Year 10 Review — All Topics

A comprehensive mixed review spanning all Year 10 topics from Terms 1–4. Questions are drawn from quadratics, non-linear relationships, polynomials, financial mathematics, trigonometry, measurement, circle geometry, deductive geometry, statistics, probability, indices and logarithms, networks, functions, coordinate geometry, and problem solving.

1. Quadratic equations. Fluency

   Solve: 3x² − 7x − 6 = 0.

   • (a) Use the quadratic formula to find the solutions (leave exact).
   • (b) State the sum and product of the roots.
   • (c) For what values of k does kx² − 6x + 3 = 0 have no real solutions?
   • (d) Sketch y = 3x² − 7x − 6, labelling the x-intercepts, y-intercept and vertex.

2. Compound interest and depreciation. Fluency

   • (a) $8 000 is invested at 5% per annum compounded quarterly. Find the value after 3 years.
   • (b) A laptop costing $2 500 depreciates at 20% per annum. Find its value after 4 years.
   • (c) How many complete years until the laptop is worth less than $500?
   • (d) What annual interest rate would double an investment in 10 years?

3. Sine rule application. Fluency

   In triangle ABC, angle A = 48°, angle B = 63° and AB = 15 cm.

   • (a) Find angle C.
   • (b) Use the sine rule to find BC.
   • (c) Find AC.
   • (d) Find the area of triangle ABC.

4. Circle geometry proof. Fluency

   O is the centre of a circle. Points A, B and C lie on the circle. Angle AOB = 100° (reflex).

   • (a) Find the non-reflex angle AOB.
   • (b) C lies on the major arc. Find angle ACB.
   • (c) D lies on the minor arc. Find angle ADB.
   • (d) What is the sum of angles ACB and ADB? What theorem does this illustrate?

5. Line of best fit. Understanding

   The line of best fit for a data set is y = −2.5x + 80, where x = age of car (years) and y = resale value ($thousands).

   • (a) Interpret the gradient in context.
   • (b) Predict the resale value of a 6-year-old car.
   • (c) According to the model, when is the car worth $0? Is this realistic?
   • (d) The data range is x = 1 to 10 years. A 15-year-old car is offered for $25 000. What does the model predict, and should you trust it?

6. Indices and logarithms. Understanding

   • (a) Simplify: (4a³b)⁻² × 8a&sup5;b³.
   • (b) Evaluate log₃ 81 + log₃(1/9).
   • (c) Solve 2^x+1 = 3^x (leave answer in logarithm form, evaluate to 2 d.p.).
   • (d) Use log laws to simplify: log(x²y) − log(xy²).

7. Functions and inverses. Understanding

   Let f(x) = 2x² − 3 (x ≥ 0) and g(x) = √((x+3)/2).

   • (a) Find f(4).
   • (b) Show that g is the inverse of f.
   • (c) State the domain and range of f, and the domain and range of f⁻¹.
   • (d) Solve f(x) = g(x).

8. Coordinate geometry. Understanding

   A circle has equation (x−2)² + (y+1)² = 25. A line has equation y = x + 2.

   • (a) State the centre and radius of the circle.
   • (b) Find the points of intersection of the line and circle.
   • (c) Find the midpoint of the chord joining the two intersection points.
   • (d) Verify that the line from the centre to the midpoint of the chord is perpendicular to the chord.

9. Optimisation problem. Problem Solving

   A farmer has 200 m of fencing. He wants to create three equal rectangular paddocks side by side (sharing internal fences).

   • (a) Let the width of each paddock be x m and the total length be L m. Write an equation linking x and L using the total fencing.
   • (b) Write the total area A(x) of the three paddocks combined as a quadratic in x.
   • (c) Find the dimensions that maximise the total area, and state the maximum area.
   • (d) State the domain of x in context and check the vertex is within it.

10. Extended mixed problem. Problem Solving

    A company sells x hundred units per week. The demand function is p(x) = 80 − 4x dollars per unit. Fixed costs are $400 per week and variable cost is $12 per unit.

    • (a) Write the revenue function R(x) and cost function C(x) (in dollars).
    • (b) Write the profit function P(x) and fully simplify.
    • (c) Find the production level (in hundreds of units) that maximises weekly profit.
    • (d) Find the maximum weekly profit and the selling price at that production level.