Higher Tier - Year 11

Year 11 Paper 6: Higher Full Course Review

Covers number, fractions and percentages, indices, ratio, linear and quadratic algebra, functions, sequences and graphs, geometry, trigonometry, mensuration, and statistics and probability.

21 questions - 100 marks - calculator allowed

Year 11 here means a typical teaching order, not a syllabus rule. No exam board defines what belongs to Year 11, and schools sequence the course differently. Check it against your own scheme of work before using it to decide what a class has covered.

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Questions

Question 1 [3 marks]

Sequences and Graphs

The nth term of a sequence is 4n + 7.

Is 150 a term in the sequence? Show your working.

Question 2 [3 marks]

Number and Calculation

Work out 6 + 4 x (9 - 5)^2 / 8.

Question 3 [3 marks]

Mensuration and Vectors

A cuboid has length 9 cm, width 6 cm and height 5 cm. Find its volume.

Question 4 [4 marks]

Angles, Polygons and Circle Theorems

A, B and C are points on the circumference of a circle, where AC is a diameter.

Angle BAC = 34 degrees. Find angle ABC and angle BCA.

Question 5 [4 marks]

Ratio and Proportion

The ratio of Maya's age to her brother's age is 3:5. Maya is 12 years old.

In how many years' time will the ratio of their ages be 5:7?

Question 6 [4 marks]

Functions and Rates of Change

The function h is defined by h(x) = (x - 3)/4 for all values of x.

Find the inverse function h^-1(x).

Question 7 [5 marks]

Indices, Surds and Standard Form

The mass of the Earth is approximately 6 x 10^24 kg.

The mass of the Moon is approximately 7.5 x 10^22 kg.

Find how many times heavier the Earth is than the Moon.

Question 8 [5 marks]

Fractions, Decimals and Percentages

A roll of ribbon is 5.4 m long.

Priya cuts off 1/3 of the ribbon for a project, then cuts 2/5 of what remains for a second project.

How many centimetres of ribbon are left?

Question 9 [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 10 [5 marks]

Quadratics and Simultaneous Equations

Solve 2x^2 - 5x - 12 = 0 by factorising.

Question 11 [4 marks]

Pythagoras and Trigonometry

The angle of elevation of the top of a tower from a point A on level ground is 28 degrees.

From a point B, which is 40 m closer to the tower than A and on the same line, the angle of elevation is 42 degrees.

Find the height of the tower, to 1 decimal place.

Question 12 [5 marks]

Indices, Surds and Standard Form

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

Question 13 [5 marks]

Quadratics and Simultaneous Equations

A ball's height h metres above the ground t seconds after being thrown is given by h = -5t^2 + 18t + 1.

Find the time at which the ball hits the ground, giving your answer to 3 significant figures.

Question 14 [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 15 [5 marks]

Quadratics and Simultaneous Equations

Solve 3x^2 + 5x - 4 = 0.

Give your answers to 2 decimal places.

Question 16 [5 marks]

Ratio and Proportion

A delivery van travels at an average speed of 54 mph and takes 1 hour 40 minutes to complete a route.

How long would the same route take at an average speed of 45 mph? Give your answer in hours.

Question 17 [6 marks]

Linear Equations and Inequalities

A cylinder has total surface area S, base radius r and height h, connected by the formula S = 2 pi r^2 + 2 pi r h.

Make r the subject of the formula, using the quadratic formula. Give your answer in terms of S, h and pi.

Question 18 [6 marks]

Statistics and Probability

The table shows the heights, in cm, of 60 plants.

120 <= h < 130: 8 plants, 130 <= h < 140: 19 plants, 140 <= h < 150: 21 plants, 150 <= h < 160: 12 plants.

Using linear interpolation, estimate the median height.

Question 19 [6 marks]

Number and Calculation

A car travels 186 km, correct to the nearest km, in a time of 3.2 hours, correct to 1 decimal place.

Find the lower bound for the car's average speed in km/h, giving your answer to 3 significant figures.

Question 20 [6 marks]

Statistics and Probability

A box contains 5 milk chocolates and 7 dark chocolates. Two chocolates are taken at random without replacement.

Find the probability that both chocolates are dark, given that at least one of the two chocolates is dark.

Question 21 [6 marks]

Ratio and Proportion

The force F between two magnets is inversely proportional to the square of the distance d between them.

When d = 3 cm, F = 40 newtons.

Find the distance d, in centimetres, when F = 2.5 newtons.

Model solutions

Mark scheme for Question 1 [3 marks]
Question 1[3 marks]
Answer or working	Marks
solving 4n + 7 = 150	M1
4n = 143, so n = 35.75	M1
no, because n is not a positive whole number	A1
Final answer: No, n = 35.75 is not a whole number

Mark scheme for Question 2 [3 marks]
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

Mark scheme for Question 3 [3 marks]
Question 3[3 marks]
Answer or working	Marks
volume = length x width x height	M1
9 x 6 x 5	M1
270 cm^3	A1

Mark scheme for Question 4 [4 marks]
Question 4[4 marks]
Answer or working	Marks
recognising angle ABC = 90 degrees, the angle in a semicircle	M1
using angle sum in triangle ABC = 180 degrees	M1
180 - 90 - 34	M1
angle BCA = 56 degrees	A1
Final answer: angle ABC = 90 degrees, angle BCA = 56 degrees

Mark scheme for Question 5 [4 marks]
Question 5[4 marks]
Answer or working	Marks
finding Maya's brother's current age: 1 part = 4, so brother = 20	M1
forming the equation (12 + x)/(20 + x) = 5/7	M1
cross-multiplying to get 7(12 + x) = 5(20 + x)	M1
x = 8 years	A1
Final answer: 8 years

