Printable · GCSE Higher · ages 14-16

Identities, equivalence and algebraic proof worksheet — GCSE Higher

Fifteen questions on "identities, equivalence and algebraic proof" — DfE statement A6. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.

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Identities, equivalence and algebraic proof worksheet — GCSE Higher

Maths · GCSE Higher · A6 · Identities, equivalence and algebraic proof · Non-calculator · 15 questions

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1.Which of these is an identity?
(a)2(3x + 1) = 6x + 2
(b)5x + 1 = 5(x + 1)
(c)7 − x = x − 7
(d)4x − 3 = 3x + 5
2.(2x + 3)(x + a) ≡ 2x² + 11x + 12 is an identity. Work out the value of a.
(a)a = 8
(b)a = 12
(c)a = −4
(d)a = 4
3.n is an integer. Which of these expressions is always an even number?
(a)n² + n
(b)n² − n + 1
(c)2n + 1
(d)n² + 1
4.A proof that the product of two consecutive even numbers is always a multiple of 8 begins: Let the two consecutive even numbers be 2n and 2n + 2, so their product is 2n(2n + 2) = 4n(n + 1). Which line correctly completes the proof?
(a)4n(n + 1) is a multiple of 4, and because n and n + 1 are consecutive integers, it must be a multiple of 8.
(b)Expanding gives 4n(n + 1) = 4n² + 4n, which is clearly a multiple of 8.
(c)4n is always a multiple of 4 for any integer n, so 4n(n + 1) is a multiple of 4 as well, which means it is a multiple of 8.
(d)n and n + 1 are consecutive integers, so one of them must be even; this makes n(n + 1) even, so 4n(n + 1) is 4 × an even number, which is a multiple of 8.
5.A market stall's cost of hiring n tables is modelled by two formulas: Formula A: C = 3(2n + 5); Formula B: C = 6n + 15, where C is in pounds. A stallholder says the two formulas always give the same cost. Work out the cost given by each formula when n = 4, and use your results to decide whether the stallholder is correct.
(a)Formula A gives £39 and Formula B gives £39, but the stallholder is wrong because the two formulas use a different number of terms.
(b)Formula A gives £29 and Formula B gives £39, so the stallholder is wrong.
(c)Formula A gives £39 and Formula B gives £39, but this is only true when n = 4, so the stallholder is wrong.
(d)Formula A gives £39 and Formula B gives £39, and since 3(2n + 5) expands to 6n + 15 for every value of n, the stallholder is correct.
6.Which expression is equivalent to 7x − 3(2x − 6)?
(a)13x − 18
(b)x − 18
(c)x + 18
(d)x + 6
7.A photo printing service has two adverts for its price. Advert A: cost in pounds = 3(2n + 4) for n photos. Advert B: cost in pounds = 6n + 12. A customer says the two adverts always charge the same amount. Is the customer correct?
(a)They always charge the same, since 3(2n + 4) = 6n + 12.
(b)It depends on the value of n, so it cannot be decided.
(c)Advert A is cheaper, since 3(2n + 4) = 6n + 4.
(d)Advert A is cheaper, since 3(2n + 4) = 2n + 7.
8.A proof sets out to show that the sum of the squares of two consecutive odd numbers, written as 2n + 1 and 2n + 3, is always 2 more than a multiple of 8. Four attempts to expand (2n + 1)² + (2n + 3)² and reach a conclusion are shown below. Which attempt correctly proves this claim?
(a)(2n + 1)² + (2n + 3)² = (4n² + 4n + 1) + (4n² + 9) = 8n² + 4n + 10 = 4(2n² + n + 2) + 2, so the sum is always 2 more than a multiple of 4.
(b)(2n + 1)² + (2n + 3)² = (4n² + 4n + 1) + (4n² + 12n + 9) = 8n² + 16n + 10 = 2(4n² + 8n + 5), and 4n² + 8n + 5 is an integer, so the sum is always even, which means it is a multiple of 8.
(c)(2n + 1)² + (2n + 3)² = (4n² + 4n + 1) + (4n² + 12n + 9) = 8n² + 16n + 10 = 8(n² + 2n + 1) + 2, and n² + 2n + 1 is an integer, so the sum is always 2 more than a multiple of 8.
(d)(2n + 1)² + (2n + 3)² = (4n² + 4n + 1) + (4n² + 12n + 9) = 8n² + 16n + 10 = 8(n² + 2n) + 10, so the sum is always 10 more than a multiple of 8.
9.A rectangular garden has width w metres and length (w + 3) metres. A gardener writes its perimeter as 2w + 3. Which statement corrects the gardener's mistake?
(a)It is 2(w + (w + 3)) = 4w + 6, not 2w + 3.
(b)It is 3w + 6, since the length doubles but not the width.
(c)It is w + (w + 3) = 2w + 3, matching the gardener.
(d)It is 4(w + 3) = 4w + 12, treating it as a square.
10.A student is proving that (n + 1)² − n² is always an odd number. Which of these correctly completes the first line of algebra?
(a)(n + 1)² − n² = 1
(b)(n + 1)² − n² = 2n + 1
(c)(n + 1)² − n² = 2n
(d)(n + 1)² − n² = n² + 2n + 1
11.Which expression is equivalent to 0.5(4x + 6) − x?
(a)3x + 3
(b)2x + 3
(c)x + 3
(d)x + 6
12.Which expression is equivalent to 3(x + 4) − 2(x − 1)?
(a)x + 14
(b)x + 10
(c)5x + 10
(d)x + 13
13.Which expression is equivalent to 3(2x − 5) + 4x?
(a)10x + 15
(b)6x − 15
(c)10x − 15
(d)10x − 5
14.A student says 4(2x − 3) is equivalent to 8x − 3. Which statement gives the correct verdict and reason?
(a)False — 4(2x − 3) = 8x − 12, not 8x − 3.
(b)True — 4(2x − 3) = 8x − 3 when the bracket is expanded.
(c)False — 4(2x − 3) = 2x − 12, only the −3 is multiplied.
(d)True — 4(2x − 3) = 8x − 3, since both are linear.
15.An equation has exactly one value of x that makes it true, but an identity is true for every value of x. Which of these best explains why 3x + 5 = 20 is an equation rather than an identity?
(a)It has an = sign, and equations always use =.
(b)3x + 5 cannot be simplified, so it must be an equation.
(c)Only x = 5 satisfies 3x + 5 = 20, not every value of x.
(d)A number on the right-hand side makes it an equation.

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