Pre-calculus Concepts Review

Year 10 Mathematics — Pre-calculus Synthesis

Name:

Date:

📋 Part 1: Multiple Choice

Circle the best answer for each question.

1. A taxi charges a flat fee of $3.50 plus $2.25 per kilometre. Which equation represents the total cost C for k kilometres?

C = 2.25 + 3.50k

C = 3.50 + 2.25k

C = 3.50 × 2.25k

C = 3.50k + 2.25

2. Two phone plans are compared: Plan A costs $20/month + $0.05 per text, and Plan B costs $35/month + $0.02 per text. At how many texts are the plans equal in cost?

300 texts

400 texts

500 texts

600 texts

3. An arithmetic sequence begins: 4, 11, 18, 25, … What is the 10th term?

4. Which correctly factors the expression x² + 7x + 12?

(x + 3)(x + 4)

(x + 2)(x + 6)

(x + 1)(x + 12)

(x − 3)(x − 4)

5. In a right triangle, the side opposite a 35° angle measures 8 m. Which expression gives the length of the hypotenuse?

h = 8 × cos(35°)

h = 8 ÷ sin(35°)

h = 8 × tan(35°)

h = 8 ÷ cos(35°)

✏️ Part 2: Short Answer & Problem Solving

Show all working. Include units where appropriate.

6. Linear Systems — Phone Plans
Plan A costs $25/month + $0.04 per text. Plan B costs $40/month + $0.01 per text.
(a) Write an equation for each plan, using t for number of texts and C for total monthly cost.

Plan A: C = _______________________________

Plan B: C = _______________________________

(b) Solve the system to find the number of texts where both plans cost the same. Show your algebraic working below.

(c) Verify your solution by substituting back into both equations.

(d) If you send 800 texts per month, which plan is cheaper? By how much?

7. Arithmetic Sequences — Weekly Savings
Maya saves $15 in Week 1 and increases her savings by $8 each week.
(a) Write the general term formula t_n for her savings in Week n.

t_n = _______________________________

(b) How much does Maya save in Week 12? Show your working.

(c) In which week will Maya first save more than $100 in a single week? Show the inequality you used.

8. Polynomial Operations & Factoring
(a) Expand and simplify: (x + 5)(x − 3)

(b) Factor the expression: x² + 2x − 15

(c) Use your factored form to solve x² + 2x − 15 = 0. State both solutions.

x = __________ or x = __________

9. Trigonometry Application
A ramp rises to a height of 1.2 m over a horizontal distance of 4.5 m.
(a) Draw and label a right triangle representing this situation. Identify the opposite, adjacent, and hypotenuse sides.

(b) Calculate the angle of inclination of the ramp (to the nearest degree). State the trigonometric ratio you used and show your working.

The angle of inclination is approximately __________ °

10. Financial Literacy — Gross & Net Pay
Jordan earns $17.50/hour and works 32 hours this week. Deductions include a $45.00 benefits charge and 22% income tax on gross pay.
(a) Write an expression for Jordan's gross pay and calculate it.

Gross pay = $ __________

(b) Calculate Jordan's total deductions and net pay.

Net pay = $ __________

(c) Write one sentence interpreting what the difference between gross and net pay means in this context.

🚪 Exit Check

Complete these three questions independently before leaving class.

11. Solve the system of equations and state what the solution represents in context:
y = 5x + 10 y = 3x + 18

12. The 3rd term of an arithmetic sequence is 19 and the common difference is 6. Find the 1st term and the 15th term.

13. A student says x² − x − 6 = 0 has solutions x = 3 and x = −2. Check whether this is correct by substituting both values. Explain your reasoning in one or two sentences.