Mark scheme for Question 6 [4 marks]
Question 6[4 marks]
Answer or working	Marks
writing x = (y - 3)/4, swapping x and y	M1
multiplying both sides by 4 to get 4x = y - 3	M1
rearranging to make y the subject, y = 4x + 3	M1
h^-1(x) = 4x + 3	A1

Mark scheme for Question 7 [5 marks]
Question 7[5 marks]
Answer or working	Marks
dividing 6 by 7.5	M1
6 / 7.5 = 0.8	M1
dividing 10^24 by 10^22 to get 10^2	M1
combining 0.8 x 10^2	M1
80 times	A1

Mark scheme for Question 8 [5 marks]
Question 8[5 marks]
Answer or working	Marks
finding 2/3 of 5.4 = 3.6 m remaining after the first cut	M1
finding 2/5 of 3.6 = 1.44 m	M1
subtracting to leave 3.6 - 1.44 = 2.16 m	M1
converting 2.16 m to centimetres	M1
216 cm	A1

Mark scheme for Question 9 [5 marks]
Question 9[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

Mark scheme for Question 10 [5 marks]
Question 10[5 marks]
Answer or working	Marks
finding two numbers with product -24 and sum -5	M1
identifying -8 and 3	M1
factorising to (2x + 3)(x - 4) = 0	M1
x = -3/2	A1
x = 4	A1
Final answer: x = -3/2 or x = 4

Mark scheme for Question 11 [4 marks]
Question 11[4 marks]
Answer or working	Marks
forming h = (x + 40) tan 28 using triangle at A, where x is the distance from B to the base	M1
forming h = x tan 42 using triangle at B	M1
setting the two expressions equal and solving for x	M1
51.9 m (1 d.p.)	A1
Final answer: 51.9 m

Mark scheme for Question 12 [5 marks]
Question 12[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 13 [5 marks]
Question 13[5 marks]
Answer or working	Marks
setting h = 0 to get -5t^2 + 18t + 1 = 0	M1
rearranging to 5t^2 - 18t - 1 = 0	M1
using the quadratic formula with a = 5, b = -18, c = -1	M1
the discriminant = 324 + 20 = 344	M1
t = 3.65 seconds (3 s.f.), rejecting the negative solution	A1
Final answer: 3.65 seconds

Mark scheme for Question 14 [5 marks]
Question 14[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

Mark scheme for Question 15 [5 marks]
Question 15[5 marks]
Answer or working	Marks
using the quadratic formula with a = 3, b = 5, c = -4	M1
the discriminant = 25 + 48 = 73	M1
x = (-5 +/- sqrt(73))/6	M1
x = 0.59	A1
x = -2.26	A1
Final answer: x = 0.59 or x = -2.26

Mark scheme for Question 16 [5 marks]
Question 16[5 marks]
Answer or working	Marks
converting 1 hour 40 minutes to 5/3 hours	M1
distance = 54 x 5/3	M1
distance = 90 miles	A1
time = 90 divided by 45	M1
2 hours	A1

Mark scheme for Question 17 [6 marks]
Question 17[6 marks]
Answer or working	Marks
rearranging to 2 pi r^2 + 2 pi h r - S = 0	M1
identifying a = 2 pi, b = 2 pi h and c = -S for the quadratic formula	M1
substituting into r = (-b +/- sqrt(b^2 - 4ac)) / (2a)	M1
simplifying b^2 - 4ac to 4 pi^2 h^2 + 8 pi S	M1
r = (-2 pi h +/- sqrt(4 pi^2 h^2 + 8 pi S)) / (4 pi)	A1
taking the positive square root, since r must be positive	A1
Final answer: r = (-2 pi h + sqrt(4 pi^2 h^2 + 8 pi S)) / (4 pi)

Mark scheme for Question 18 [6 marks]
Question 18[6 marks]
Answer or working	Marks
cumulative frequencies 8, 27, 48, 60	M1
identifying n/2 = 30 lies in the class 140 <= h < 150	M1
using linear interpolation: 140 + (30 - 27)/21 x 10	M1
(30 - 27)/21 = 3/21	M1
140 + 1.428...	A1
141.4 cm (1 d.p.)	A1
Final answer: 141.4 cm

Mark scheme for Question 19 [6 marks]
Question 19[6 marks]
Answer or working	Marks
identifying the lower bound of the distance as 185.5 km	M1
identifying the upper bound of the time as 3.25 hours	M1
recognising lower bound speed requires the smallest distance divided by the largest time	M1
using speed = distance / time with these bounds	M1
185.5 / 3.25 = 57.0769...	M1
57.1 km/h (3 s.f.)	A1
Final answer: 57.1 km/h

Mark scheme for Question 20 [6 marks]
Question 20[6 marks]
Answer or working	Marks
P(both milk) = 5/12 x 4/11 = 20/132	M1
P(at least one dark) = 1 - 20/132 = 112/132	M1
P(both dark) = 7/12 x 6/11 = 42/132	M1
using conditional probability P(both dark \| at least one dark) = P(both dark)/P(at least one dark)	M1
42/132 divided by 112/132 = 42/112	M1
3/8	A1

Mark scheme for Question 21 [6 marks]
Question 21[6 marks]
Answer or working	Marks
F = k/d^2	M1
40 = k/9	M1
k = 360	A1
forming 2.5 = 360/d^2	M1
d^2 = 360 / 2.5 = 144	M1
d = 12 cm	A1
Final answer: 12 cm