Foundation Tier - Year 10

OCR-Style Year 10 Foundation Paper 2

An OCR-style Year 10 Foundation Paper 2: 100 marks, 90 minutes, non-calculator, matching OCR J560's published paper format. Covers the same nine Year 10 topics as the board-agnostic set: number and calculation, fractions, decimals and percentages, ratio and proportion, indices and standard form, linear algebra, graphs and coordinates, sequences, angles and geometrical reasoning, and area, volume and measures. OCR's non-calculator paper is the middle paper of a sitting rather than the first, verified against OCR's own published question papers.

41 questions - 100 marks

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.

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Questions

Question 1 [1 marks]

Area, Volume and Measures

A cube has edge length 4 cm. Which of the following is the total surface area of the cube?

Question 2 [1 marks]

Linear Algebra

Solve x - 5 = 9

Question 3 [1 marks]

Ratio and Proportion

Simplify the ratio 35:14.

Question 4 [1 marks]

Indices and Standard Form

Write the number 45000 in standard form.

Question 5 [2 marks]

Angles and Geometrical Reasoning

Lines l and m are parallel and are crossed by a straight transversal. Angle y is in the alternate position to the 47 degree angle marked between l and m.

Question 6 [3 marks]

Fractions, Decimals and Percentages

(Non-calculator) Work out 17.5% of 240.

Question 7 [3 marks]

Sequences

Three sequences are shown below. Sequence A: 7, 11, 15, 19 Sequence B: 4, 12, 36, 108 Sequence C: 3, 6, 11, 18

State whether Sequence A is arithmetic, geometric, or neither of these. Give a reason for your answer.

State whether Sequence B is arithmetic, geometric, or neither of these. Give a reason for your answer.

State whether Sequence C is arithmetic, geometric, or neither of these. Give a reason for your answer.

Question 8 [1 marks]

Graphs and Coordinates

Write down the gradient of the line with equation y = 6 - x.

Question 9 [1 marks]

Indices and Standard Form

Simplify (r^3)^4.

Question 10 [1 marks]

Graphs and Coordinates

Which pair of simultaneous equations below has no solution, because the two lines are parallel?

Question 11 [2 marks]

Linear Algebra

Solve 7x - 4 = 3x + 20

Question 12 [1 marks]

Number and Calculation

Write down all the factors of 24.

Question 13 [1 marks]

Ratio and Proportion

A ratio is 8:9. Write down the second quantity as a fraction of the first quantity, giving your fraction in its simplest form.

Question 14 [2 marks]

Linear Algebra

Make n the subject of the formula m = (n + 4) / 5. Show each step.

Question 15 [2 marks]

Number and Calculation

Three warning lights flash at regular intervals: the red light every 15 seconds, the amber light every 20 seconds and the green light every 24 seconds. All three flash together at exactly 9:00:00 am. Work out the time when all three will next flash together.

Question 16 [2 marks]

Linear Algebra

Make m the subject of the formula: k = 3m + 2n

Question 17 [1 marks]

Number and Calculation

Work out 17 x 17.

Question 18 [1 marks]

Ratio and Proportion

Given that m:n = 4:7, write n in terms of m as a linear function in the form n = km.

Question 19 [2 marks]

Linear Algebra

Factorise x^2 + 3x - 40

Question 20 [2 marks]

Number and Calculation

Work out the smallest positive integer that is a multiple of 4, 6 and 9.

Question 21 [3 marks]

Fractions, Decimals and Percentages

Work out 4 1/2 ÷ 3/8. Give your answer in simplest form.

Question 22 [2 marks]

Linear Algebra

Solve x / 3 + 4 = 9 Show your working.

Question 23 [2 marks]

Ratio and Proportion

A fruit bowl contains 8 apples, 6 bananas and 10 oranges. Write the ratio apples : bananas : oranges in its simplest form.

Question 24 [4 marks]

Graphs and Coordinates

Complete the table of values for y = 4/x. x : -4 , -2 , -1 , 1 , 2 , 4 y : -1 , _ , _ , _ , _ , 1

Question 25 [2 marks]

Linear Algebra

Solve 7 - x = 2x + 1 Show your working.

Question 26 [2 marks]

Angles and Geometrical Reasoning

Work out the value of x.

Question 27 [2 marks]

Linear Algebra

Solve 5x - 7 = 18

Question 28 [2 marks]

Number and Calculation

Write 18 as a product of its prime factors.

Question 29 [3 marks]

Ratio and Proportion

Three graphs, A, B and C, each show y plotted against x for x > 0. Graph A is a straight line through the origin. Graph B is a curve that starts high up on the y-axis, decreases steeply, and gets closer and closer to the x-axis without ever touching it, and never touches the y-axis either. Graph C is a straight line that crosses the y-axis above the origin (it does not pass through the origin).

