum, infimum and epsilon arguments

Greatest lower and least upper bounds, and proofs that work with arbitrarily small tolerances.

  1. DSE 2020 Q32Suppose u:ℝ→ℝ₊ is strictly increasing with supremum (least upper bound) α∈ℝ. Then the function x↦ u(x)/(α-u(x)) is
  2. JNU 2017 Q49a and b are real numbers. If a < b + ε for all real numbers ε > 0, then show that a ≤ b.
  3. JNU 2010 Q51If x < y + ε, for all ε > 0, then
  4. JNU 2010 Q80Consider the set A = {x | 0 < x < 1}. What is the minimum number that belongs to set A?