1
Write each of the following using correct algebraic notation, simplified where possible.
Non-calculator
ORIGINAL
(i) p × p × p (ii) 5 × q (iii) m ÷ n (iv) 2 × a × a × b
(2)
(Total for Question 1 is 2 marks)
Pearson Edexcel GCSE (9–1) Mathematics
Algebra — topic question pack
1MA1
Original practice paper written in the style of Edexcel papers — not a Pearson publication. This pack is untiered and can include both Foundation and Higher content.
Name: __________________________________
Class: __________________________________
Answer every question. Show all stages of your working. The total number of marks is 87.
This topic pack mixes calculator and non-calculator practice. Each question states whether a calculator may be used.
Question paper
2
You are given the formulae v = u + at and s = ut + ½at². In this question u = 6, a = −3 and t = 4.
Either calculator policy
ORIGINAL
(a)Work out the value of v.
(2)
(b)Work out the value of s.
(2)
(Total for Question 2 is 4 marks)
3
Here are four statements. A: 5y + 2 B: V = IR C: 3(x − 2) = 3x − 6 D: 2x + 7 = 15
Non-calculator
ORIGINAL
One of the statements is an expression, one is an equation, one is a formula and one is an identity. Match each of A, B, C and D to the correct word.
(2)
(Total for Question 3 is 2 marks)
4
A is the point (−4, 5) and B is the point (2, −3). M is the midpoint of the line segment AB.
Non-calculator
ORIGINAL
(a)Work out the coordinates of M.
(2)
(b)The point C has coordinates (−4, −2). Write down the number of the quadrant in which C lies.
(1)
(Total for Question 4 is 3 marks)
5
The graph of y = k^x, where k > 0, passes through the point (2, 49).
Non-calculator
ORIGINAL
(a)Find the value of k.
(2)
(b)Write down the coordinates of the point where the graph crosses the y-axis.
(1)
(Total for Question 5 is 3 marks)
6
The distance-time graph for Sara's cycle ride is made of three straight sections: from (0 min, 0 km) up to (30 min, 12 km); a horizontal section from (30 min, 12 km) to (45 min, 12 km); then a straight section from (45 min, 12 km) down to (90 min, 0 km).
Either calculator policy
ORIGINAL
(a)What does the horizontal section of the graph tell you about Sara's ride?
(1)
(b)Use the graph to work out Sara's speed, in km/h, on her journey home.
(2)
(Total for Question 6 is 3 marks)
7
A car starts from rest. On a velocity-time graph, its motion is a straight line from (0 s, 0 m/s) to (8 s, 20 m/s). After that the car travels at a constant 20 m/s for a further 12 seconds.
Either calculator policy
ORIGINAL
(a)Work out the acceleration of the car during the first 8 seconds.
(2)
(b)Work out the total distance travelled by the car during the 20 seconds.
(3)
(Total for Question 7 is 5 marks)
8
A sequence has first term 5. The term-to-term rule is 'multiply by 2 and then subtract 3'.
Non-calculator
ORIGINAL
Work out the first four terms of the sequence.
(2)
(Total for Question 8 is 2 marks)
9
This question is about manipulating algebraic expressions.
Non-calculator
ORIGINAL
(a)Expand and simplify (x + 5)(x − 3).
(2)
(b)Factorise fully x² − 9x + 20.
(2)
(Total for Question 9 is 4 marks)
10
Simplify each of the following. Write each answer as a single power or term where possible.
Non-calculator
ORIGINAL
(a)p⁶ × p⁴
(1)
(b)(y⁵)³
(1)
(c)12a⁵b³ ÷ 3a²b
(1)
(Total for Question 10 is 3 marks)
11
The sizes of the three angles of a triangle, in degrees, are x + 20, 2x and 3x − 20.
Non-calculator
ORIGINAL
(a)Form an equation in x and solve it to find the value of x.
(3)
(b)Write down the size of the largest angle.
(1)
(c)Is the triangle acute-angled, right-angled or obtuse-angled? Give a reason.
(1)
(Total for Question 11 is 5 marks)
12
Solve the quadratic equation by factorising.
Non-calculator
ORIGINAL
x² − 7x + 10 = 0
(3)
(Total for Question 12 is 3 marks)
13
Solve the equation.
Non-calculator
ORIGINAL
5x + 7 = 2x + 22
(3)
(Total for Question 13 is 3 marks)
14
A is the point (1, 4) and B is the point (4, 13).
Non-calculator
ORIGINAL
(a)Work out the gradient of the line AB.
(2)
(b)Find the equation of the line AB in the form y = mx + c.
(2)
(Total for Question 14 is 4 marks)
15
This question is about a linear inequality.
Non-calculator
ORIGINAL
(a)Solve the inequality 4x − 3 > 17.
(2)
(b)Write down the smallest integer value of x that satisfies 4x − 3 > 17.
(1)
(Total for Question 15 is 3 marks)
16
Prove, using algebra, that the sum of any three consecutive integers is always a multiple of 3.
Non-calculator
ORIGINAL
Show a full algebraic proof.
