Higher Tier - Grades 4-9

IGCSE Maths Higher Paper 2

Covers Number and Calculation, Fractions, Decimals and Percentages, Indices, Surds and Standard Form and 9 more.

12 questions - 60 marks - calculator allowed

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Questions

Question 1 [3 marks]

Pythagoras and Trigonometry

In triangle ABC, angle A = 40 degrees, angle B = 65 degrees, and side a (opposite angle A) = 12 cm.

Find the length of side b (opposite angle B), to 1 decimal place.

Question 2 [4 marks]

Sequences and Graphs

The straight line L has equation y = 2x - 7.

Find the equation of the line parallel to L that passes through (3, 4).

Question 3 [4 marks]

Statistics and Probability

A bag contains 4 red counters and 6 blue counters.

A counter is drawn at random, its colour noted, and then replaced. A second counter is then drawn at random.

Find the probability that both counters are the same colour.

Question 4 [5 marks]

Quadratics and Simultaneous Equations

Solve 3x^2 + 11x - 4 = 0 by factorising.

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]

Linear Equations and Inequalities

n is an integer.

-3 < 2n - 5 <= 7.

Find all possible values of n.

Question 7 [5 marks]

Functions and Rates of Change

A curve has equation y = x^3 - 12x + 5.

Find the range of values of x for which the curve has a negative gradient.

Question 8 [5 marks]

Indices, Surds and Standard Form

Solve 9^x = 1/3, giving x as a fraction.

Question 9 [6 marks]

Number and Calculation

The recurring decimal 0.41666... means the digit 6 repeats forever after the initial 41.

Convert 0.41666... to a fraction in its simplest form.

Question 10 [6 marks]

Fractions, Decimals and Percentages

An investment of 5000 pounds grows to 5788.13 pounds after 3 years of compound interest.

Find the annual rate of interest, to 1 decimal place.

Question 11 [6 marks]

Ratio and Proportion

Two cyclists start 63 km apart and cycle towards each other along the same road.

Cyclist A travels at 15 km/h and cyclist B travels at 12 km/h.

Find how long it takes for the two cyclists to meet, giving your answer in hours and minutes.

Question 12 [6 marks]

Angles, Polygons and Circle Theorems

A chord AB of length 24 cm is drawn in a circle of radius 13 cm, centre O.

Find the perpendicular distance from O to AB, and find angle AOB, giving your answer to 1 decimal place.

Model solutions

Mark scheme for Question 1 [3 marks]
Question 1[3 marks]
Answer or working	Marks
using the sine rule b/sin B = a/sin A	M1
b = 12 x sin 65 / sin 40	M1
16.9 cm	A1

Mark scheme for Question 2 [4 marks]
Question 2[4 marks]
Answer or working	Marks
identifying gradient 2 from a parallel line	M1
using y = 2x + c	M1
substituting (3, 4) to get 4 = 6 + c	M1
y = 2x - 2	A1

Mark scheme for Question 3 [4 marks]
Question 3[4 marks]
Answer or working	Marks
P(red) = 4/10 and P(blue) = 6/10, unchanged after replacement	M1
P(both red) = 4/10 x 4/10 = 16/100	M1
P(both blue) = 6/10 x 6/10 = 36/100	M1
52/100, or 13/25	A1
Final answer: 13/25

Mark scheme for Question 4 [5 marks]
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

Mark scheme for Question 5 [5 marks]
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

Mark scheme for Question 6 [5 marks]
Question 6[5 marks]
Answer or working	Marks
adding 5 to all three parts of the inequality	M1
-3 + 5 < 2n <= 7 + 5, giving 2 < 2n <= 12	M1
dividing all three parts by 2	M1
1 < n <= 6	A1
n = 2, 3, 4, 5 or 6	A1

Mark scheme for Question 7 [5 marks]
Question 7[5 marks]
Answer or working	Marks
differentiating to get dy/dx = 3x^2 - 12	M1
setting dy/dx < 0	M1
dividing by 3 to get x^2 < 4	M1
taking the square root, giving -2 < x < 2	M1
-2 < x < 2	A1

Mark scheme for Question 8 [5 marks]
Question 8[5 marks]
Answer or working	Marks
writing 9 as 3^2	M1
writing 1/3 as 3^-1	M1
forming the equation 3^(2x) = 3^-1	M1
equating powers to get 2x = -1	M1
x = -1/2	A1

Mark scheme for Question 9 [6 marks]
Question 9[6 marks]
Answer or working	Marks
setting x = 0.41666...	M1
10x = 4.1666...	M1
100x = 41.6666...	M1
subtracting to get 90x = 37.5	M1
x = 375/900	A1
simplifying to 5/12	A1
Final answer: 5/12

Mark scheme for Question 10 [6 marks]
Question 10[6 marks]
Answer or working	Marks
forming 5000 x r^3 = 5788.13, where r is the multiplier	M1
r^3 = 5788.13 / 5000 = 1.157626	M1
taking the cube root of both sides	M1
r = 1.05 (to 3 s.f.)	M1
identifying the multiplier corresponds to a 5% increase	A1
5.0%	A1

Mark scheme for Question 11 [6 marks]
Question 11[6 marks]
Answer or working	Marks
combined speed = 15 + 12 = 27 km/h	M1
time = distance / combined speed	M1
63 / 27 = 7/3 hours	M1
converting 7/3 hours to 2 1/3 hours	M1
2 hours	A1
20 minutes	A1
Final answer: 2 hours 20 minutes

Mark scheme for Question 12 [6 marks]
Question 12[6 marks]
Answer or working	Marks
recognising the perpendicular from the centre bisects the chord, giving half the chord = 12 cm	M1
using Pythagoras to find the perpendicular distance: sqrt(13^2 - 12^2)	M1
the perpendicular distance = 5 cm	A1
using trigonometry in the right-angled triangle formed, such as cos(AOM) = 5/13	M1
doubling the angle found to find angle AOB	M1
angle AOB = 134.8 degrees (1 d.p.)	A1
Final answer: perpendicular distance = 5 cm, angle AOB = 134.8 degrees