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CG PET Vectors & 3D Geometry — practice questions — page 3

Explore 108 free Vectors & 3D Geometry questions in CG PET Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Vectors & 3D Geometry questions and exercises with answers

Showing questions 61–90 of 108.

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 →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 definition of conditional probabilityRead the answer and worked solution →True or false: conditional probabilityRead the answer and worked solution →Independent or mutually exclusive?Read the answer and worked solution →Match each rule to its nameRead the answer and worked solution →IndependenceRead the answer and worked solution →Order the steps for a Bayes' theorem questionRead the answer and worked solution →Complete the conditions for a partitionRead the answer and worked solution →True or false: independence versus exclusivityRead the answer and worked solution →

Worked practice questions

Flip and pair: function with its principal range

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Matching pairs
  1. A
    sin⁻¹ x
    B
    [−π/2, π/2]
  2. A
    cos⁻¹ x
    B
    [0, π]
  3. A
    cosec⁻¹ x
    B
    [−π/2, π/2] − {0}
  4. A
    sec⁻¹ x
    B
    [0, π] − {π/2}
  5. A
    tan⁻¹ x
    B
    (−π/2, π/2)
  6. A
    cot⁻¹ x
    B
    (0, π)
Learning the six ranges as pairs is the fastest route through most exam questions on this chapter. Source: syllabus Maths Part I, Ch 2 Inverse Trigonometric Functions.

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Which principal ranges sit around zero, and which run 0 to π?

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Items
  1. Item
    sin⁻¹ x
    Category
    Centred on 0 (−π/2 to π/2)
  2. Item
    cosec⁻¹ x
    Category
    Centred on 0 (−π/2 to π/2)
  3. Item
    tan⁻¹ x
    Category
    Centred on 0 (−π/2 to π/2)
  4. Item
    cos⁻¹ x
    Category
    Runs 0 to π
  5. Item
    sec⁻¹ x
    Category
    Runs 0 to π
  6. Item
    cot⁻¹ x
    Category
    Runs 0 to π
Categories
  1. Item
    Centred on 0 (−π/2 to π/2)
  2. Item
    Runs 0 to π
sin⁻¹, cosec⁻¹ and tan⁻¹ are centred on zero, running from −π/2 to π/2. cos⁻¹, sec⁻¹ and cot⁻¹ instead run from 0 to π. Source: syllabus Maths Part I, Ch 2 Inverse Trigonometric Functions.

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Tap every function whose principal range has a point removed

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Questions
  1. Items
    1. Text
      cosec⁻¹ x removes 0
      Matches the category
      Yes
    2. Text
      sec⁻¹ x removes π/2
      Matches the category
      Yes
    3. Text
      sin⁻¹ x
      Matches the category
      No
    4. Text
      cos⁻¹ x
      Matches the category
      No
    5. Text
      tan⁻¹ x
      Matches the category
      No
    6. Text
      cot⁻¹ x
      Matches the category
      No
    Category
    Principal range has a point cut out
  2. Items
    1. Text
      tan⁻¹ x (−π/2, π/2)
      Matches the category
      Yes
    2. Text
      cot⁻¹ x (0, π)
      Matches the category
      Yes
    3. Text
      sin⁻¹ x [−π/2, π/2]
      Matches the category
      No
    4. Text
      cos⁻¹ x [0, π]
      Matches the category
      No
    Category
    Principal range is an OPEN interval
cosec⁻¹ drops 0 and sec⁻¹ drops π/2, because cosec and sec are undefined there. The other four ranges are unbroken. Source: syllabus Maths Part I, Ch 2 Inverse Trigonometric Functions.

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