State which graph, A, B or C, could show y directly proportional to x.

State which graph, A, B or C, could show y inversely proportional to x.

Explain why graph C cannot show y directly proportional to x, or y inversely proportional to x.

Question 30 [3 marks]

Linear Algebra

Solve 5(x + 3) = 2(x + 12) Show your working.

Question 31 [3 marks]

Angles and Geometrical Reasoning

Enlarge triangle T with vertices (1, 1), (3, 1) and (1, 2) by a scale factor of 2 from the origin. Give the coordinates of the enlarged triangle's vertices.

Question 32 [2 marks]

Linear Algebra

Expand and simplify (x - 5)^2

Question 33 [3 marks]

Angles and Geometrical Reasoning

v is the column vector (k, 4), where k > 0, and |v| = 5. Find the value of k.

Question 34 [2 marks]

Linear Algebra

Solve 3(2x - 1) = 15 Show your working.

Question 35 [4 marks]

Ratio and Proportion

y is directly proportional to x^3. When x = 2, y = 40.

Find a formula for y in terms of x.

Work out the value of y when x = 4.

Question 36 [5 marks]

Linear Algebra

A car park charges according to the formula C = 2.50 + 1.20h, where C is the total cost in £ and h is the number of hours parked.

Make h the subject of the formula.

Sasha pays £8.50 to park her car. Work out how many hours she parked for.

Question 37 [5 marks]

Sequences

The table shows the number of dots used to make a sequence of patterns. Pattern number (n): 1, 2, 3, 4, 5 Number of dots: 2, 7, 12, 17, ?

Complete the table by finding the number of dots in Pattern 5.

Find an expression, in terms of n, for the number of dots in Pattern n.

Find the pattern number that has 92 dots.

Question 38 [5 marks]

Linear Algebra

A wedding venue charges a fixed booking fee of £150 plus £40 per guest. The total cost, C (£), for n guests is given by the formula C = 150 + 40n

Make n the subject of the formula.

A couple have a budget of £3000 for the venue. Work out the maximum number of guests they can invite.

Question 39 [5 marks]

Graphs and Coordinates

Two courier companies charge for delivering parcels. SwiftPost charges C = 2n + 30 and ParcelGo charges C = 5n + 9, where C is the cost in pounds and n is the number of parcels. The graphs of both charges, for n from 0 to 15, are shown on the grid.

Write down the number of parcels for which the two companies charge the same amount, and state this cost.

A customer needs to send 15 parcels. By using the graphs or otherwise, determine which company is cheaper and find the difference in cost.

Question 40 [7 marks]

Sequences

Arjun is training for a marathon. In week 1, he runs 8 km. Each week after that, he runs 3 km more than the week before.

Work out how far Arjun runs in week 6.

Find an expression, in terms of n, for the distance, in km, Arjun runs in week n.

Arjun wants to know the first week in which he will run at least 50 km. Find this week number.

Question 41 [3 marks]

Linear Algebra

Solve (2x - 1) / 3 + (x + 1) / 2 = 6 Show your working.

Model solutions

Mark scheme for Question 1 [1 mark]
Question 1[1 mark]
Answer or working	Marks
C (96 cm^2)	B1
Final answer: C) 96 cm^2

Mark scheme for Question 2 [1 mark]
Question 2[1 mark]
Answer or working	Marks
x = 14	B1cao

Mark scheme for Question 3 [1 mark]
Question 3[1 mark]
Answer or working	Marks
5:2	B1cao

Mark scheme for Question 4 [1 mark]
Question 4[1 mark]
Answer or working	Marks
4.5 x 10^4	B1oe

Mark scheme for Question 5 [2 marks]
Question 5[2 marks]
Answer or working	Marks
y = 47	B1cao
correct reason: alternate angles are equal (l parallel to m)	B1
Final answer: y = 47 degrees

Mark scheme for Question 6 [3 marks]
Question 6[3 marks]
Answer or working	Marks
10% of 240 = 24	M1
5% = 12 and 2.5% = 6 (oe building blocks)	M1
42	A1

Mark scheme for Question 7 [3 marks]
Question 7[3 marks]
Answer or working	Marks
arithmetic, since the terms have a common difference of 4	B1oe
geometric, since the terms have a common ratio of 3	B1oe
neither, since the differences (3, 5, 7) are not constant and the ratios are not constant	B1oe
Final answer: Arithmetic (common difference of 4) \| Geometric (common ratio of 3) \| Neither (it is a quadratic sequence)

Mark scheme for Question 8 [1 mark]
Question 8[1 mark]
Answer or working	Marks
-1	B1cao

