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Bihar Board Class 12 (Commerce) Calculus — practice questions — page 2

Explore 42 free Calculus questions in Bihar Board Class 12 (Commerce) Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Calculus questions and exercises with answers

Showing questions 31–42 of 42.

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 →Match each calculus ideaRead the answer and worked solution →Tap every use of the derivativeRead the answer and worked solution →True or false: calculusRead the answer and worked solution →

Worked practice questions

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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Order the steps for finding a principal value

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Steps
  1. Let y be the inverse value you are asked for
  2. Rewrite it as the direct trigonometric equation
  3. Find an angle that satisfies that equation
  4. Check the angle lies inside the principal value branch
  5. If it does not, shift to the equivalent angle that does
This is the method your syllabus uses in Examples 1 and 2: name the value, convert to the direct function, find the angle then check it lies in the principal branch. Source: syllabus Maths Part I, Ch 2 Inverse Trigonometric Functions.

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