Mathematics
Number and algebra — Topics 1-2
- 1.
A laptop is reduced by 15% to a sale price of $612. Calculate its price before the reduction.
[3 marks] - 2.
Solve simultaneously: 2x + y = 11 and x − y = 1. Show your elimination or substitution method.
[4 marks] · no calculator - 3.
Simplify fully: 3(2x − 5) − 2(x − 4).
[3 marks] · no calculator - 4.
$5000 is invested at 4% per year compound interest. Calculate the value of the investment after 3 years.
[3 marks] - 5.
Factorise fully: 6x² − 15x.
[2 marks] · no calculator - 6.
Express 0.(45) — the recurring decimal 0.454545... — as a fraction in its simplest form.
[4 marks] · no calculator - 7.
Solve the inequality 3x − 7 ≤ 11, and represent the solution on a number line.
[3 marks] · no calculator - 8.
Write 384 as a product of its prime factors, giving your answer in index form.
[3 marks] · no calculator - 9.
Make r the subject of the formula V = (4/3)πr³.
[3 marks] · no calculator - 10.
A number increased by 20% is then decreased by 20%. Determine whether the final result is greater than, less than, or equal to the original number, showing your reasoning with an example.
[4 marks] - 11.
A jacket costs $250. It is first reduced by 20% in a sale, then 8% sales tax is added to the reduced price. Calculate the final price.
[3 marks] - 12.
Write 0.000473 in standard form.
[2 marks] · no calculator - 13.
Calculate (3.2 × 10⁵) × (4 × 10³), giving your answer in standard form.
[3 marks] · no calculator - 14.
Simplify fully: (2x³)⁴ ÷ (4x⁵).
[3 marks] · no calculator - 15.
The nth term of a sequence is given by Uₙ = 5n − 3. Find U₁ and U₁₀, and determine whether 152 is a term of this sequence.
[4 marks] - 16.
A geometric sequence has first term 3 and common ratio 2. Find the 6th term and the sum of the first 6 terms.
[4 marks] - 17.
Solve x² − 5x + 3 = 0 using the quadratic formula, giving your answers correct to 2 decimal places.
[4 marks] - 18.
Solve simultaneously: y = x + 1 and y = x² − 5.
[5 marks] · no calculator - 19.
y is directly proportional to x². When x = 3, y = 45. Find y when x = 5.
[4 marks] - 20.
p is inversely proportional to q. When q = 4, p = 15. Find p when q = 10.
[4 marks] - 21.
$720 is shared between A, B and C in the ratio 2 : 3 : 4. Calculate B's share.
[3 marks] - 22.
A length is measured as 15 cm, correct to the nearest centimetre. Write down the upper and lower bounds of this length.
[2 marks] · no calculator - 23.
Factorise fully: 2x² + 7x + 3.
[3 marks] · no calculator - 24.
Simplify fully: (x² − 9)/(x² + x − 6).
[4 marks] · no calculator - 25.
Make x the subject of the formula y = (3x + 2)/(x − 1).
[5 marks] · no calculator - 26.
A jacket costs 68 after a 15% discount. Calculate its original price. All prices use the same currency.
[3 marks] · no calculator - 27.
Adult tickets cost 12 and child tickets cost 7. Nine tickets cost 93 in total. Form simultaneous equations and find how many of each ticket were bought.
[4 marks] · no calculator - 28.
Write 0.272727... as a fraction in its simplest form, showing an algebraic method.
[3 marks] · no calculator - 29.
Solve x^2 - x - 6 < 0. Explain why the endpoints are not included.
[4 marks] · no calculator - 30.
A rectangle has sides x cm and (x + 3) cm and area 40 cm^2. Form an equation, find its dimensions and calculate its perimeter.
[5 marks] · no calculator - 31.
An investment grows by 8% in each of two years, then falls by 10%. Calculate the overall percentage change to two decimal places and explain why 8 + 8 - 10 is not correct.
[5 marks]