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AP Intermediate 2nd Year Probability & Statistics — practice questions

Explore 8 free Probability & Statistics questions in AP Intermediate 2nd Year Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Probability & Statistics questions and exercises with answers

Showing questions 1–8 of 8.

Simplify: arcsin(sin(x)) for x ∈ [π/2, π]:Read the answer and worked solution →Solve 2 arctan(x) = π/3:Read the answer and worked solution →Range of f(x) = arcsin(x) + arccos(x):Read the answer and worked solution →d/dx [arctan(x²)] =Read the answer and worked solution →arctan(x) - arctan(y) = (for xy > -1):Read the answer and worked solution →cos(2 arctan(x)) =Read the answer and worked solution →d/dx arcsin(sqrt(x)) =Read the answer and worked solution →Number of solutions to arctan(x) = π/3 in real x:Read the answer and worked solution →

Worked practice questions

Simplify: arcsin(sin(x)) for x ∈ [π/2, π]:

  1. -x
  2. π - x (since sin(π-x) = sin x and π - x ∈ [0, π/2])
  3. x
  4. π + x
Show answer and solution

Correct answer: B. π - x (since sin(π-x) = sin x and π - x ∈ [0, π/2])

For x ∈ [π/2, π]: sin x = sin(π - x), and π - x ∈ [0, π/2] which IS in arcsin's principal range. **So arcsin(sin x) = π - x.**

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Solve 2 arctan(x) = π/3:

  1. x = 0
  2. x = tan(π/6) = 1/sqrt(3)
  3. x = 1
  4. x = sqrt(3)
Show answer and solution

Correct answer: B. x = tan(π/6) = 1/sqrt(3)

arctan(x) = π/6 → x = tan(π/6) = 1/√3 ≈ 0.577.

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Range of f(x) = arcsin(x) + arccos(x):

  1. Empty
  2. [0, π]
  3. {π/2} (constant function, equals π/2 always on [-1, 1])
  4. [-π/2, π/2]
Show answer and solution

Correct answer: C. {π/2} (constant function, equals π/2 always on [-1, 1])

For all x ∈ [-1, 1]: arcsin(x) + arccos(x) = π/2 (constant identity). **Hence range is single point {π/2}.**

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