IGCSE Maths Higher Paper 3
Covers Number and Calculation, Fractions, Decimals and Percentages, Indices, Surds and Standard Form and 9 more.
Questions
Question 1 [3 marks]
Sequences and Graphs
Find the midpoint of the line segment joining the points (2, -5) and (8, 3).
Question 2 [3 marks]
Functions and Rates of Change
The function g is defined by g(x) = 2x^2 - 3.
Find g(-4).
Question 3 [3 marks]
Indices, Surds and Standard Form
Simplify (x^7 y^3)/(x^3 y^-2).
Give your answer using positive indices.
Question 4 [5 marks]
Quadratics and Simultaneous Equations
Solve 3x^2 + 11x - 4 = 0 by factorising.
Question 5 [5 marks]
Pythagoras and Trigonometry
A triangular field has vertices A, B and C.
Angle BAC = 55 degrees, angle ABC = 72 degrees, and side BC = 84 m.
Find the length of side AC, to the nearest metre.
Question 6 [5 marks]
Angles, Polygons and Circle Theorems
A ship sails from port A on a bearing of 062 degrees to port B, then continues on a bearing of 130 degrees to port C.
Find the bearing of port A from port B, and find the angle ABC.
Question 7 [4 marks]
Indices, Surds and Standard Form
Expand and simplify (3 + sqrt(5))(2 - sqrt(5)).
Give your answer in the form a + b sqrt(5), where a and b are integers.
Question 8 [5 marks]
Fractions, Decimals and Percentages
A savings account pays compound interest at 2.75% per year.
1500 pounds is invested.
Find the value of the investment after 5 years, to the nearest penny.
Question 9 [5 marks]
Number and Calculation
P = 6.4 x 10^5 and Q = 3.7 x 10^4.
Work out P + Q, giving your answer in standard form.
Question 10 [5 marks]
Mensuration and Vectors
A cone has base radius 4.5 cm and height 10 cm.
Find its volume, to 3 significant figures, using volume = (1/3) pi r^2 h.
Question 11 [5 marks]
Ratio and Proportion
The pressure P of a gas is inversely proportional to its volume V.
When V = 8, P = 30.
Find P when V = 12.
Question 12 [6 marks]
Statistics and Probability
The table shows the masses, in kg, of 90 parcels.
0 <= m < 5: 18 parcels, 5 <= m < 10: 34 parcels, 10 <= m < 15: 26 parcels, 15 <= m < 20: 12 parcels.
Using linear interpolation, estimate the number of parcels with a mass greater than 12 kg.
Question 13 [6 marks]
Linear Equations and Inequalities
A cafe sells coffees and teas. On Monday, 12 coffees and 8 teas were sold for a total of 54.20 pounds.
On Tuesday, 15 coffees and 5 teas were sold for a total of 54.50 pounds.
Find the price of one coffee and the price of one tea.
Model solutions
| Question 1[3 marks] | |
|---|---|
| Answer or working | Marks |
| adding the x-coordinates and dividing by 2 | M1 |
| adding the y-coordinates and dividing by 2 | M1 |
| (5, -1) | A1 |
| Question 2[3 marks] | |
|---|---|
| Answer or working | Marks |
| substituting x = -4 | M1 |
| 2 x (-4)^2 | M1 |
| g(-4) = 29 | A1 |
| Final answer: 29 | |
| Question 3[3 marks] | |
|---|---|
| Answer or working | Marks |
| subtracting powers of x to get x^4 | M1 |
| subtracting powers of y to get y^5 | M1 |
| x^4 y^5 | A1 |
| Question 4[5 marks] | |
|---|---|
| Answer or working | Marks |
| finding two numbers with product -12 (3 x -4) and sum 11 | M1 |
| identifying 12 and -1 | M1 |
| factorising to (3x - 1)(x + 4) = 0 | M1 |
| x = 1/3 | A1 |
| x = -4 | A1 |
| Final answer: x = 1/3 or x = -4 | |
| Question 5[5 marks] | |
|---|---|
| Answer or working | Marks |
| finding angle ACB = 180 - 55 - 72 = 53 degrees | M1 |
| using the sine rule AC/sin B = BC/sin A | M1 |
| AC = 84 x sin 72 / sin 55 | M1 |
| 97.5 m before rounding | A1 |
| 98 m (nearest metre) | A1 |
| Final answer: 98 m | |
| Question 6[5 marks] | |
|---|---|
| Answer or working | Marks |
| recognising the back bearing differs from the original bearing by 180 degrees | M1 |
| 062 + 180 = 242, the bearing of A from B | M1 |
| 242 degrees | A1 |
| finding angle ABC as the difference between the bearing of A from B and the bearing of C from B | M1 |
| angle ABC = 242 - 130 = 112 degrees | A1 |
| Final answer: bearing of A from B = 242 degrees, angle ABC = 112 degrees | |
| Question 7[4 marks] | |
|---|---|
| Answer or working | Marks |
| multiplying out to get 6 - 3 sqrt(5) + 2 sqrt(5) - 5 | M1 |
| simplifying the surd terms to -sqrt(5) | M1 |
| simplifying the integer terms to 1 | M1 |
| 1 - sqrt(5) | A1 |
| Question 8[5 marks] | |
|---|---|
| Answer or working | Marks |
| identifying the multiplier 1.0275 | M1 |
| using value = 1500 x 1.0275^5 | M1 |
| evaluating 1.0275^5 correctly to at least 4 decimal places | M1 |
| evaluating 1500 x their power before rounding | M1 |
| 1717.91 pounds | A1 |
| Question 9[5 marks] | |
|---|---|
| Answer or working | Marks |
| converting to the same power of 10, such as P = 64 x 10^4 | M1 |
| identifying Q = 3.7 x 10^4 | M1 |
| 64 x 10^4 + 3.7 x 10^4 = 67.7 x 10^4 | M1 |
| 677000 | A1 |
| 6.77 x 10^5 | A1 |
| Question 10[5 marks] | |
|---|---|
| Answer or working | Marks |
| using volume = 1/3 pi r^2 h | M1 |
| 1/3 x pi x 4.5^2 x 10 | M1 |
| 67.5 pi | A1 |
| 212.058... before rounding | M1 |
| 212 cm^3 | A1 |
| Question 11[5 marks] | |
|---|---|
| Answer or working | Marks |
| P = k/V | M1 |
| 30 = k/8 | M1 |
| k = 240 | A1 |
| P = 240/12 | M1 |
| 20 | A1 |
| Question 12[6 marks] | |
|---|---|
| Answer or working | Marks |
| cumulative frequencies 18, 52, 78, 90 | M1 |
| identifying 12 kg lies in the class 10 <= m < 15 | M1 |
| using linear interpolation: cumulative frequency at 12 = 52 + (12 - 10)/5 x 26 | M1 |
| evaluating 52 + 10.4 = 62.4 | M1 |
| subtracting from the total: 90 - 62.4 = 27.6 | M1 |
| 28 parcels (nearest whole number) | A1 |
| Final answer: 28 parcels | |
| Question 13[6 marks] | |
|---|---|
| Answer or working | Marks |
| forming the equations 12x + 8y = 54.20 and 15x + 5y = 54.50 | M1 |
| multiplying the first equation by 5 and the second by 8 to align the y terms | M1 |
| subtracting to eliminate y, giving 60x = 165 | M1 |
| x = 2.75 (one coffee costs 2.75 pounds) | A1 |
| substituting x = 2.75 into one of the original equations | M1 |
| y = 2.65 (one tea costs 2.65 pounds) | A1 |
| Final answer: coffee = 2.75 pounds, tea = 2.65 pounds | |