OpenRevise

Exam questions · Maths · Further Algebra

Algebraic Fractions and Proof

  • 6 exam questions
  • 17 marks
  • 9 quick checks
  1. 1 Prove [3 marks]

    Prove that the sum of any three consecutive odd numbers is a multiple of 3.

    Show answerHide answer

    Model answer

    Let the numbers be \(2n + 1\), \(2n + 3\) and \(2n + 5\). Their sum is \(6n + 9 = 3(2n + 3)\), which is a multiple of 3.

    Mark scheme

    • \(2n + 1\), \(2n + 3\), \(2n + 5\) — M1
    • \(6n + 9\) — M1
    • \(3(2n + 3)\) with a conclusion — A1
  2. 2 Simplify [3 marks]

    Simplify \(\dfrac{3x^2 + 6x}{x^2 - 4}\).

    Show answerHide answer

    Model answer

    \(\dfrac{3x(x + 2)}{(x - 2)(x + 2)} = \dfrac{3x}{x - 2}\).

    Mark scheme

    • \(3x(x + 2)\) — M1
    • \((x - 2)(x + 2)\) — M1
    • \(\dfrac{3x}{x - 2}\) — A1
  3. 3 Solve [3 marks]

    Solve \(\dfrac{x + 2}{5} + \dfrac{x - 3}{2} = 1\).

    Show answerHide answer

    Model answer

    Multiply every term by 10: \(2(x + 2) + 5(x - 3) = 10\). Then \(7x - 11 = 10\), so \(x = 3\).

    Mark scheme

    • \(2(x + 2) + 5(x - 3) = 10\) — M1
    • \(7x - 11 = 10\) — M1
    • \(x = 3\) — A1
  4. 4 Show that [2 marks]

    Show that the statement “\(n^2 + n + 11\) is always prime” is not true.

    Show answerHide answer

    Model answer

    When \(n = 11\), \(121 + 11 + 11 = 143 = 11 \times 13\), which is not prime. So the statement is not true.

    Mark scheme

    • A counter-example, such as \(n = 11\), substituted — M1
    • \(143 = 11 \times 13\) with a conclusion — A1
  5. 5 Prove [3 marks]

    Prove that \((n + 2)^2 - n^2\) is a multiple of 4 for every integer \(n\).

    Show answerHide answer

    Model answer

    \((n + 2)^2 - n^2 = n^2 + 4n + 4 - n^2 = 4n + 4 = 4(n + 1)\), which is a multiple of 4.

    Mark scheme

    • \(n^2 + 4n + 4\) — M1
    • \(4n + 4\) — M1
    • \(4(n + 1)\) with a conclusion — A1
  6. 6 Simplify [3 marks]

    Simplify \(\dfrac{x^2 - 3x - 10}{x^2 - 4}\).

    Show answerHide answer

    Model answer

    \(\dfrac{(x - 5)(x + 2)}{(x - 2)(x + 2)} = \dfrac{x - 5}{x - 2}\).

    Mark scheme

    • \((x - 5)(x + 2)\) or \((x - 2)(x + 2)\) — M1
    • Both factorised — M1
    • \(\dfrac{x - 5}{x - 2}\) — A1

Quick check

  1. 1

    What may be cancelled in an algebraic fraction?

    1. AAny terms that appear on the top and bottom
    2. BFactors that multiply the whole top and the whole bottom
    3. COnly numbers
    4. DOnly letters
    Show answerHide answer

    B: Factors that multiply the whole top and the whole bottom

    Terms that are added or subtracted cannot be cancelled.

  2. 2

    Simplify \(\dfrac{x^2 - 9}{x + 3}\).

    1. A\(x - 3\)
    2. B\(x + 3\)
    3. C\(x - 9\)
    4. D\(\dfrac{x - 9}{1}\)
    Show answerHide answer

    A: \(x - 3\)

    \(\dfrac{(x - 3)(x + 3)}{x + 3} = x - 3\).

  3. 3

    Which statement about \(\dfrac{x + 3}{3}\) is correct?

    1. AIt simplifies to \(x\)
    2. BIt simplifies to \(x + 1\)
    3. CIt simplifies to 1
    4. DThe 3s cannot be cancelled because the 3 on top is added
    Show answerHide answer

    D: The 3s cannot be cancelled because the 3 on top is added

    Only factors can be cancelled, not terms.

  4. 4

    Solve \(\dfrac{x - 1}{3} + \dfrac{x + 2}{6} = 2\).

    1. A\(x = 3\)
    2. B\(x = 5\)
    3. C\(x = 4\)
    4. D\(x = 6\)
    Show answerHide answer

    C: \(x = 4\)

    Multiply by 6: \(2(x - 1) + (x + 2) = 12\), so \(3x = 12\).

  5. 5

    Which expression is an odd number for any whole number \(n\)?

    1. A\(2n\)
    2. B\(2n + 1\)
    3. C\(n + 1\)
    4. D\(n^2\)
    Show answerHide answer

    B: \(2n + 1\)

    \(2n\) is even, so adding 1 makes it odd.

  6. 6

    What is the sum of three consecutive whole numbers \(n\), \(n + 1\) and \(n + 2\)?

    1. A\(3n + 3\)
    2. B\(3n\)
    3. C\(3n + 2\)
    4. D\(n + 3\)
    Show answerHide answer

    A: \(3n + 3\)

    \(n + n + 1 + n + 2 = 3n + 3 = 3(n + 1)\).

  7. 7

    Which value of \(n\) is a counter-example to “\(n^2 + n + 1\) is always prime”?

    1. A\(n = 1\)
    2. B\(n = 2\)
    3. C\(n = 3\)
    4. D\(n = 4\)
    Show answerHide answer

    D: \(n = 4\)

    \(16 + 4 + 1 = 21 = 3 \times 7\), which is not prime.

  8. 8

    Simplify \(\dfrac{x^2 + 5x + 6}{x^2 + 3x + 2}\).

    1. A\(\dfrac{x + 2}{x + 1}\)
    2. B\(\dfrac{x + 3}{x + 2}\)
    3. C\(\dfrac{x + 3}{x + 1}\)
    4. D\(\dfrac{5x + 6}{3x + 2}\)
    Show answerHide answer

    C: \(\dfrac{x + 3}{x + 1}\)

    \(\dfrac{(x + 2)(x + 3)}{(x + 1)(x + 2)} = \dfrac{x + 3}{x + 1}\).

  9. 9

    Expand and simplify \((n + 1)^2 - (n - 1)^2\).

    1. A\(2\)
    2. B\(4n\)
    3. C\(2n\)
    4. D\(4n + 2\)
    Show answerHide answer

    B: \(4n\)

    \(n^2 + 2n + 1 - n^2 + 2n - 1 = 4n\).