Edexcel GCSE Maths · A1–A7

Notation, vocabulary and manipulation

50 maths questions with worked answers with answers for A1–A7 Notation, vocabulary and manipulation in Edexcel GCSE Maths (spec 1MA1). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official Pearson material.

All 50 questions and answers

Question 1 · Notation · A1
Simplify 3a × 4b.
Answer:
Multiply the numbers and write the letters together.
12ab.
Question 2 · Notation · A1
Simplify 5x + 3y − 2x + y.
Answer:
Collect like terms: 5x − 2x = 3x and 3y + y = 4y.
3x + 4y.
Question 3 · Notation · A1
Simplify 12p²q ÷ 4pq.
Answer:
12 ÷ 4 = 3, p² ÷ p = p, q ÷ q = 1.
3p.
Question 4 · Index laws · A1
Simplify y⁵ × y³.
Answer:
Add the indices.
y⁸.
Question 5 · Index laws · A1
Simplify (2m³)².
Answer:
Square the 2 and multiply the indices.
4m⁶.
Question 6 · Index laws · A1 · Higher only
Simplify 15x⁴y²5x⁶y.
Answer:
15 ÷ 5 = 3, x⁴ ÷ x⁶ = x⁻², y² ÷ y = y.
3yx².
Question 7 · Substitution · A2
Find the value of 3a² − 2b when a = 4 and b = 5.
Answer:
3 × 4² = 3 × 16 = 48 and 2 × 5 = 10.
48 − 10 = 38.
Question 8 · Substitution · A2
Find the value of p + q2 when p = −3 and q = 11.
Answer:
p + q = 8.
8 ÷ 2 = 4.
Question 9 · Substitution · A2
Use v = u + at to find v when u = 5, a = 3 and t = 7.
Answer:
v = 5 + 3 × 7.
v = 5 + 21 = 26.
Question 10 · Substitution · A2
Use E = 12mv² to find E when m = 6 and v = 4.
Answer:
v² = 16.
E = 0.5 × 6 × 16 = 48.
Question 11 · Vocabulary · A3
State whether 4(x + 2) = 4x + 8 is an equation, a formula or an identity.
Answer:
It is true for every value of x.
It is an identity.
Question 12 · Vocabulary · A3
State whether 3x + 1 = 10 is an equation or an identity.
Answer:
It is true only for x = 3.
It is an equation.
Question 13 · Vocabulary · A3
In A = πr², what is A called?
Answer:
A is the subject of the formula.
It is the subject.
Question 14 · Vocabulary · A3
How many terms are there in 5x² − 3x + 7?
Answer:
The terms are 5x², −3x and 7.
There are 3 terms.
Question 15 · Expanding · A4
Expand 6(2x − 5).
Answer:
Multiply each term by 6.
12x − 30.
Question 16 · Expanding · A4
Expand and simplify 3(x + 4) − 2(x − 1).
Answer:
3x + 12 − 2x + 2.
x + 14.
Question 17 · Expanding · A4
Expand and simplify (x + 5)(x + 3).
Answer:
x² + 3x + 5x + 15.
x² + 8x + 15.
Question 18 · Expanding · A4
Expand and simplify (x − 6)(x + 2).
Answer:
x² + 2x − 6x − 12.
x² − 4x − 12.
Question 19 · Expanding · A4
Expand and simplify (x − 4)².
Answer:
(x − 4)(x − 4) = x² − 4x − 4x + 16.
x² − 8x + 16.
Question 20 · Expanding · A4 · Higher only
Expand and simplify (2x + 3)(3x − 1).
Answer:
6x² − 2x + 9x − 3.
6x² + 7x − 3.
Question 21 · Expanding · A4 · Higher only
Expand and simplify (x + 1)(x + 2)(x + 3).
Answer:
(x + 1)(x + 2) = x² + 3x + 2.
(x² + 3x + 2)(x + 3) = x³ + 6x² + 11x + 6.
Question 22 · Factorising · A4
Factorise 15y − 20.
Answer:
HCF of 15 and 20 is 5.
5(3y − 4).
Question 23 · Factorising · A4
Factorise fully 8a²b + 12ab².
Answer:
HCF is 4ab.
4ab(2a + 3b).
Question 24 · Factorising · A4
Factorise x² + 7x + 12.
Answer:
Find two numbers that multiply to 12 and add to 7: 3 and 4.
(x + 3)(x + 4).
Question 25 · Factorising · A4
Factorise x² − 2x − 15.
Answer:
Two numbers that multiply to −15 and add to −2: −5 and 3.
(x − 5)(x + 3).
Question 26 · Difference of two squares · A4
Factorise x² − 49.
Answer:
x² − 7².
(x + 7)(x − 7).
