Year 10 Paper 6: Full Year 10 Review
Covers number and calculation, fractions, decimals and percentages, ratio and proportion, indices and standard form, linear algebra, graphs and coordinates, sequences, and angles and geometrical reasoning.
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 [1 marks]
Indices and Standard Form
Write the number 45000 in standard form.
Question 2 [2 marks]
Fractions, Decimals and Percentages
Work out 23% of £560.
Question 3 [2 marks]
Number and Calculation
A number, n, is rounded to the nearest whole number. The result is 15. Write down the error interval for n.
Question 4 [5 marks]
Linear Algebra
a = 4 and b = -3. Work out the value of each expression.
a + b
ab
a - 2b
Question 5 [1 marks]
Graphs and Coordinates
Find the gradient of the straight line joining the points (-3, -4) and (1, -4).
Question 6 [2 marks]
Number and Calculation
Find the LCM of 8 and 12.
Question 7 [2 marks]
Linear Algebra
Expand and simplify 2(x + 6) + 5(x - 2)
Question 8 [2 marks]
Number and Calculation
Work out sqrt(81) + 2^3.
Question 9 [3 marks]
Sequences
The nth term of a sequence is n^2 + 4n + 3. Find the value of n for which the term is 63.
Question 10 [3 marks]
Angles and Geometrical Reasoning
Column vectors are written in the form (x, y), where x is the top (horizontal) number and y is the bottom (vertical) number. a = (5, -2) and b = (-3, 4).
Write down 2a as a column vector.
Work out a + b as a column vector.
Work out a - b as a column vector.
Question 11 [3 marks]
Number and Calculation
Find the smallest positive integer n such that 360 x n is a perfect cube.
Question 12 [3 marks]
Fractions, Decimals and Percentages
After a pay rise of 3%, Kwame's monthly salary is £2,729.50. Work out his monthly salary before the rise.
Question 13 [3 marks]
Graphs and Coordinates
Use your table of values from Question 3 for y = x^3 - 4.
Draw the graph of y = x^3 - 4 for -2 <= x <= 2 on the grid provided.
Write down the coordinates of the point where the graph crosses the y-axis.
Question 14 [4 marks]
Ratio and Proportion
The table shows some values of x and y, where y is directly proportional to x^2. x: 2, 4, 6 y: 12, 48, ?
Find the value of k in y = kx^2.
Complete the table by finding the missing value of y when x = 6.
Question 15 [5 marks]
Graphs and Coordinates
The line L has equation y = ax + 4 and passes through the point (6, -2). Find the value of a. Then state the equation of the line perpendicular to L that passes through (6, -2).
State the equation of the line perpendicular to L through (6, -2).
Question 16 [6 marks]
Linear Algebra
These formulae involve a square or a square root.
The area of a circle of radius r is A = pi * r^2. Make r the subject of the formula.
v^2 = u^2 + 2as. Make u the subject of the formula.
Question 17 [6 marks]
Graphs and Coordinates
(a) Rearrange 3x - 2y = 6 to make y the subject. (b) Rearrange x + y = 7 to make y the subject. (c) By drawing suitable graphs on the grid, solve the simultaneous equations 3x - 2y = 6 and x + y = 7.
Question 18 [7 marks]
Angles and Geometrical Reasoning
The front elevation of a step-shaped prism is an L-shape, with these measurements: overall width 10 cm, overall height 8 cm, the height of the lower step (across the full width) is 3 cm, the width of the upper part is 4 cm, and the upper part rises a further 5 cm above the lower step. The prism is 6 cm deep (front to back).
Work out the area of the front elevation.
Work out the perimeter of the front elevation.
The prism is 6 cm deep. Work out the total surface area of the solid.
Question 19 [4 marks]
Indices and Standard Form
Simplify (5 + sqrt(3))/(sqrt(3) - 1) fully, writing your answer in the form a + bsqrt(3), where a and b are integers.
Question 20 [3 marks]
Number and Calculation
A padlock code has 3 digits. Each digit can be any number from 0 to 9 and digits may repeat, but the first digit cannot be 0 and the last digit cannot be 0. Work out the number of different codes possible.
Question 21 [3 marks]
Indices and Standard Form
Simplify sqrt(x^3) / x^(1/4). Give your answer as a single power of x.
Question 22 [4 marks]
Sequences
Here are the first five terms of a sequence. 3, 8, 15, 24, 35
Work out the first differences between consecutive terms.
Work out the second differences, and use this to explain why the sequence is quadratic.
