IGCSE Maths Higher Paper 1
Covers Number and Calculation, Fractions, Decimals and Percentages, Indices, Surds and Standard Form and 9 more.
Questions
Question 1 [3 marks]
Ratio and Proportion
The cost of petrol is directly proportional to the number of litres bought.
35 litres costs 54.25 pounds. Find the cost of 52 litres.
Question 2 [3 marks]
Number and Calculation
Work out 6 + 4 x (9 - 5)^2 / 8.
Question 3 [5 marks]
Sequences and Graphs
A geometric sequence has first term 40 and common ratio 1/2.
Find the 5th term of the sequence, and find the sum to infinity of the sequence.
Question 4 [5 marks]
Pythagoras and Trigonometry
A cuboid has length 8 cm, width 5 cm and height 6 cm.
Find the length of the diagonal that runs from one corner of the cuboid to the opposite corner, to 1 decimal place.
Question 5 [5 marks]
Mensuration and Vectors
A window is made from a rectangle measuring 120 cm by 80 cm, topped with a semicircle whose diameter equals the 80 cm width of the rectangle.
Find the total area of the window, to the nearest cm^2.
Question 6 [5 marks]
Quadratics and Simultaneous Equations
Solve 3x^2 + 11x - 4 = 0 by factorising.
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 [4 marks]
Fractions, Decimals and Percentages
A shop increases the price of a jacket by 12%, then later reduces the new price by 12%.
The final price is 98.56 pounds. Find the original price.
Question 9 [4 marks]
Statistics and Probability
120 students were asked whether they play a musical instrument.
Of the 65 boys, 28 play an instrument. Of the 55 girls, 31 play an instrument.
A student is chosen at random from all 120 students. Find the probability that the student is a girl who does not play an instrument.
Question 10 [5 marks]
Functions and Rates of Change
A curve has equation y = x^3 - 6x^2 + 9x + 2.
Find the x-coordinates of the stationary points of the curve, using differentiation.
Question 11 [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 12 [6 marks]
Angles, Polygons and Circle Theorems
The interior angle of a regular polygon is 5 times the size of its exterior angle.
Find the number of sides of the polygon.
Question 13 [6 marks]
Linear Equations and Inequalities
The formula for the time period T of a pendulum of length L is T = 2 pi sqrt(L/g), where g is the gravitational field strength.
Make L the subject of the formula, then find the value of L when T = 2 seconds and g = 9.8 m/s^2,
giving your answer to 3 significant figures.
Model solutions
| Question 1[3 marks] | |
|---|---|
| Answer or working | Marks |
| finding the cost per litre, 54.25 / 35 = 1.55 | M1 |
| multiplying by 52 | M1 |
| 80.60 pounds | A1 |
| Question 2[3 marks] | |
|---|---|
| Answer or working | Marks |
| evaluating the bracket (9 - 5) = 4 and squaring to get 16 | M1 |
| multiplying by 4 and dividing by 8 to get 8 | M1 |
| 14 | A1 |
| Question 3[5 marks] | |
|---|---|
| Answer or working | Marks |
| using the nth term formula a r^(n-1) | M1 |
| 5th term = 40 x (1/2)^4 | M1 |
| 2.5 | A1 |
| using the sum to infinity formula a/(1 - r) | M1 |
| sum to infinity = 80 | A1 |
| Final answer: 5th term = 2.5, sum to infinity = 80 | |
| Question 4[5 marks] | |
|---|---|
| Answer or working | Marks |
| using the 3D Pythagoras formula d = sqrt(l^2 + w^2 + h^2) | M1 |
| 8^2 + 5^2 + 6^2 = 125 | M1 |
| taking the square root of 125 | M1 |
| 11.18033... | A1 |
| 11.2 cm | A1 |
| Question 5[5 marks] | |
|---|---|
| Answer or working | Marks |
| the rectangle area = 120 x 80 = 9600 cm^2 | M1 |
| the semicircle radius = 40 cm | M1 |
| the semicircle area = 1/2 x pi x 40^2 | M1 |
| the semicircle area = 2513.27... cm^2 | A1 |
| the total area = 12113 cm^2 | A1 |
| Final answer: 12113 cm^2 | |
| Question 6[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 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[4 marks] | |
|---|---|
| Answer or working | Marks |
| the multiplier for a 12% increase, 1.12 | M1 |
| the multiplier for a 12% decrease, 0.88 | M1 |
| the combined multiplier 1.12 x 0.88 = 0.9856 | M1 |
| original price = 100 pounds | A1 |
| Final answer: 100 pounds | |
| Question 9[4 marks] | |
|---|---|
| Answer or working | Marks |
| finding the number of girls who do not play an instrument: 55 - 31 = 24 | M1 |
| identifying the total number of students is 120 | M1 |
| forming the probability 24/120 | M1 |
| 1/5 | A1 |
| Question 10[5 marks] | |
|---|---|
| Answer or working | Marks |
| differentiating to get dy/dx = 3x^2 - 12x + 9 | M1 |
| setting dy/dx = 0 | M1 |
| dividing by 3 to get x^2 - 4x + 3 = 0 | M1 |
| factorising to (x - 1)(x - 3) = 0 | M1 |
| x = 1 and x = 3 | A1 |
| Question 11[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 12[6 marks] | |
|---|---|
| Answer or working | Marks |
| using interior angle + exterior angle = 180 degrees | M1 |
| forming the equation e + 5e = 180, where e is the exterior angle | M1 |
| solving 6e = 180 to get e = 30 degrees | M1 |
| using the exterior angles of a polygon sum to 360 degrees | M1 |
| number of sides = 360 / 30 | M1 |
| 12 sides | A1 |
| Question 13[6 marks] | |
|---|---|
| Answer or working | Marks |
| dividing both sides by 2 pi to get T/(2 pi) = sqrt(L/g) | M1 |
| squaring both sides to get T^2/(4 pi^2) = L/g | M1 |
| multiplying both sides by g | M1 |
| L = (g T^2)/(4 pi^2) | A1 |
| substituting T = 2 and g = 9.8 into the formula | M1 |
| L = 0.993 m (3 s.f.) | A1 |
| Final answer: L = (g T^2)/(4 pi^2); L = 0.993 m | |