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PUCET (Punjab) Number Theory — practice questions — page 3

Explore 90 free Number Theory questions in PUCET (Punjab) 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 61–90 of 90.

Complete the rule about restricting a domainRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Principal values worked in the chapterRead the answer and worked solution →Order the steps for finding a principal valueRead the answer and worked solution →True or false: principal valuesRead the answer and worked solution →Match each identity to the condition it needsRead the answer and worked solution →Assertion and reason: identitiesRead the answer and worked solution →Match each mathematician to the year your syllabus givesRead the answer and worked solution →Trace how trigonometry travelledRead the answer and worked solution →True or false: the historical noteRead the answer and worked solution →Complete the arc length relationRead the answer and worked solution →Match each identity to its nameRead the answer and worked solution →The Pythagorean identitiesRead the answer and worked solution →Odd function or even function?Read the answer and worked solution →Complete the degree and radian conversionRead the answer and worked solution →True or false: identities and parityRead the answer and worked solution →Match each sum or difference formula to its resultRead the answer and worked solution →Signs in the expansionRead the answer and worked solution →Which formula does this belong to?Read the answer and worked solution →True or false: periodicityRead the answer and worked solution →Complete the complementary angle rulesRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Order the steps: convert degrees to radiansRead the answer and worked solution →Flip and pair: identity with its valueRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Tap every correct identityRead the answer and worked solution →Radians and degreesRead the answer and worked solution →True or false: reading the identitiesRead the answer and worked solution →

Worked practice questions

Complete the rule about restricting a domain

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Missing parts
  1. Answer
    one-one and onto
    Choices
    1. one-one and onto
    2. continuous
    3. increasing
  2. Answer
    bijective
    Choices
    1. bijective
    2. periodic
    3. bounded
  3. Answer
    principal value branch
    Choices
    1. principal value branch
    2. natural domain
    3. range of definition
Complete the statement
An inverse exists only when a function is {0}. Trigonometric functions are not, so the domain is restricted until the function becomes {1}, and the agreed choice is called the {2}.
This sentence is the argument of the whole chapter in one line. Source: syllabus Maths Part I, Ch 2 Inverse Trigonometric Functions.

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Three of these are wrong. Find the one that holds.

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Questions
  1. Question
    Three of these statements about sin⁻¹ are false.
    Choices
    1. Item
      Its principal range is [−π/2, π/2]
      Correct
      Yes
    2. Explanation
      That is the range of cos⁻¹, not sin⁻¹.
      Item
      Its principal range is [0, π]
    3. Explanation
      That is (sin x)⁻¹. The −1 here means inverse function.
      Item
      It equals 1/sin x
    4. Explanation
      Sine only outputs [−1, 1], so the inverse accepts only that.
      Item
      It is defined for every real x
  2. Question
    Three of these statements about sec⁻¹ are false.
    Choices
    1. Item
      Its principal range is [0, π] − {π/2}
      Correct
      Yes
    2. Explanation
      sec is undefined at π/2, so that point must be removed.
      Item
      Its principal range is [0, π]
    3. Explanation
      That is where sec⁻¹ is NOT defined. Its domain is R − (−1, 1).
      Item
      Its domain is [−1, 1]
    4. Explanation
      cosec⁻¹ removes 0. sec⁻¹ removes π/2.
      Item
      It removes 0 from its range
Each wrong option is a real exam trap: a swapped range, a reciprocal confusion, or an unrestricted domain. Source: syllabus Maths Part I, Ch 2 Inverse Trigonometric Functions.

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Principal values worked in the chapter

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Questions
  1. Question
    Principal value of sin⁻¹(1/√2)
    Choices
    1. π/3
    2. π/6
    3. π/4
    Correct choice
    π/4
  2. Question
    Principal value of cot⁻¹(−1/√3)
    Choices
    1. 2π/3
    2. −π/3
    3. π/3
    Correct choice
    2π/3
  3. Question
    Why is cot⁻¹(−1/√3) not −π/3?
    Choices
    1. cot(−π/3) is undefined
    2. −π/3 is outside (0, π)
    3. −π/3 is not a real number
    Correct choice
    −π/3 is outside (0, π)
Both values are computed in full in your syllabus Examples 1 and 2: sin⁻¹(1/√2) = π/4 because sin(π/4) = 1/√2 and π/4 lies in [−π/2, π/2] cot⁻¹(−1/√3) = 2π/3 because cot(2π/3) = −1/√3 and 2π/3 lies in (0, π). Source: syllabus Maths Part I, Ch 2 Inverse Trigonometric Functions.

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