AQA GCSE Maths · 3.2.1

Notation, vocabulary and manipulation

50 maths questions with worked answers with answers for 3.2.1 Notation, vocabulary and manipulation in AQA GCSE Maths (spec 8300). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official AQA material.

All 50 questions and answers

Question 1 · Notation · A1
Write a × b × b using index notation.
Answer:
The factor b occurs twice, so b × b = b².
Write a × b² without a multiplication sign: ab².
Question 2 · Notation · A1
Write 4 × y without a multiplication sign.
Answer:
Place the numerical coefficient before the variable.
4 × y = 4y.
Question 3 · Notation · A1
Write x + x + x + x + x as one term.
Answer:
There are five copies of x.
x + x + x + x + x = 5x.
Question 4 · Substitution · A2
Find 3x + 4 when x = 5.
Answer:
Substitute x = 5: 3 × 5 + 4.
15 + 4 = 19.
Question 5 · Substitution · A2
Find a² − b when a = −3 and b = 2.
Answer:
Substitute a = −3 and b = 2.
(−3)² − 2 = 9 − 2 = 7.
Question 6 · Substitution · A2
Find 2p − q when p = −4 and q = 5.
Answer:
Substitute p = −4 and q = 5.
2 × (−4) − 5 = −8 − 5 = −13.
Question 7 · Substitution · A2
Find 2a² + 3b when a = 2 and b = −1.
Answer:
Substitute a = 2 and b = −1.
2 × 2² + 3 × (−1) = 8 − 3 = 5.
Question 8 · Vocabulary · A3
In 7x + 3, what is the coefficient of x?
Answer:
The coefficient is the number multiplying x.
In 7x, that number is 7.
Question 9 · Vocabulary · A3
What is the constant term in 5y − 8?
Answer:
A constant term has no variable.
In 5y − 8, the constant term is −8.
Question 10 · Vocabulary · A3
Is 3x + 2 an expression or an equation?
Answer:
There is no equals sign and no statement that two quantities are equal.
3x + 2 is an expression.
Question 11 · Vocabulary · A3
Is 4x − 1 = 11 an expression or an equation?
Answer:
The equals sign states that two expressions are equal.
4x − 1 = 11 is an equation.
Question 12 · Like terms · A4
Simplify 4x + 7x − 3x.
Answer:
Collect the coefficients of x: 4 + 7 − 3 = 8.
The result is 8x.
Question 13 · Like terms · A4
Simplify 5a + 3b − 2a + b.
Answer:
Collect a-terms: 5a − 2a = 3a.
Collect b-terms: 3b + b = 4b.
Result: 3a + 4b.
Question 14 · Like terms · A4
Simplify 6x² + 2x − 4x² + 5x.
Answer:
Collect x²-terms: 6x² − 4x² = 2x².
Collect x-terms: 2x + 5x = 7x.
Result: 2x² + 7x.
Question 15 · Brackets · A4
Expand 4(x + 3).
Answer:
Multiply each term in the bracket by 4.
4(x + 3) = 4x + 12.
Question 16 · Brackets · A4
Expand −3(2y − 5).
Answer:
Multiply each term in the bracket by −3.
−3(2y − 5) = −6y + 15.
Question 17 · Brackets · A4
Simplify 2(x + 4) + 3(x − 1).
Answer:
Expand: 2x + 8 + 3x − 3.
Collect like terms: 5x + 5.
Question 18 · Factors · A4
Factorise 6x + 18 fully.
Answer:
The highest common factor is 6.
6x ÷ 6 = x and 18 ÷ 6 = 3.
Therefore 6x + 18 = 6(x + 3).
Question 19 · Factors · A4
Factorise 8y² − 12y fully.
Answer:
The highest common factor is 4y.
8y² ÷ 4y = 2y and −12y ÷ 4y = −3.
Therefore 8y² − 12y = 4y(2y − 3).
Question 20 · Factors · A4
Factorise 5a − 10b.
Answer:
Both terms have a factor of 5.
5a − 10b = 5(a − 2b).
Question 21 · Indices · A4
Simplify x³ × x⁵.
Answer:
Multiply powers of the same base by adding the indices.
x³ × x⁵ = x⁸.
Question 22 · Indices · A4
Simplify y⁹ ÷ y⁴, for y ≠ 0.
Answer:
Divide powers of the same non-zero base by subtracting the indices.
y⁹ ÷ y⁴ = y⁵.
Question 23 · Indices · A4
Simplify (2x³)².
Answer:
Square the coefficient and multiply the index by 2.
(2x³)² = 2²x⁶ = 4x⁶.
Question 24 · Quadratics · A4
Expand (x + 3)(x + 5).
Answer:
Multiply each term: x² + 5x + 3x + 15.
Collect the middle terms: x² + 8x + 15.
Question 25 · Quadratics · A4
Expand (x − 4)(x + 2).
Answer:
Multiply each term: x² + 2x − 4x − 8.
Collect the middle terms: x² − 2x − 8.
Question 26 · Quadratics · A4
Factorise x² + 7x + 12.
Answer:
Find two numbers that multiply to 12 and add to 7: 3 and 4.
x² + 7x + 12 = (x + 3)(x + 4).
Question 27 · Quadratics · A4
Factorise x² − 9.