Work out the next term in the sequence.
Question 23 [2 marks]
Number and Calculation
Explain why rounding 148 to 1 significant figure gives 100, and not 150.
Question 24 [4 marks]
Sequences
A quadratic sequence has nth term n^2 - 7n + 15.
Work out the 5th term of the sequence.
Find the smallest value in the sequence, and state the term number(s) at which it occurs.
Model solutions
| Question 1[1 mark] | |
|---|---|
| Answer or working | Marks |
| 4.5 x 10^4 | B1oe |
| Question 2[2 marks] | |
|---|---|
| Answer or working | Marks |
| 0.23 x 560 | M1oe |
| 128.80 (pounds) | A1 |
| Final answer: £128.80 | |
| Question 3[2 marks] | |
|---|---|
| Answer or working | Marks |
| one correct bound identified, 14.5 or 15.5 | M1oe |
| 14.5 <= n < 15.5 | A1cao |
| Question 4[5 marks] | |
|---|---|
| Answer or working | Marks |
| 1 | B1cao |
| 4 x (-3) seen | M1oe |
| -12 | A1cao |
| 2 x (-3) (= -6) seen | M1oe |
| 10 | A1cao |
| Final answer: 1 | -12 | 10 | |
| Question 5[1 mark] | |
|---|---|
| Answer or working | Marks |
| 0 | B1cao |
| Question 6[2 marks] | |
|---|---|
| Answer or working | Marks |
| 8 = 2^3 and 12 = 2^2 x 3, or a correct list of multiples of each number | M1 |
| 24 | A1cao |
| Question 7[2 marks] | |
|---|---|
| Answer or working | Marks |
| 2x + 12 + 5x - 10 (both brackets expanded correctly) | M1oe |
| 7x + 2 | A1cao |
| Question 8[2 marks] | |
|---|---|
| Answer or working | Marks |
| correct method e.g. sqrt(81)=9 and 2^3=8 shown | M1 |
| 17 | A1cao |
| Question 9[3 marks] | |
|---|---|
| Answer or working | Marks |
| form equation n^2 + 4n + 3 = 63 | M1oe |
| rearrange and factorise to (n + 10)(n - 6) = 0 or equivalent method | M1 |
| n = 6 cao (positive integer solution) | A1 |
| Final answer: 6 | |
| Question 10[3 marks] | |
|---|---|
| Answer or working | Marks |
| (10, -4) | B1cao |
| (2, 2) | B1cao |
| (8, -6) | B1cao |
| Final answer: (10, -4) | (2, 2) | (8, -6) | |
| Question 11[3 marks] | |
|---|---|
| Answer or working | Marks |
| prime factorise 360 = 2^3 x 3^2 x 5^1 | M1 |
| choose multipliers to make exponents multiples of 3: need 3^1 and 5^2 | M1 |
| n = 3 x 25 = 75 | A1cao |
| Final answer: 75 | |
| Question 12[3 marks] | |
|---|---|
| Answer or working | Marks |
| correct multiplier 1.03 | M1oe |
| complete method 2729.50 / 1.03 | M1 |
| £2,650 | A1cao |
| Question 13[3 marks] | |
|---|---|
| Answer or working | Marks |
| all 5 points from the table plotted correctly, ft candidate's table from Q3 | B1 |
| single smooth curve drawn through all the points, correct increasing s-shape with no turning points | B1 |
| (0, -4) | B1cao |
| Final answer: Smooth curve through (-2,-12), (-1,-5), (0,-4), (1,-3), (2,4) | (0, -4) | |
| Question 14[4 marks] | |
|---|---|
| Answer or working | Marks |
| uses a pair of values, e.g. 12 = k*2^2, or 48 = k*4^2 | M1 |
| k = 3 | A1cao |
| substitutes x = 6 into y = 3x^2 (ft their k) | M1 |
| y = 108 | A1cao |
| Final answer: k = 3 | y = 108 | |
| Question 15[5 marks] | |
|---|---|
| Answer or working | Marks |
| substitute (6, -2) into y = ax + 4: -2 = 6a + 4 and form 6a = -6 | M1 |
| a = -1 | A1cao |
| use perpendicular gradient product = -1: gradient of L is -1 so perpendicular gradient = 1 | M1 |
| use point (6, -2) to form equation y = 1x + c and solve for c: -2 = 6 + c | M1 |
| y = x - 8 | A1cao |
| Final answer: a = -1 | y = x - 8 | |
| Question 16[6 marks] | |
|---|---|
| Answer or working | Marks |
| divides both sides by pi, e.g. r^2 = A/pi | M1 |
