Identities, equivalence and algebraic proof worksheet — GCSE Higher

1.n is an integer. Which of these expressions is always an even number?

2.Two expressions are 4(x + 3) and 4x + 3. A student checks whether they are equivalent by substituting x = 2. Which statement correctly interprets the result?

3.(2x + 3)(x + a) ≡ 2x² + 11x + 12 is an identity. Work out the value of a.

4.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?

5.A student attempts to prove that the product of two consecutive integers is always even: (i) Let the two consecutive integers be n and n + 1. (ii) Since n(n + 1) is even, one of n and n + 1 must be an even number. (iii) Therefore, n(n + 1) is even. At which statement does the proof first assume the very fact it is trying to prove?

6.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?

7.A student attempts to prove that the sum of any three consecutive integers is a multiple of 3. Line 1: Let the three consecutive integers be n, n + 1 and n + 2. Line 2: Their sum is n + (n + 1) + (n + 2) = 3n + 2. Line 3: 3n + 2 leaves a remainder of 2 when divided by 3, so it is not a multiple of 3. Line 4: So the sum of three consecutive integers is not always a multiple of 3. Which line contains the FIRST error?

8.A student is asked whether 3(x − 4) = 3x − 4 is an identity. Which statement gives the correct verdict and reason?

9.A student is proving that (n + 1)² − n² is always an odd number. Which of these correctly completes the first line of algebra?

10.Which line of algebra shows that the sum of two consecutive odd numbers is always a multiple of 4?

11.Which expression is equivalent to 3(x + 4) − 2(x − 1)?

12.A student claims: 'For every positive integer n, n² + n + 1 is a prime number.' Which value of n shows that this claim is false?

13.Which of these is an identity?

14.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?

15.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?