AP problems

412 problems for AP practice, each with a worked solution. Try each one on paper first, then check your answer.

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Type:AnyConceptualNumerical

1.

Find the exact value of sin 105°. Then give it to three decimal places. New

High School (11–12) · Compound and Multiple Angles, Trigonometric IdentitiesBoard Exam, AP, A-Level, IBStemPy Lab

Easy

2.

sin A = 4/5 with π/2 < A < π, and cos B = -5/13 with π < B < 3π/2. New

High School (11–12) · Compound and Multiple Angles, Trigonometric IdentitiesBoard Exam, AP, A-Level, IBStemPy Lab

Medium

3.

cos x = -5/13, and x lies between π/2 and π. New

High School (11–12) · Compound and Multiple Angles, Trigonometric IdentitiesBoard Exam, AP, A-Level, IBStemPy Lab

Medium

4.

The point P on the unit circle at angle x has cos x = -20/29, and P lies below the x-axis. New

High School (11–12) · Trigonometric Functions and Graphs, Trigonometric IdentitiesBoard Exam, AP, A-Level, IBStemPy Lab

Easy

5.

An angle x lies between π/2 and π, and tan x = -3/4. Find sin x and cos x. New

High School (11–12) · Trigonometric Functions and Graphs, Trigonometric IdentitiesBoard Exam, AP, A-Level, IBStemPy Lab

Medium

6.

For which values of x with 0 ≤ x < 2π is sin x = cos x? New

High School (11–12) · Trigonometric Functions and Graphs, Trigonometric IdentitiesBoard Exam, AP, A-Level, IBStemPy Lab

Hard

7.

Find cos(-23π/4) exactly, and as a decimal to 3 places. New

High School (11–12) · Trigonometric Functions and GraphsBoard Exam, APStemPy Lab

Easy

8.

An angle x has sec x < 0 and cot x > 0. In which quadrant does x lie? New

High School (11–12) · Trigonometric Functions and GraphsBoard Exam, APStemPy Lab

Easy

9.

The depth of water in a harbour is h = 6 + 2.5 sin (π t)/6 metres, t hours after midnight. New

High School (11–12) · Trigonometric Functions and GraphsBoard Exam, AP, A-Level, IBStemPy Lab

Medium

10.

For y = -2 cos(3x + π), find the amplitude, the period (to 3 decimal places) and the range. New

High School (11–12) · Trigonometric Functions and GraphsBoard Exam, AP, A-Level, IBStemPy Lab

Medium

11.

The areas of five circles were measured as 3.18 cm², 12.49 cm², 28.11 cm², 50.39 cm² and 78.42 cm²…

High School (11–12) · Functions as Models of DataAPStemPy Lab

Medium

12.

A parabola fitted to stopping distances measured between 20 and 80 km/h is d = 0.006v² + 0.15v +…

High School (11–12) · Functions as Models of DataAPStemPy Lab

Medium

13.

Let f(x) = x + 5 and g(x) = 4x - 1. Find (g ° f)(x) and (f ° g)(x), and their values at x = 3.

High School (11–12) · Composite and Inverse FunctionsBoard Exam, AP, A-Level, IBStemPy Lab

Easy

14.

Let f(x) = 1/(x + 2) and g(x) = √x. Find (g ° f)(x) and (f ° g)(x), with their domains.

High School (11–12) · Composite and Inverse FunctionsBoard Exam, AP, A-Level, IBStemPy Lab

Hard

15.

Let f(x) = 2x + 3 and g(x) = ax + b. For which a and b is g ° f = f ° g?

High School (11–12) · Composite and Inverse FunctionsBoard Exam, AP, A-Level, IBStemPy Lab

Hard

16.

Write h(x) = 1/((3x - 1)²) as a composite of three simple functions.

High School (11–12) · Composite and Inverse FunctionsBoard Exam, APStemPy Lab

Easy

17.

Find the inverse of f(x) = (2x + 5)/(x - 3). Give the domain and the range of f.

High School (11–12) · Composite and Inverse FunctionsBoard Exam, AP, A-Level, IBStemPy Lab

Medium

18.

Show that f(x) = 4 - 3x is one-one. Show that g(x) = x² + 6x is not one-one on ℝ.

High School (11–12) · Composite and Inverse FunctionsBoard Exam, AP, A-Level, IBStemPy Lab

Medium

19.

Let f(x) = x - 4 and g(x) = 2x. Find (g ° f)⁻¹, and check that it equals f⁻¹ ° g⁻¹ but not g⁻¹ °…

High School (11–12) · Composite and Inverse FunctionsBoard Exam, AP, A-Level, IBStemPy Lab

Medium

20.

A function is defined by f(x) = 2x + 1, x < 0; x², 0 ≤ x ≤ 3; 12 - x, x > 3.

High School (11–12) · Graphs of FunctionsBoard Exam, APStemPy Lab

Medium

21.

Describe how to get the graph of y = -3(x + 2)² + 4 from the graph of y = x².

High School (11–12) · Graphs of FunctionsBoard Exam, APStemPy Lab

Medium

22.

The point (3, 4) is on the graph of y = f(x).

High School (11–12) · Graphs of FunctionsBoard Exam, AP, A-Level, IBStemPy Lab

Medium

23.

Describe how to get the graph of y = √(9 - x) from the graph of y = √x.

High School (11–12) · Graphs of FunctionsBoard Exam, AP, A-Level, IBStemPy Lab

Hard

24.

Describe the graph of y = 1/(x + 1) - 2. Give the asymptotes, the domain and the range.

High School (11–12) · Graphs of FunctionsBoard Exam, AP, A-Level, IBStemPy Lab

Medium

25.

Say whether each function is even, odd or neither. 1/x³; x⁴ + x²; x² + x; x|x|

High School (11–12) · Graphs of FunctionsBoard Exam, AP, IBStemPy Lab

Easy