ACARA Year 9 Maths Worksheet — Australian Grade 9 (Age 14)

Instructions

Read each question carefully and show your working in the space provided. Calculators are permitted for Section 2. Make sure to round your answers as instructed.

Section 1: Number & Algebra - Indices and Surds

1. Simplify the following expressions, leaving your answer in index form:
   1. m⁴ × m⁷
   2. (p⁵)³
   3. 12k⁸ ÷ 4k²


2. Evaluate the following (find their numerical value):
   1. 5^-2
   2. 64^1/2
   3. 27^1/3


3. Simplify the following surd: √48

Section 2: Measurement & Geometry - Pythagoras & Trigonometry

For this section, you may use a calculator. Round answers to two decimal places where necessary.

4. In a right-angled triangle, the two shorter sides are 5 cm and 12 cm. Find the length of the hypotenuse.


5. A right-angled triangle has a hypotenuse of 10 m and one shorter side of 6 m. What is the length of the other shorter side?


6. In a right-angled triangle, an angle is 35°. The side opposite this angle is 8 cm long. Find the length of the hypotenuse.


7. A right-angled triangle has an adjacent side of 15 cm and a hypotenuse of 20 cm. Find the size of the angle between these two sides (to the nearest degree).

Section 3: Algebra - Equations & Factorisation

8. Solve the following equation for x:
   5(x - 3) = 2x + 6


9. Factorise the following quadratic expression:
   x² + 7x + 10


10. Factorise this 'difference of two squares' expression:
    a² - 81

Section 4: Statistics

11. Consider the following set of numbers: 5, 9, 2, 7, 5, 10, 4
    Calculate the:
    1. Mean
    2. Median
    3. Mode

Answer Key

1. 1. m¹¹
   2. p¹⁵
   3. 3k⁶
2. 1. 1/25 or 0.04
   2. 8
   3. 3
3. √48 = √(16 × 3) = 4√3
4. c² = a² + b² → c² = 5² + 12² = 25 + 144 = 169. So, c = √169 = 13 cm.
5. a² = c² - b² → a² = 10² - 6² = 100 - 36 = 64. So, a = √64 = 8 m.
6. sin(θ) = Opposite/Hypotenuse → sin(35°) = 8/H → H = 8 / sin(35°) → H ≈ 13.95 cm.
7. cos(θ) = Adjacent/Hypotenuse → cos(θ) = 15/20 = 0.75 → θ = cos^-1(0.75) → θ ≈ 41°.
8. 5x - 15 = 2x + 6 → 3x = 21 → x = 7.
9. (x + 5)(x + 2)
10. (a - 9)(a + 9)
11. Ordered data: 2, 4, 5, 5, 7, 9, 10
    1. Mean: (2+4+5+5+7+9+10) / 7 = 42 / 7 = 6
    2. Median: The middle number is 5.
    3. Mode: The most frequent number is 5.