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Grade 12Common Core (CCSS) · Math: geometry

Math · Grade 12

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Triangle area, sine and cosine rules · Grade 12

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Worked examples

Derivatives of polynomials

  1. Differentiate term by term: axⁿ becomes n·a·xⁿ⁻¹, constants become 0.
  2. Put the x-value into f′(x).
  3. Example: f(x) = x³ + 2x gives f′(x) = 3x² + 2, so f′(1) = 5.

Arithmetic sequences

  1. Each term grows by d.
  2. aₙ = a₁ + (n − 1)·d; sum = n·(a₁ + aₙ)/2.
  3. Example: a₁ = 3, d = 4 gives a₁₀ = 3 + 9·4 = 39.

Definite integrals

  1. Find an antiderivative: axⁿ becomes a·xⁿ⁺¹/(n + 1).
  2. Calculate F(upper limit) − F(0).
  3. Example: ∫₀² 2x dx = [x²] = 4 − 0 = 4.

Trigonometry in right triangles

  1. sin v = opposite/hypotenuse, cos v = adjacent/hypotenuse.
  2. Multiply the hypotenuse by sin v or cos v.
  3. Example: 10 · sin 30° = 5 cm.

Logarithms

  1. log_b(n) asks: which power of b gives n?
  2. Write n as a power of b.
  3. Example: log₂(8) = 3 because 2³ = 8.

Exponential growth and decay

  1. The growth factor is 1 + p/100 (or 1 − p/100 for a decrease).
  2. Multiply the start value by the factor to the power of the years.
  3. Example: 1,000 · 1.05³ ≈ 1,158.

Polynomial equations

  1. x³ = a: take the cube root of both sides.
  2. x³ = kx: move everything to one side and factor out x, so x = 0 is one solution.
  3. (x − a)(x − b)(x − c) = 0: the zero-product property gives x = a, x = b and x = c.

The circle's equation

  1. The circle x² + y² = r² has centre at the origin and radius r.
  2. The radius is the distance between centre (h, k) and a point (a, b): r = √((a − h)² + (b − k)²).
  3. Example: centre (0, 0) through (3, 4) gives r = √(9 + 16) = 5.

Statistics: mean, median and standard deviation

  1. Mean = sum / count. Median = middle value when sorted.
  2. Range = largest − smallest.
  3. Standard deviation: square each deviation from the mean, average them and take the square root.

Vectors

  1. Length: |(a, b)| = √(a² + b²).
  2. Sum: add the x- and y-coordinates separately.
  3. Dot product: u · v = u₁v₁ + u₂v₂. Example: (2, 3) · (4, −1) = 8 − 3 = 5.
1.

Find f′(-2): f(x) = 5x² − 4x − 5

2.

Find the term: a₁ = -10, d = 4. a₇ = ?

3.

Calculate the integral: ∫₀¹ (6x + 1) dx =

4.

Find the opposite side: hypotenuse 10 cm, angle 60° (1 decimal)

5.

Calculate: log(1,000,000) =

6.

Find the value: 5000 grows by 10 % per year for 2 years (whole number)

7.

Find the value: 10,000 grows by 10 % per year for 2 years (whole number)

8.

Solve: x³ = 216

9.

Find f′(0): f(x) = 4x² − 6x + 6

10.

The circle x² + y² = 100. Find the radius.

11.

Calculate the integral: ∫₀⁴ (4x) dx =

12.

Find the mean and median: 2, 11, 3, 20, 19

13.

Calculate: log(100,000) =

14.

Find the length of the vector: v = (-6, 8)

15.

Find the value: 2000 grows by 2 % per year for 2 years (whole number)

16.

Calculate the integral: ∫₀² (8x) dx =

17.

Find f′(4): f(x) = 3x² − 9x − 1

18.

Calculate the integral: ∫₀³ (4x + 6) dx =

19.

Calculate the integral: ∫₀³ (4x + 6) dx =

20.

Calculate the integral: ∫₀⁴ (6x + 6) dx =

Answer key

  1. 1. -24
  2. 2. 14
  3. 3. 4
  4. 4. 8.7 cm
  5. 5. 6
  6. 6. 6050
  7. 7. 12,100
  8. 8. x = 6
  9. 9. -6
  10. 10. r = 10
  11. 11. 32
  12. 12. 11; median 11
  13. 13. 5
  14. 14. |v| = 10
  15. 15. 2081
  16. 16. 16
  17. 17. 15
  18. 18. 36
  19. 19. 36
  20. 20. 72

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Area
Geometry
Exercise type
Triangle area, sine and cosine rules
Grade
12 (ages 17–18)
Difficulty
Easy
Number of tasks
20
Task format
Calculations
Paper size
US Letter
1. Choose content
Area
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Practise trigonometry in non-right triangles.

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