(3)
(Total for Question 16 is 3 marks)
17
Prove that the sum of the squares of any two consecutive integers is always an odd number.
Non-calculator
ORIGINAL
Show a full algebraic proof.
(3)
(Total for Question 17 is 3 marks)
18
You are given the formula y = 3x − 7.
Non-calculator
ORIGINAL
Rearrange the formula to make x the subject.
(2)
(Total for Question 18 is 2 marks)
19
A student is sketching the graph of the straight line with equation y = 2x − 3.
Non-calculator
ORIGINAL
(a)Write down the coordinates of the point where the line crosses the y-axis.
(1)
(b)Work out the coordinates of the point where the line crosses the x-axis.
(2)
(Total for Question 19 is 3 marks)
20
Simplify t⁹ ÷ t⁴.
Non-calculator
ORIGINAL
Write your answer as a single power of t.
(1)
(Total for Question 20 is 1 mark)
21
Solve these equations.
Non-calculator
ORIGINAL
(a)3x + 7 = 22
(2)
(b)x/4 − 2 = 3
(2)
(Total for Question 21 is 4 marks)
22
Here are the first four terms of an arithmetic sequence: 5, 8, 11, 14
Non-calculator
ORIGINAL
(a)Find an expression, in terms of n, for the nth term of the sequence.
(2)
(b)Work out the 20th term of the sequence.
(1)
(c)Is 100 a term of this sequence? You must justify your answer.
(2)
(Total for Question 22 is 5 marks)
23
This question is about number sequences.
Non-calculator
ORIGINAL
(a)Here are the first four square numbers: 1, 4, 9, 16. Write down the next two square numbers.
(1)
(b)Here is a Fibonacci-type sequence: 2, 3, 5, 8, ... Write down the next term.
(1)
(c)Write down the 5th triangular number.
(1)
(Total for Question 23 is 3 marks)
24
A rectangle has length (x + 4) cm and width x cm. The perimeter of the rectangle is 28 cm.
Non-calculator
ORIGINAL
(a)Form an equation in x and solve it to find the value of x.
(3)
(b)Work out the area of the rectangle.
(2)
(Total for Question 24 is 5 marks)
25
A taxi company works out the cost, £C, of a journey of m miles using the formula C = 3 + 2m.
Either calculator policy
ORIGINAL
(a)Work out the cost of a journey of 7 miles.
(2)
(b)A journey costs £27. Work out the number of miles for this journey.
(2)
(Total for Question 25 is 4 marks)
26
A bag contains n red counters and 4 blue counters. Two counters are taken at random without replacement. The probability that both counters are blue is 1/6.
Calculator
ORIGINAL
Work out the total number of counters in the bag. Show your algebraic working.
(5)
(Total for Question 26 is 5 marks)
Pearson Edexcel GCSE (9–1) Mathematics
Mark scheme — Algebra
1
Answer: (i) p³ (ii) 5q (iii) m/n (iv) 2a²b
B2 for all four correct (p³, 5q, m/n, 2a²b). B1 for any two or three correct.
2 marks
2
(a)
Answer: −6
M1 for 6 + (−3) × 4. A1 for −6.
(b)
Answer: 0
M1 for 6 × 4 + ½ × (−3) × 4² (= 24 − 24). A1 for 0.
4 marks
3
Answer: A: expression, B: formula, C: identity (true for every value of x), D: equation
B2 for all four correct (A expression, B formula, C identity, D equation). B1 for any two or three correct.
2 marks
4
(a)
Answer: (−1, 1)
M1 for a correct midpoint method, e.g. ((−4 + 2)/2, (5 + (−3))/2). A1 for (−1, 1).
(b)
Answer: The third quadrant (bottom-left).
B1 for the third quadrant (accept 'bottom-left').
3 marks
5
(a)
Answer: 7
M1 for k² = 49. A1 for k = 7.
(b)
Answer: (0, 1)
B1 for (0, 1).
3 marks
6
(a)
Answer: She was not moving (resting/stationary) for 15 minutes, 12 km from home.
B1 for stationary / not moving / resting (accept any clear statement that distance from home is not changing).
(b)
Answer: 16 km/h
M1 for using the final section of the graph: 12 km in 45 minutes, e.g. 12 ÷ 0.75 or 12 ÷ 45 × 60 (gradient magnitude of the final section). A1 for 16.
3 marks
7
(a)
Answer: 2.5 m/s²
M1 for 20 ÷ 8 (gradient). A1 for 2.5 (m/s²).
(b)
Answer: 320 m
M1 for area of the triangle ½ × 8 × 20 (= 80). M1 for area of the rectangle 12 × 20 (= 240). A1 for 320 (m).
5 marks
8
Answer: 5, 7, 11, 19
M1 for correctly applying the rule at least once (e.g. 5 × 2 − 3 = 7). A1 for all four terms 5, 7, 11, 19.
2 marks
9
(a)
Answer: x² + 2x − 15
M1 for at least three of the four terms correct (x² + 5x − 3x − 15). A1 for x² + 2x − 15.
(b)
Answer: (x − 4)(x − 5)
M1 for (x ± 4)(x ± 5) or a pair of factors of 20 that add to give the middle term. A1 for (x − 4)(x − 5).
