Year 10 Paper 2: Algebra, Graphs and Geometry
Covers linear algebra, sequences and graphs, angles, polygons and circle theorems, ratio and proportion, and statistics and probability.
Year 10 here means a typical teaching order, not a syllabus rule. No exam board defines what belongs to Year 10, and schools sequence the course differently. Check it against your own scheme of work before using it to decide what a class has covered.
Questions
Question 1 [2 marks]
Linear Equations and Inequalities
Solve x/4 - 3 = 9.
Question 2 [3 marks]
Statistics and Probability
The numbers of goals scored by a football team in six matches are 2, 0, 3, 1, 4 and 2.
Find the mean and the range of the number of goals.
Question 3 [2 marks]
Linear Equations and Inequalities
Solve 6x - 11 = 31.
Question 4 [3 marks]
Ratio and Proportion
A fruit drink is made from mango juice and orange juice in the ratio 3:4.
Freya makes 490 ml of the drink.
How many millilitres of orange juice does she need?
Question 5 [3 marks]
Sequences and Graphs
Find the midpoint of the line segment joining the points (2, -5) and (8, 3).
Question 6 [3 marks]
Ratio and Proportion
A fruit squash is made from concentrate and water in the ratio 1:6.
Yuki uses 150 ml of concentrate. Find the total volume of squash she makes, in millilitres.
Question 7 [4 marks]
Angles, Polygons and Circle Theorems
Triangle PQR is isosceles, with PQ = PR. Angle QPR = 2x degrees and angle PQR = 3x degrees.
Find the value of x, then find the size of angle QPR.
Question 8 [4 marks]
Sequences and Graphs
The first four terms of a sequence are 5, 9, 13, 17.
Find an expression for the nth term, then use it to determine whether 205 is a term of the sequence.
Question 9 [3 marks]
Linear Equations and Inequalities
Solve the inequality 4x + 7 < 2x + 19.
Question 10 [4 marks]
Ratio and Proportion
In a school, the ratio of Year 10 students to Year 11 students is 5:4.
The ratio of Year 11 students to Year 12 students is 2:3.
Find the ratio of Year 10 to Year 11 to Year 12 students, in its simplest form.
Question 11 [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 12 [4 marks]
Ratio and Proportion
y is directly proportional to x^3.
When x = 2, y = 40.
Find y when x = 5.
Question 13 [4 marks]
Statistics and Probability
A spinner has sections labelled A, B, C and D.
P(A) = 0.3, P(B) = 0.25 and P(C) = 0.2.
Find P(D).
Question 14 [4 marks]
Angles, Polygons and Circle Theorems
In parallelogram WXYZ, angle W = (4x + 10) degrees and angle X = (2x + 26) degrees, where W and X are adjacent vertices.
Find the value of x, then find angle W.
Question 15 [4 marks]
Ratio and Proportion
A map has a scale of 1:20000.
The distance between two farms on the map is 5.4 cm.
Find the real distance between the farms in kilometres.
Question 16 [4 marks]
Linear Equations and Inequalities
Solve (2x + 5)/3 = x - 4.
Question 17 [5 marks]
Angles, Polygons and Circle Theorems
A, B and C are points on the circumference of a circle with centre O.
B is on the major arc AC and angle AOC (the non-reflex angle at the centre) = 136 degrees.
Find angle ABC and angle OAC.
