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UPCET / UPSEE Number Theory — practice questions — page 2

Explore 90 free Number Theory questions in UPCET / UPSEE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Number Theory questions and exercises with answers

Showing questions 31–60 of 90.

Equivalence relation requires:Read the answer and worked solution →For sets A, B: A × B = B × A iff:Read the answer and worked solution →Range of f(x) = sin(x) from R to R:Read the answer and worked solution →|A| = 5, |B| = 3. Number of relations from A to B:Read the answer and worked solution →For f: A → B and g: B → C, (g ∘ f)⁻¹ =Read the answer and worked solution →Function f(x) = (x-1)/(x+2) on R - {-2}. f is bijection from R - {-2} to R - {?}:Read the answer and worked solution →Equivalence classes of 'congruent mod 3' on Z:Read the answer and worked solution →Number of one-one and onto (bijective) functions from {1,2,3} to {1,2,3}:Read the answer and worked solution →For function f(x) = ax + b (a ≠ 0), inverse is:Read the answer and worked solution →For f: R → R, f(x) = x³, f is:Read the answer and worked solution →Distributive law: A ∩ (B ∪ C) =Read the answer and worked solution →Inverse image f⁻¹(B) under f: X → Y for B ⊆ Y:Read the answer and worked solution →Number of equivalence relations on set {a, b, c}:Read the answer and worked solution →Match the trig identity to its resultRead the answer and worked solution →Sort each inverse function into its domainRead the answer and worked solution →Tap every function whose domain is [−1, 1]Read the answer and worked solution →Match each inverse function to its domainRead the answer and worked solution →Match each inverse function to its principal value branchRead the answer and worked solution →Flip and pair: function with its principal rangeRead the answer and worked solution →Which principal ranges sit around zero, and which run 0 to π?Read the answer and worked solution →Tap every function whose principal range has a point removedRead the answer and worked solution →Name the principal value branchRead the answer and worked solution →Domain checkRead the answer and worked solution →Put the reasoning in order: why an inverse needs a restricted domainRead the answer and worked solution →True or false: existence of inversesRead the answer and worked solution →True or false: reading the tableRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Complete the summary tableRead the answer and worked solution →

Worked practice questions

Equivalence relation requires:

  1. Just reflexive
  2. REFLEXIVE + SYMMETRIC + TRANSITIVE
  3. Just transitive
  4. Anti-symmetric
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Correct answer: B. REFLEXIVE + SYMMETRIC + TRANSITIVE

**Equivalence relation:** aRa (reflexive), aRb ⇒ bRa (symmetric), aRb ∧ bRc ⇒ aRc (transitive). Partitions set into disjoint equivalence classes. e.g., 'congruence mod n', 'same age'.

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For sets A, B: A × B = B × A iff:

  1. Same cardinality
  2. Never
  3. A = B (or one of them empty)
  4. Always
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Correct answer: C. A = B (or one of them empty)

A × B and B × A consist of ordered pairs. A × B = B × A iff every (a, b) in A × B also belongs to B × A. Requires A = B (or both empty). **Hence Cartesian product is generally NOT commutative.**

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Range of f(x) = sin(x) from R to R:

  1. R
  2. Random
  3. [0, 1]
  4. [-1, 1] (sine bounded)
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Correct answer: D. [-1, 1] (sine bounded)

sin x ∈ [-1, 1] for all real x. f is many-one (periodic), so not bijection from R to R. Restricting domain to [-π/2, π/2] makes it bijection [-π/2, π/2] → [-1, 1], inverse = arcsin.

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