Mark scheme for Question 9 [1 mark]
Question 9[1 mark]
Answer or working	Marks
r^12	B1cao

Mark scheme for Question 10 [1 mark]
Question 10[1 mark]
Answer or working	Marks
C	B1
Final answer: C) 2y = 4x + 6 and y = 2x - 1

Mark scheme for Question 11 [2 marks]
Question 11[2 marks]
Answer or working	Marks
correct collection of terms, 4x = 24	M1oe
x = 6	A1cao

Mark scheme for Question 12 [1 mark]
Question 12[1 mark]
Answer or working	Marks
All factors listed: 1, 2, 3, 4, 6, 8, 12, 24	B1cao
Final answer: 1, 2, 3, 4, 6, 8, 12, 24 (any order accepted)

Mark scheme for Question 13 [1 mark]
Question 13[1 mark]
Answer or working	Marks
9/8	B1cao

Mark scheme for Question 14 [2 marks]
Question 14[2 marks]
Answer or working	Marks
multiply both sides by 5 or show method leading to n + 4 = 5m	M1
n = 5m - 4	A1cao

Mark scheme for Question 15 [2 marks]
Question 15[2 marks]
Answer or working	Marks
Finds LCM of 15, 20 and 24 (method shown)	M1
9:02:00 am cao (accept 120 seconds or 2 minutes after 9:00:00 am)	A1
Final answer: 9:02:00 am (120 seconds later)

Mark scheme for Question 16 [2 marks]
Question 16[2 marks]
Answer or working	Marks
subtracts 2n from both sides, e.g. k - 2n = 3m	M1
m = (k - 2n)/3 oe	A1cao
Final answer: m = (k - 2n)/3

Mark scheme for Question 17 [1 mark]
Question 17[1 mark]
Answer or working	Marks
289	B1cao

Mark scheme for Question 18 [1 mark]
Question 18[1 mark]
Answer or working	Marks
n = 7/4 m oe (k = 7/4)	B1
Final answer: n = 7/4 m

Mark scheme for Question 19 [2 marks]
Question 19[2 marks]
Answer or working	Marks
two numbers with sum 3 and product -40 identified (8 and -5)	M1oe
(x + 8)(x - 5)	A1cao

Mark scheme for Question 20 [2 marks]
Question 20[2 marks]
Answer or working	Marks
Correct method: prime factors or pairwise LCMs (4 = 2^2, 6 = 23, 9 = 3^2 so LCM = 2^2 3^2)	M1
36	A1cao

Mark scheme for Question 21 [3 marks]
Question 21[3 marks]
Answer or working	Marks
convert 4 1/2 to improper fraction 9/2	M1
divide by 3/8 using reciprocal: 9/2 × 8/3 shown	M1
12	A1cao

Mark scheme for Question 22 [2 marks]
Question 22[2 marks]
Answer or working	Marks
x / 3 = 5	M1oe
x = 15	A1cao

Mark scheme for Question 23 [2 marks]
Question 23[2 marks]
Answer or working	Marks
identify a common factor of 8, 6 and 10 (e.g. 2)	M1oe
4:3:5	A1cao

Mark scheme for Question 24 [4 marks]
Question 24[4 marks]
Answer or working	Marks
y = -2 at x = -2	B1
y = -4 at x = -1	B1
y = 4 at x = 1	B1
y = 2 at x = 2	B1
Final answer: x=-2: y=-2; x=-1: y=-4; x=1: y=4; x=2: y=2

Mark scheme for Question 25 [2 marks]
Question 25[2 marks]
Answer or working	Marks
7 = 3x + 1 or 6 = 3x oe (collect x terms onto one side)	M1
x = 2	A1cao

Mark scheme for Question 26 [2 marks]
Question 26[2 marks]
Answer or working	Marks
3x + 5 = 65 oe (corresponding angles are equal)	M1
x = 20	A1cao

Mark scheme for Question 27 [2 marks]
Question 27[2 marks]
Answer or working	Marks
5x = 25	M1oe
x = 5	A1cao

Mark scheme for Question 28 [2 marks]
Question 28[2 marks]
Answer or working	Marks
correct first split of 18 shown, e.g. 2 x 9, or a factor tree with at least one prime factor correct	M1
2 x 3^2 oe, fully correct in index form	A1
Final answer: 2 x 3^2

Mark scheme for Question 29 [3 marks]
Question 29[3 marks]
Answer or working	Marks
A	B1
B	B1
states that graph C does not pass through the origin, so it cannot be y = kx (oe); or states that it is a straight line, not a reciprocal curve, so it cannot be y = k/x (oe) - either correct reason is sufficient	B1
Final answer: Graph A \| Graph B \| Graph C is a straight line not through the origin, so it cannot be direct proportion (y = kx must pass through the origin); and it is not a curve of the form y = k/x, so it cannot be inverse proportion either.