Model solutions
| Question 1[2 marks] | |
|---|---|
| Answer or working | Marks |
| adding 3 to both sides to get x/4 = 12 | M1 |
| x = 48 | A1 |
| Question 2[3 marks] | |
|---|---|
| Answer or working | Marks |
| summing the six values to get 12 | M1 |
| mean = 2 | A1 |
| range = 4 | A1 |
| Final answer: mean 2, range 4 | |
| Question 3[2 marks] | |
|---|---|
| Answer or working | Marks |
| adding 11 to both sides to get 6x = 42 | M1 |
| x = 7 | A1 |
| Question 4[3 marks] | |
|---|---|
| Answer or working | Marks |
| one ratio part = 490 / 7 = 70 | M1 |
| multiplying by 4 | M1 |
| 280 ml | A1 |
| Question 5[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 6[3 marks] | |
|---|---|
| Answer or working | Marks |
| recognising 150 ml represents 1 part of the ratio | M1 |
| total parts = 1 + 6 = 7 | M1 |
| 1050 ml | A1 |
| Question 7[4 marks] | |
|---|---|
| Answer or working | Marks |
| recognising angle PRQ = angle PQR = 3x, since triangle PQR is isosceles | M1 |
| using angles in a triangle sum to 180 degrees: 2x + 3x + 3x = 180 | M1 |
| solving 8x = 180 | M1 |
| x = 22.5, so angle QPR = 45 degrees | A1 |
| Final answer: x = 22.5; angle QPR = 45 degrees | |
| Question 8[4 marks] | |
|---|---|
| Answer or working | Marks |
| identifying the common difference 4 | M1 |
| comparing with 4n to find the adjustment +1, giving 4n + 1 | M1 |
| solving 4n + 1 = 205 | M1 |
| n = 51, so 205 is a term of the sequence | A1 |
| Final answer: 4n + 1; yes, 205 is the 51st term | |
| Question 9[3 marks] | |
|---|---|
| Answer or working | Marks |
| collecting terms in x on one side, such as 2x + 7 < 19 | M1 |
| 2x < 12 | M1 |
| x < 6 | A1 |
| Question 10[4 marks] | |
|---|---|
| Answer or working | Marks |
| scaling the second ratio so the Year 11 parts match: 2:3 becomes 4:6 | M1 |
| combining to get Year 10 : Year 11 : Year 12 = 5 : 4 : 6 | M1 |
| checking the ratio cannot be simplified further | M1 |
| 5:4:6 | A1 |
| Question 11[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 |
| Question 12[4 marks] | |
|---|---|
| Answer or working | Marks |
| y = kx^3 | M1 |
| 40 = 8k, so k = 5 | M1 |
| substituting x = 5 into y = 5x^3 | M1 |
| 625 | A1 |
| Question 13[4 marks] | |
|---|---|
| Answer or working | Marks |
| probabilities summing to 1 | M1 |
| 0.3 + 0.25 + 0.2 = 0.75 | M1 |
| 1 - 0.75 | M1 |
| 0.25 | A1 |
| Question 14[4 marks] | |
|---|---|
| Answer or working | Marks |
| using adjacent angles in a parallelogram sum to 180 degrees | M1 |
| forming (4x + 10) + (2x + 26) = 180 | M1 |
| solving 6x + 36 = 180 to get x = 24 | M1 |
| angle W = 106 degrees | A1 |
| Final answer: x = 24; angle W = 106 degrees | |
| Question 15[4 marks] | |
|---|---|
| Answer or working | Marks |
| multiplying 5.4 by 20000 | M1 |
| 108000 cm | A1 |
| converting cm to km by dividing by 100000 | M1 |
| 1.08 km | A1 |
| Question 16[4 marks] | |
|---|---|
| Answer or working | Marks |
| multiplying both sides by 3 to get 2x + 5 = 3(x - 4) | M1 |
| expanding to 2x + 5 = 3x - 12 | M1 |
| collecting terms to get 17 = x | M1 |
| x = 17 | A1 |
| Question 17[5 marks] | |
|---|---|
| Answer or working | Marks |
| recognising the angle at the centre is twice the angle at the circumference on the same arc | M1 |
| angle ABC = 136 / 2 | M1 |
| angle ABC = 68 degrees | A1 |
| using triangle OAC is isosceles since OA = OC, so base angles = (180 - 136)/2 | M1 |
| angle OAC = 22 degrees | A1 |
| Final answer: angle ABC = 68 degrees, angle OAC = 22 degrees | |