Question 27 · Factorising · A4 · Higher only
Factorise 2x² + 7x + 3.
Answer:
ac = 6; numbers 6 and 1 add to 7. 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3).
(2x + 1)(x + 3).
Question 28 · Factorising · A4 · Higher only
Factorise 3x² − 10x − 8.
Answer:
ac = −24; numbers −12 and 2. 3x² − 12x + 2x − 8 = 3x(x − 4) + 2(x − 4).
(3x + 2)(x − 4).
Question 29 · Difference of two squares · A4 · Higher only
Factorise 9x² − 25y².
Answer:
(3x)² − (5y)².
(3x + 5y)(3x − 5y).
Question 30 · Algebraic fractions · A4 · Higher only
Simplify x² − 9x² + 5x + 6.
Answer:
Factorise: (x + 3)(x − 3)(x + 3)(x + 2).
x − 3x + 2.
Question 31 · Algebraic fractions · A4 · Higher only
Write 2x + 3x + 1 as a single fraction.
Answer:
Common denominator x(x + 1): 2(x + 1) + 3xx(x + 1).
5x + 2x(x + 1).
Question 32 · Algebraic fractions · A4 · Higher only
Simplify 4x + 8x² − 4.
Answer:
4(x + 2)(x + 2)(x − 2).
4x − 2.
Question 33 · Rearranging · A5
Make x the subject of y = 3x − 7.
Answer:
Add 7: y + 7 = 3x.
x = y + 73.
Question 34 · Rearranging · A5
Make r the subject of C = 2πr.
Answer:
Divide both sides by 2π.
r = C2π.
Question 35 · Rearranging · A5 · Higher only
Make r the subject of A = πr².
Answer:
r² = Aπ.
r = √(Aπ).
Question 36 · Rearranging · A5 · Higher only
Make t the subject of s = at + bt.
Answer:
st = at + b, so st − at = b.
t(s − a) = b, so t = bs − a.
Question 37 · Rearranging · A5 · Higher only
Make y the subject of x = y + 2y − 3.
Answer:
xy − 3x = y + 2, so xy − y = 3x + 2.
y(x − 1) = 3x + 2, so y = 3x + 2x − 1.
Question 38 · Proof · A6
Show that 3(x + 4) − 2(x + 1) ≡ x + 10.
Answer:
3x + 12 − 2x − 2.
= x + 10, so the identity holds.
Question 39 · Proof · A6 · Higher only
Prove that the sum of two consecutive odd numbers is a multiple of 4.
Answer:
Let the numbers be 2n + 1 and 2n + 3.
Sum = 4n + 4 = 4(n + 1), which is a multiple of 4.
Question 40 · Proof · A6 · Higher only
Prove that (n + 3)² − (n − 3)² is a multiple of 12.
Answer:
(n² + 6n + 9) − (n² − 6n + 9) = 12n.
12n = 12 × n, a multiple of 12.
Question 41 · Proof · A6 · Higher only
Prove that the square of an odd number is always odd.
Answer:
Let the odd number be 2n + 1. (2n + 1)² = 4n² + 4n + 1.
= 2(2n² + 2n) + 1, which is odd.
Question 42 · Counterexample · A6
Give a counterexample to "all prime numbers are odd".
Answer:
2 is prime.
2 is even, so the statement is false.
Question 43 · Functions · A7
f(x) = 4x − 1. Find f(3).
Answer:
4 × 3 − 1.
f(3) = 11.
Question 44 · Functions · A7 · Higher only
f(x) = 2x + 5 and g(x) = x². Find fg(3).
Answer:
g(3) = 9.
f(9) = 2 × 9 + 5 = 23.
Question 45 · Functions · A7 · Higher only
f(x) = 3x − 2. Find f⁻¹(x).
Answer:
Let y = 3x − 2, so x = y + 23.
f⁻¹(x) = x + 23.
Question 46 · Functions · A7 · Higher only
f(x) = x + 4 and g(x) = 3x. Find gf(x).
Answer:
Apply f first: x + 4.
gf(x) = 3(x + 4) = 3x + 12.
Question 47 · Functions · A7 · Higher only
f(x) = x − 12. Solve f(x) = 6.
Answer:
x − 12 = 6, so x − 1 = 12.
x = 13.
Question 48 · Notation · A1
Write an expression for the total cost in pence of n pens at 45p each and 3 rulers at r pence each.
Answer:
Pens cost 45n and rulers cost 3r.
45n + 3r.
Question 49 · Expanding · A4
Expand x(3x − 2).
Answer:
Multiply each term by x.
3x² − 2x.
Question 50 · Expanding · A4
Expand and simplify (x + 7)(x − 7).
Answer:
x² − 7x + 7x − 49.
x² − 49.

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