Notation, Vocabulary and Manipulation

1. 1

   Simplify $5x + 3y - 2x + y$

   • A

     $3x + 4y$

   • B

     $7x + 2y$

   • C

     $3x + 2y$

   • D

     $7x + 4y$

   Check answer

   Answer & explanation

   Correct: option A

   Collect like terms: $(5x - 2x) + (3y + y) = 3x + 4y$.

2. 2

   Expand and simplify $(x + 5)(x - 2)$

   • A

     $x^2 + 3x - 10$

   • B

     $x^2 - 3x - 10$

   • C

     $x^2 + 3x + 10$

   • D

     $x^2 + 7x - 10$

   Check answer

   Answer & explanation

   Correct: option A

   FOIL: First $x^2$, Outer $-2x$, Inner $+5x$, Last $-10$. Collecting: $x^2 + 3x - 10$.

3. 3

   Factorise fully $12x^2 - 8x$

   • A

     $4x(3x - 2)$

   • B

     $2x(6x - 4)$

   • C

     $4(3x^2 - 2x)$

   • D

     $x(12x - 8)$

   Check answer

   Answer & explanation

   Correct: option A

   HCF of $12x^2$ and $8x$ is $4x$. Dividing through: $4x(3x - 2)$. Options B, C and D do not take out the full HCF.

4. 4

   Find the value of $2a^2 - 3b$ when $a = -3$ and $b = 4$

   • A

     $6$

   • B

     $-30$

   • C

     $30$

   • D

     $-6$

   Check answer

   Answer & explanation

   Correct: option A

   $2(-3)^2 - 3(4) = 2(9) - 12 = 18 - 12 = 6$. Note $(-3)^2 = 9$, not $-9$.

5. 5

   Factorise $x^2 - 16$

   • A

     $(x + 4)(x - 4)$

   • B

     $(x - 4)^2$

   • C

     $(x + 8)(x - 2)$

   • D

     $(x + 4)^2$

   Check answer

   Answer & explanation

   Correct: option A

   $x^2 - 16 = x^2 - 4^2$ is a difference of two squares: $(x + 4)(x - 4)$.

6. 6

   Make $t$ the subject of $v = u + at$

   • A

     $t = \dfrac{v - u}{a}$

   • B

     $t = v - ua$

   • C

     $t = \dfrac{v + u}{a}$

   • D

     $t = \dfrac{v}{u + a}$

   Check answer

   Answer & explanation

   Correct: option A

   Subtract $u$: $v - u = at$. Divide by $a$: $t = \dfrac{v - u}{a}$.

7. 7

   Which of the following is an identity?

   • A

     $(x + 2)^2 \equiv x^2 + 4x + 4$

   • B

     $x^2 + 4 = 0$

   • C

     $2x + 1 = 5$

   • D

     $x^2 = 4$

   Check answer

   Answer & explanation

   Correct: option A

   $(x+2)^2 = x^2 + 4x + 4$ is true for every value of $x$, making it an identity. The others are equations that hold only for specific values.

8. 8

   Factorise $x^2 + 7x + 10$

   • A

     $(x + 2)(x + 5)$

   • B

     $(x + 1)(x + 10)$

   • C

     $(x + 7)(x + 10)$

   • D

     $(x - 2)(x - 5)$

   Check answer

   Answer & explanation

   Correct: option A

   Need two numbers that multiply to $10$ and add to $7$: $2$ and $5$. So $x^2 + 7x + 10 = (x + 2)(x + 5)$.

Exam-style multi-part questions. Try each part on paper, then reveal the model answer to compare your working.

1. Q15 marks total

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   Scenario

   Expand and simplify $(2x + 3)^2 - (x - 1)(x + 4)$

   1. (a)

      Expand $(2x + 3)^2$

      [2]
      Show answer

      Model answer · 2 marks

      $(2x + 3)^2 = 4x^2 + 12x + 9$

   2. (b)

      Expand $(x - 1)(x + 4)$

      [1]
      Show answer

      Model answer · 1 mark

      $x^2 + 4x - x - 4 = x^2 + 3x - 4$

   3. (c)

      Hence simplify the full expression

      [2]
      Show answer

      Model answer · 2 marks

      $4x^2 + 12x + 9 - (x^2 + 3x - 4) = 4x^2 + 12x + 9 - x^2 - 3x + 4 = 3x^2 + 9x + 13$

   Examiner note

   A common error in part (c) is to incorrectly distribute the negative sign, writing $-x^2 + 3x - 4$ instead of $-x^2 - 3x + 4$.

2. Q25 marks total

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   Scenario

   Make $x$ the subject of $y = \dfrac{3x + 1}{x - 2}$

   1. (a)

      Multiply both sides by $(x - 2)$ and expand

      [2]
      Show answer

      Model answer · 2 marks

      $y(x - 2) = 3x + 1 \implies yx - 2y = 3x + 1$

   2. (b)

      Collect all terms in $x$ on one side and factorise

      [2]
      Show answer

      Model answer · 2 marks

      $yx - 3x = 1 + 2y \implies x(y - 3) = 2y + 1$

   3. (c)

      Write down the final expression for $x$

      [1]
      Show answer

      Model answer · 1 mark

      $x = \dfrac{2y + 1}{y - 3}$

   Examiner note

   This is a Higher-tier question. The key step is recognising that $x$ appears on both sides after expanding, so it must be collected and factorised before isolating.

3. Q33 marks total

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   Scenario

   Prove that $(n + 3)^2 - (n + 1)^2 \equiv 4(n + 2)$ for all values of $n$.

   1. (a)

      Expand both squared brackets on the left-hand side

      [1]
      Show answer

      Model answer · 1 mark

      $(n+3)^2 = n^2 + 6n + 9$ and $(n+1)^2 = n^2 + 2n + 1$

   2. (b)

      Subtract and simplify to complete the proof

      [2]
      Show answer

      Model answer · 2 marks

      $n^2 + 6n + 9 - (n^2 + 2n + 1) = 4n + 8 = 4(n + 2) = \text{RHS} \checkmark$

   Examiner note

   The final line must include a clear conclusion (e.g. '= RHS' or 'which proves the identity') to gain the last mark.