| square roots both sides, dependent on the previous method mark | dM1 |
| r = sqrt(A/pi) oe, cao (positive root only, since r is a length) | A1 |
| subtracts 2as from both sides, e.g. u^2 = v^2 - 2as | M1 |
| square roots both sides, dependent on the previous method mark | dM1 |
| u = sqrt(v^2 - 2as) oe | A1cao |
| Final answer: r = sqrt(A/pi) | u = sqrt(v^2 - 2as) | |
| Question 17[6 marks] | |
|---|---|
| Answer or working | Marks |
| y = 1.5x - 3 (oe, e.g. y = (3x - 6) / 2) | B1 |
| y = 7 - x | B1 |
| draws y = 1.5x - 3 correctly, ft from (a), e.g. through (0,-3), (2,0), (4,3) | M1 |
| draws y = 7 - x correctly, ft from (b), e.g. through (0,7), (4,3), (7,0) | M1 |
| identifies intersection at (4,3) | A1 |
| states solution x = 4, y = 3 | A1ft |
| Final answer: y = 1.5x - 3 | y = 7 - x | x = 4, y = 3 | |
| Question 18[7 marks] | |
|---|---|
| Answer or working | Marks |
| splitting into two rectangles, e.g. 10 x 3 (= 30) and 4 x 5 (= 20) | M1 |
| 50 cm^2 | A1cao |
| identifying the missing horizontal length, 10 - 4 = 6 cm, and attempting to sum all six sides (10 + 3 + 6 + 5 + 4 + 8) | M1 |
| 36 cm | A1cao |
| 2 x 50 ft (their area from part a), the two L-shaped faces (= 100) | M1 |
| 36 x 6 ft (their perimeter from part b, x depth) (= 216) | M1 |
| 316 cm^2 cao | A1ft |
| Final answer: 50 cm^2 | 36 cm | 316 cm^2 | |
| Question 19[4 marks] | |
|---|---|
| Answer or working | Marks |
| multiplies numerator and denominator by (sqrt(3) + 1) | M1 |
| denominator simplifies to 3 - 1 = 2 | M1 |
| numerator simplifies to 8 + 6sqrt(3) | A1 |
| 4 + 3sqrt(3) cao (a = 4, b = 3) | A1 |
| Final answer: 4 + 3sqrt(3) (a = 4, b = 3) | |
| Question 20[3 marks] | |
|---|---|
| Answer or working | Marks |
| identifies 9 choices for the first digit and 9 choices for the last digit (excluding 0 in each case) | M1 |
| (dep) 9 * 10 * 9 | M1oe |
| 810 | A1cao |
| Question 21[3 marks] | |
|---|---|
| Answer or working | Marks |
| sqrt(x^3) written as x^(3/2) | M1 |
| indices subtracted: 3/2 - 1/4 | M1 |
| x^(5/4) oe (accept x^1.25) | A1 |
| Final answer: x^(5/4) | |
| Question 22[4 marks] | |
|---|---|
| Answer or working | Marks |
| 5, 7, 9, 11 all correct | B1cao |
| second differences all equal to 2, ft from part (a) | B1 |
| correct explanation, e.g. a constant (non-zero) second difference means the sequence is quadratic | B1 |
| 48 cao, ft from their pattern of differences | B1 |
| Final answer: 5, 7, 9, 11 | Second differences are 2, 2, 2. Since the second differences are constant, the sequence is quadratic. | 48 | |
| Question 23[2 marks] | |
|---|---|
| Answer or working | Marks |
| identifying that the first significant figure is the hundreds digit, 1, and all following digits become 0 for 1 sf | B1 |
| correct reasoning that the next digit (4, the tens digit) is less than 5, so the 1 is not rounded up, giving 100; 150 is not a rounding to 1 significant figure | B1 |
| Final answer: 100, because the first significant figure (1) is followed by a 4, which is less than 5, so it rounds down; 150 has two significant figures, not one. | |
| Question 24[4 marks] | |
|---|---|
| Answer or working | Marks |
| 5 | B1cao |
| finding the vertex of the quadratic at n=3.5 using n=-b/(2a) (or an equivalent method) | M1 |
| evaluating the sequence at n=3 and n=4 (both give 3) | M1 |
| smallest value is 3, occurring at both n=3 and n=4 | A1 |
| Final answer: 5 | Smallest value is 3, at n=3 and n=4 | |