Answer:
Recognise a difference of squares: x² − 3².
x² − 9 = (x − 3)(x + 3).
Question 28 · Quadratics · A4 · Higher only
Factorise 2x² + 7x + 3.
Answer:
Split the middle term: 2x² + 6x + x + 3.
Factor by grouping: 2x(x + 3) + 1(x + 3).
Result: (2x + 1)(x + 3).
Question 29 · Quadratics · A4 · Higher only
Expand (x + 1)(x − 1)(x + 2).
Answer:
First (x + 1)(x − 1) = x² − 1.
Then (x² − 1)(x + 2) = x³ + 2x² − x − 2.
Question 30 · Algebraic fractions · A4 · Higher only
Simplify x² − 9x + 3, stating any restriction.
Answer:
Factor the numerator: x² − 9 = (x − 3)(x + 3).
Cancel x + 3 to obtain x − 3, provided x ≠ −3.
Question 31 · Algebraic fractions · A4 · Higher only
Simplify 2x + 3x, for x ≠ 0.
Answer:
The denominators are the same, so add the numerators.
2x + 3x = 5x, where x ≠ 0.
Question 32 · Algebraic fractions · A4 · Higher only
Simplify 1x + 12, for x ≠ 0.
Answer:
Use the common denominator 2x: 1x = 22x and 12 = x2x.
Add: x + 22x, where x ≠ 0.
Question 33 · Formulae · A5
Make x the subject of y = x + 7.
Answer:
Subtract 7 from both sides.
y − 7 = x, so x = y − 7.
Question 34 · Formulae · A5
Make a the subject of P = 2a + 5.
Answer:
Subtract 5: P − 5 = 2a.
Divide by 2: a = P − 52.
Question 35 · Formulae · A5
Make h the subject of A = bh2, where b ≠ 0.
Answer:
Multiply by 2: 2A = bh.
Divide by b: h = 2Ab, where b ≠ 0.
Question 36 · Formulae · A5
Make t the subject of v = u + at, where a ≠ 0.
Answer:
Subtract u: v − u = at.
Divide by a: t = v − ua, where a ≠ 0.
Question 37 · Identities · A6
Explain why 3(x + 2) = 3x + 6 is an identity.
Answer:
Expand the left side: 3(x + 2) = 3x + 6.
The two sides agree for every x, so this is an identity.
Question 38 · Identities · A6 · Higher only
Explain why x² + x is even for every integer x.
Answer:
Factor x² + x = x(x + 1).
One of two consecutive integers is even, so their product is even.
Question 39 · Functions · A7
A function multiplies by 3 and adds 2. What is its output for input 4?
Answer:
Multiply the input by 3: 4 × 3 = 12.
Add 2: 12 + 2 = 14.
Question 40 · Functions · A7
A machine doubles its input then subtracts 5. What input gives output 13?
Answer:
Reverse the operations: add 5 to 13, then divide by 2.
(13 + 5) ÷ 2 = 9.
Question 41 · Functions · A7 · Higher only
If f(x) = 3x − 4, find f(5).
Answer:
Substitute 5 for x.
f(5) = 3 × 5 − 4 = 11.
Question 42 · Functions · A7 · Higher only
If f(x) = 2x + 3, find f⁻¹(x).
Answer:
Set y = 2x + 3 and rearrange: x = y − 32.
Replace y by x to write f⁻¹(x) = x − 32.
Question 43 · Functions · A7 · Higher only
If f(x) = 2x + 1 and g(x) = x², find f(g(3)).
Answer:
First g(3) = 3² = 9.
Then f(g(3)) = f(9) = 2 × 9 + 1 = 19.
Question 44 · Functions · A7 · Higher only
If f(x) = x + 2 and g(x) = 3x, find g(f(x)).
Answer:
First f(x) = x + 2.
Substitute into g: g(f(x)) = 3(x + 2) = 3x + 6.
Question 45 · Proof · A6 · Higher only
Show algebraically that the sum of two consecutive integers is odd.
Answer:
Let the integers be n and n + 1.
Their sum is n + (n + 1) = 2n + 1, which is odd.
Question 46 · Proof · A6 · Higher only
Show algebraically that the square of an even integer is divisible by 4.
Answer:
Write the even integer as 2n, where n is an integer.
Its square is (2n)² = 4n², a multiple of 4.
Question 47 · Error check · A4
A student expands 2(x + 5) as 2x + 5. Correct it.
Answer:
Multiply both terms in the bracket by 2.
2(x + 5) = 2x + 10.
Question 48 · Error check · A4
A student simplifies x + x² to 2x³. Explain the error.
Answer:
x and x² have different powers and are unlike terms.
They cannot be added as one term, so the expression remains x + x².
Question 49 · Error check · A4
Simplify 3x × 4x correctly.
Answer:
Multiply the coefficients: 3 × 4 = 12.
Multiply the variables: x × x = x².
Result: 12x².
Question 50 · Formulae · A5
If C = 5n + 2, calculate C when n = 6.
Answer:
Substitute n = 6: C = 5 × 6 + 2.
C = 30 + 2 = 32.

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