Mark scheme for Question 30 [3 marks]
Question 30[3 marks]
Answer or working	Marks
5x + 15 = 2x + 24 (expand both brackets correctly)	M1
3x = 9 oe (dep, collect terms correctly)	dM1
x = 3	A1cao

Mark scheme for Question 31 [3 marks]
Question 31[3 marks]
Answer or working	Marks
correct method: multiply each coordinate by 2 (apply scale factor to coordinates)	M1
show multiplication for at least one vertex or evidence applied correctly to all vertices, e.g. (1,1)->(2,2) and (3,1)->(6,2)	M1
(2, 2), (6, 2), (2, 4)	A1cao

Mark scheme for Question 32 [2 marks]
Question 32[2 marks]
Answer or working	Marks
writes as (x - 5)(x - 5) and attempts to expand, at least 3 of 4 terms correct	M1
x^2 - 10x + 25	A1cao

Mark scheme for Question 33 [3 marks]
Question 33[3 marks]
Answer or working	Marks
k^2 + 4^2 = 5^2	M1oe
dep: k^2 = 9	M1
k = 3 cao (rejects k = -3 since k > 0)	A1
Final answer: k = 3

Mark scheme for Question 34 [2 marks]
Question 34[2 marks]
Answer or working	Marks
6x - 3 = 15 oe (expand correctly)	M1
x = 3	A1cao

Mark scheme for Question 35 [4 marks]
Question 35[4 marks]
Answer or working	Marks
finds k = 40/2^3 (= 40/8)	M1oe
y = 5x^3	A1oe
substitutes x = 4 into their formula	M1
y = 320	A1cao
Final answer: y = 5x^3 \| y = 320

Mark scheme for Question 36 [5 marks]
Question 36[5 marks]
Answer or working	Marks
subtracts 2.50 from both sides, e.g. C - 2.50 = 1.20h	M1
h = (C - 2.50)/1.20 oe	A1cao
substitutes C = 8.50 into the formula from part (a), ft, e.g. h = (8.50 - 2.50)/1.20	M1
simplifies to h = 6/1.20	M1oe
5 (hours)	A1cao
Final answer: h = (C - 2.50)/1.20 \| 5 hours

Mark scheme for Question 37 [5 marks]
Question 37[5 marks]
Answer or working	Marks
22	B1cao
common difference of 5 used correctly, e.g. 5n + c	M1oe
5n - 3 oe	A1cao
5n - 3 = 92 oe, or 5n = 95 (ft their expression)	M1
Pattern 19	A1cao
Final answer: 22 \| 5n - 3 \| Pattern 19

Mark scheme for Question 38 [5 marks]
Question 38[5 marks]
Answer or working	Marks
subtracts 150 from both sides, e.g. C - 150 = 40n	M1
n = (C - 150)/40 oe	A1cao
substitutes C = 3000 into the formula from part (a), ft, e.g. n = (3000 - 150)/40	M1
evaluates n = 71.25 (or 2850/40) and recognises the number of guests must be a whole number that does not exceed the budget	M1
71 (guests)	A1cao
Final answer: n = (C - 150)/40 \| 71 guests

Mark scheme for Question 39 [5 marks]
Question 39[5 marks]
Answer or working	Marks
n = 7 parcels	B1
cost = GBP 44	B1
substitutes n = 15 into both C = 2n + 30 and C = 5n + 9	M1
SwiftPost = GBP 60 and ParcelGo = GBP 84	A1
concludes SwiftPost is cheaper, by GBP 24 (ft from their two costs)	A1
Final answer: n = 7 parcels, cost = GBP 44 \| SwiftPost is cheaper by GBP 24

Mark scheme for Question 40 [7 marks]
Question 40[7 marks]
Answer or working	Marks
8 + 5 x 3 oe, or lists terms up to week 6	M1
23 km	A1cao
common difference of 3 used correctly, e.g. 3n + c	M1oe
3n + 5 oe	A1cao
3n + 5 >= 50 oe (ft their expression)	M1
3n >= 45 oe, leading to n >= 15	M1
week 15	A1cao
Final answer: 23 km \| 3n + 5 \| Week 15

Mark scheme for Question 41 [3 marks]
Question 41[3 marks]
Answer or working	Marks
multiplies every term by 6, the LCM of 3 and 2, e.g. 2(2x - 1) + 3(x + 1) = 36	M1oe
expands and collects terms to reach 7x + 1 = 36 oe	dM1dep
x = 5	A1cao