Year 11 General Maths — Units 1 & 2 Combined Review

25 questions spanning all Unit 1 and Unit 2 topics: Consumer Arithmetic, Shape and Measurement, Linear Equations, Applications of Linear Equations, Trigonometry, Matrices, and Univariate Data. Aligned to QCAA General Mathematics 2025.

Fluency
Consumer Arithmetic: A refrigerator has a marked price of $1 490. It is offered for sale with a 15% discount. Calculate the sale price.

Discount = 15% × $1 490 = 0.15 × 1490 = $223.50

Sale price = $1 490 − $223.50 = $1 266.50
Fluency
Consumer Arithmetic: Caleb earns a weekly wage of $1 280 and pays 32.5% income tax on his earnings. Calculate his weekly after-tax income (take-home pay).

Tax = 32.5% × $1 280 = $416

Take-home = $1 280 − $416 = $864
Fluency
Shape and Measurement: A rectangular swimming pool is 12 m long, 6 m wide, and 1.8 m deep. Calculate the volume of water needed to fill it. Give your answer in kilolitres (1 kL = 1 m³).

Volume = length × width × depth = 12 × 6 × 1.8 = 129.6 m³ = 129.6 kL
Fluency
Linear Equations: Solve for x: 3x − 7 = 2(x + 4)

3x − 7 = 2x + 8

3x − 2x = 8 + 7

x = 15
Fluency
Consumer Arithmetic — Compound Interest: An investment of $8 000 earns compound interest at 4.5% p.a. for 3 years. Calculate the final value using A = P(1 + r)ⁿ.

A = 8 000 × (1 + 0.045)³ = 8 000 × (1.045)³

(1.045)³ = 1.045 × 1.045 × 1.045 ≈ 1.14117

A = 8 000 × 1.14117 ≈ $9 129.34
Understanding
Shape and Measurement — Similarity: Two similar triangles have corresponding sides in the ratio 3 : 5. The area of the smaller triangle is 27 cm². Find the area of the larger triangle.

For similar figures, the ratio of areas = (ratio of sides)².

Area ratio = (3/5)² = 9/25

Larger area = 27 × (25/9) = 27 × 25/9 = 3 × 25 = 75 cm²
Understanding
Linear Equations — Graphing: Find the gradient and y-intercept of the line 4x − 2y + 6 = 0. Sketch a brief description of the line (where it crosses axes).

Rearrange: −2y = −4x − 6 → y = 2x + 3

Gradient = 2 (rises 2 units for every 1 unit across)

y-intercept = 3 (crosses y-axis at (0, 3))

x-intercept: set y = 0: 0 = 2x + 3 → x = −1.5, so crosses x-axis at (−1.5, 0).
Understanding
Consumer Arithmetic — Comparison: Two savings accounts both hold $5 000 for 2 years. Account A earns 6% p.a. simple interest. Account B earns 5.8% p.a. compound interest. Which account grows to a larger value?

Account A (simple): I = Prn = 5000 × 0.06 × 2 = $600. Total = $5 600

Account B (compound): A = 5000 × (1.058)² = 5000 × 1.11936 ≈ $5 596.84

Account A yields more ($5 600 vs $5 596.84). For short time periods at similar rates, simple interest can slightly outperform compound interest.
Understanding
Shape and Measurement — Surface Area: A closed cylinder has radius 5 cm and height 12 cm. Calculate the total surface area. (Use π ≈ 3.14159)

Total SA = 2πr² + 2πrh = 2π(5)² + 2π(5)(12)

= 2π(25) + 2π(60) = 50π + 120π = 170π

= 170 × 3.14159 ≈ 534.07 cm²
Understanding
Linear Equations — Simultaneous: Solve the system:
3x + 2y = 16
x − y = 2

From equation 2: x = y + 2. Substitute into equation 1:

3(y + 2) + 2y = 16 → 3y + 6 + 2y = 16 → 5y = 10 → y = 2

x = 2 + 2 = 4

x = 4, y = 2. Check: 3(4)+2(2)=16 ✓ and 4−2=2 ✓
Fluency
Applications of Linear Equations — Break-even: A business has fixed costs of $2 400 per month and variable costs of $18 per unit. It sells each unit for $30. Write the cost and revenue functions, then find the break-even quantity.