4 marks
10
(a)
Answer: p¹⁰
B1 for p¹⁰.
(b)
Answer: y¹⁵
B1 for y¹⁵.
(c)
Answer: 4a³b²
B1 for 4a³b² (accept the three factors handled correctly: 12÷3 = 4, a^(5−2) = a³, b^(3−1) = b²).
3 marks
11
(a)
Answer: x = 30
M1 for (x + 20) + 2x + (3x − 20) = 180. M1 for 6x = 180. A1 for x = 30.
(b)
Answer: 70°
B1 for 70 (from 3x − 20 with x = 30; the three angles are 50°, 60°, 70°).
(c)
Answer: Acute-angled, because its largest angle is 70°, which is less than 90° (so all three angles are less than 90°).
B1 for acute-angled with a reason referring to the largest angle (70°) being less than 90° (ft their angles).
5 marks
12
Answer: x = 2 or x = 5
M1 for (x − 2)(x − 5) [factors of 10 that add to −7]. M1 for setting each factor to 0. A1 for both x = 2 and x = 5.
3 marks
13
Answer: x = 5
M1 for collecting the x terms (3x = ...) or the number terms correctly. M1 for 3x = 15. A1 for x = 5.
3 marks
14
(a)
Answer: 3
M1 for (13 − 4)/(4 − 1). A1 for 3.
(b)
Answer: y = 3x + 1
M1 for substituting a point into y = 3x + c (e.g. 4 = 3×1 + c), ft their gradient. A1 for y = 3x + 1.
4 marks
15
(a)
Answer: x > 5
M1 for 4x > 20. A1 for x > 5.
(b)
Answer: 6
B1 for 6 (ft their inequality, provided it is a strict inequality of the form x > 5).
3 marks
16
Answer: Let the three consecutive integers be n, n + 1 and n + 2. Their sum is n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1). Since n + 1 is an integer, 3(n + 1) is 3 × an integer, so the sum is always a multiple of 3.
M1 for representing three consecutive integers algebraically (e.g. n, n + 1, n + 2). M1 for a correct simplified sum 3n + 3. A1 for factorising to 3(n + 1) with a conclusion that this is a multiple of 3.
3 marks
17
Answer: Let the two consecutive integers be n and n + 1. The sum of their squares is n² + (n + 1)² = n² + n² + 2n + 1 = 2n² + 2n + 1 = 2(n² + n) + 1. Since 2(n² + n) is even, adding 1 makes the result odd, so the sum of the squares is always odd.
M1 for n and n + 1 with (n + 1)² expanded to n² + 2n + 1. M1 for a correct simplified expression 2n² + 2n + 1. A1 for writing it as 2(n² + n) + 1 and concluding it is odd.
3 marks
18
Answer: x = (y + 7)/3
M1 for a correct first step (y + 7 = 3x). A1 for x = (y + 7)/3.
2 marks
19
(a)
Answer: (0, −3)
B1 for (0, −3).
(b)
Answer: (1.5, 0)
M1 for setting 2x − 3 = 0. A1 for (1.5, 0) (accept x = 1.5 clearly identified as the x-axis crossing).
3 marks
20
Answer: t⁵
B1 for t⁵.
1 mark
21
(a)
Answer: x = 5
M1 for 3x = 15 (subtracting 7 correctly). A1 for x = 5.
(b)
Answer: x = 20
M1 for x/4 = 5 (adding 2 correctly). A1 for x = 20.
4 marks
22
(a)
Answer: 3n + 2
M1 for a common difference of 3 used, i.e. 3n seen. A1 for 3n + 2.
(b)
Answer: 62
B1 for 62 (3 × 20 + 2, ft their nth term).
(c)
Answer: No. Setting 3n + 2 = 100 gives 3n = 98, so n = 32.67, which is not a whole number, so 100 is not a term.
M1 for solving 3n + 2 = 100 to reach n = 98/3 (or testing terms around 100). A1 for a correct 'no' with justification that n is not an integer.
5 marks
23
(a)
Answer: 25, 36
B1 for 25 and 36 (both required).
(b)
Answer: 13
B1 for 13 (8 + 5).
(c)
Answer: 15
B1 for 15 (the triangular numbers are 1, 3, 6, 10, 15, ...).
3 marks
24
(a)
Answer: x = 5
M1 for a correct perimeter equation, e.g. 2(x + 4) + 2x = 28 (or 4x + 8 = 28). M1 for 4x = 20. A1 for x = 5.
(b)
Answer: 45 cm²
M1 for using length 9 and width 5 (ft their x). A1 for 45.
5 marks
25
(a)
Answer: £17
M1 for 3 + 2 × 7. A1 for 17.
(b)
Answer: 12 miles
M1 for 27 = 3 + 2m leading to 2m = 24. A1 for 12.
4 marks
26
Answer: 9
M1 for 4/(n + 4) × 3/(n + 3) = 1/6. M1 for (n + 4)(n + 3) = 72. M1 for n² + 7n - 60 = 0. A1 for n = 5, rejecting n = -12. A1 for the total 9.
5 marks