Cost function: C(x) = 2400 + 18x

Revenue function: R(x) = 30x

Break-even: 30x = 2400 + 18x → 12x = 2400 → x = 200 units
Fluency
Trigonometry: In a right-angled triangle, the hypotenuse is 13 cm and one leg is 5 cm. Find the angle opposite the 5 cm side. Give your answer in degrees to 1 d.p.

sin(θ) = opposite/hypotenuse = 5/13

θ = sin⁻¹(5/13) = sin⁻¹(0.3846...) ≈ 22.6°
Fluency
Matrices: Find AB where A =

2 1

3 0

and B =

1

4

A is 2×2, B is 2×1. AB is defined (2×1 result).

Row 1: 2(1)+1(4) = 6 Row 2: 3(1)+0(4) = 3

AB =

6

3
Fluency
Univariate Data: Find the mean, median and mode of: 5, 8, 6, 8, 9, 4, 8, 7, 6, 9

Ordered: 4, 5, 6, 6, 7, 8, 8, 8, 9, 9

Mean = 70/10 = 7

Median = (7+8)/2 = 7.5

Mode = 8 (appears 3 times)
Understanding
Applications of Linear Equations — Piecewise: A mobile plan charges $0.30 per minute for the first 100 minutes per month, then $0.15 per minute for all additional minutes. Write the piecewise function C(x) for total cost, where x is the number of minutes used. Calculate C(80) and C(150).

C(x) = {0.30x for 0 ≤ x ≤ 100
30 + 0.15(x − 100) for x > 100}

C(80) = 0.30 × 80 = $24.00

C(150) = 30 + 0.15(150 − 100) = 30 + 0.15(50) = 30 + 7.50 = $37.50
Understanding
Trigonometry — Sine Rule: In triangle PQR, angle P = 42°, angle Q = 65°, and side p = 18 cm (opposite angle P). Find the length of side q (opposite angle Q).

Sine rule: p/sin P = q/sin Q

18/sin(42°) = q/sin(65°)

q = 18 × sin(65°)/sin(42°) = 18 × 0.9063/0.6691 ≈ 24.4 cm
Understanding
Matrices — Solving Systems: Use the matrix inverse method to solve:
2x + 3y = 12
x − y = 1

A =

2 3

1 −1

, B =

12

1

det(A) = −2 − 3 = −5. A⁻¹ = −⅕

−1 −3

−1 2

= ⅕

1 3

1 −2

X = ⅕

12+3

12−2

= ⅕

15

10

x = 3, y = 2. Check: 2(3)+3(2)=12 ✓ and 3−2=1 ✓
Understanding
Univariate Data — Spread: Dataset: 15, 18, 20, 22, 24, 28, 65.
1. Find Q1, Q3 and IQR.
2. Is 65 an outlier? Show the fence calculation.
3. Which measure of centre (mean or median) is more appropriate for this dataset?
(a) n = 7. Median = 22 (4th). Lower half: 15, 18, 20 → Q1 = 18. Upper half: 24, 28, 65 → Q3 = 28. IQR = 10.

(b) Upper fence = 28 + 1.5(10) = 43. Since 65 > 43, 65 is an outlier.

(c) Mean = (15+18+20+22+24+28+65)/7 = 192/7 ≈ 27.4. Median = 22. The median (22) is more appropriate because the outlier (65) pulls the mean up to 27.4, above all values except 65 itself. The median better represents the typical value.
Problem Solving
Trigonometry — Cosine Rule: A ship travels 80 km on a bearing of 040°, then changes course and travels 60 km on a bearing of 110°. How far is the ship from its starting point? (The angle between the two legs is 110° − 040° = 70°.)

The interior angle between the two sides at the turning point:
The first bearing is 040° and the second is 110°. The angle between the two directions of travel = 110° − 40° = 70° (the exterior angle at the turn). The interior angle of the triangle at the turning point = 180° − 70° = 110°.

Using cosine rule: c² = a² + b² − 2ab cos(C)

c² = 80² + 60² − 2(80)(60)cos(110°)

= 6400 + 3600 − 9600 × cos(110°)

cos(110°) = −cos(70°) ≈ −0.3420

c² = 10000 − 9600 × (−0.3420) = 10000 + 3283.2 = 13283.2

c = √13283.2 ≈ 115.3 km from start.
Problem Solving
Applications of Linear Equations — Step Graph: A parking station charges: $4.00 for the first hour or part thereof, $3.00 for the 2nd hour, $2.50 per hour for hours 3–5, and $2.00 per hour for each hour after 5.
1. Calculate the cost for a stay of 3.5 hours.
2. Calculate the cost for a stay of 7 hours.
3. What is the maximum parking cost per day (10 hours maximum)?
(a) 3.5 hours: Hours 1–1: $4.00. Hour 2: $3.00. Hours 3–3.5 (counts as hour 3): $2.50. Total = 4.00 + 3.00 + 2.50 = $9.50

(b) 7 hours: Hour 1: $4.00. Hour 2: $3.00. Hours 3, 4, 5: 3 × $2.50 = $7.50. Hours 6, 7: 2 × $2.00 = $4.00. Total = 4.00 + 3.00 + 7.50 + 4.00 = $18.50

(c) 10 hours (maximum): Hours 1–5 cost = 4.00 + 3.00 + 3 × 2.50 = $14.50. Hours 6–10: 5 × $2.00 = $10.00. Total = $14.50 + $10.00 = $24.50
Problem Solving
Matrices — Application: A school canteen sells sandwiches ($6) and pies ($4). On Monday and Tuesday:

Monday: 35 sandwiches and 20 pies sold. Tuesday: 28 sandwiches and 32 pies sold.
1. Write a 2×2 sales matrix S (rows = days, columns = items) and a 2×1 price column vector P.
2. Calculate revenue for each day using SP.
3. What was the total revenue over both days?
(a) S =

35 20

28 32

, P =

6

4

(b) SP =

35×6 + 20×4

28×6 + 32×4

=

$290

$296

Monday revenue: $290. Tuesday revenue: $296.

(c) Total = $290 + $296 = $586
Problem Solving
Univariate Data — Normal Distribution: The mean mass of apples from an orchard is μ = 165 g, with standard deviation σ = 12 g. Masses are approximately normally distributed.
1. Between what masses do 95% of apples fall?
2. An apple has mass 189 g. Calculate its z-score and comment on whether it is unusual.
3. A supermarket will only stock apples with mass between 141 g and 189 g. What percentage of the orchard’s apples meet this requirement?
(a) μ ± 2σ = 165 ± 24 → between 141 g and 189 g.

(b) z = (189 − 165)/12 = 24/12 = 2. An apple at z = 2 is 2 standard deviations above the mean. Only 2.5% of apples are heavier than this, so it is at the upper edge of the normal range — large but not unusual.

(c) 141 = μ − 2σ and 189 = μ + 2σ. The range 141–189 g corresponds to μ ± 2σ, which contains 95% of apples.
Problem Solving
Multi-topic: A community hall charges a flat hire fee of $200 plus $15 per person attending. A rival venue charges a flat fee of $80 plus $25 per person.
1. Write a cost function for each venue (C in terms of n, number of people).
2. For what number of attendees are the two venues equal in cost? (Set up and solve a linear equation.)
3. Which venue is cheaper for 20 people? For 8 people?
4. Sketch a description of what the two linear graphs would look like and identify the intersection point.
(a) Venue A: C_A(n) = 200 + 15n Venue B: C_B(n) = 80 + 25n

(b) Set equal: 200 + 15n = 80 + 25n → 120 = 10n → n = 12 people.

(c) 20 people: A = 200+15(20) = $500; B = 80+25(20) = $580. Venue A cheaper for 20 people.
8 people: A = 200+15(8) = $320; B = 80+25(8) = $280. Venue B cheaper for 8 people.

(d) Both functions are linear. C_A has y-intercept 200 (steeper start) and gradient 15 (shallow slope). C_B has y-intercept 80 (cheaper start) and gradient 25 (steeper slope). They intersect at (12, 380). For n < 12: Venue B is cheaper. For n > 12: Venue A is cheaper.

2	3
1	−1

−1	−3
−1	2

1	3
1	−2