OpenRevise

Exam questions · Maths · Further Algebra

Algebraic Fractions and Proof

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

    Show that the sum of three consecutive even numbers is a multiple of 6.

    Show answerHide answer

    Model answer

    Let the numbers be \(2n\), \(2n + 2\) and \(2n + 4\). Their sum is \(6n + 6 = 6(n + 1)\), which is a multiple of 6.

    Mark scheme

    • \(2n\), \(2n + 2\), \(2n + 4\) — M1
    • \(6n + 6\) — M1
    • \(6(n + 1)\) with a conclusion — Q1
  2. 2 Simplify [3 marks]

    Simplify \(\dfrac{x^2 + 5x}{x^2 - 25}\).

    Show answerHide answer

    Model answer

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

    Mark scheme

    • \(x(x + 5)\) — M1
    • \((x - 5)(x + 5)\) — M1
    • \(\dfrac{x}{x - 5}\) — A1
  3. 3 Solve [3 marks]

    Solve \(\dfrac{x - 2}{3} + \dfrac{x + 1}{2} = 4\).

    Show answerHide answer

    Model answer

    Multiply every term by 6: \(2(x - 2) + 3(x + 1) = 24\). Then \(5x - 1 = 24\), so \(x = 5\).

    Mark scheme

    • \(2(x - 2) + 3(x + 1) = 24\) — M1
    • \(5x - 1 = 24\) — M1
    • \(x = 5\) — A1
  4. 4 Show that [2 marks]

    Edith says, “\(n^2 > n\) for every number \(n\).” Show that Edith is wrong.

    Show answerHide answer

    Model answer

    When \(n = 1\), \(n^2 = 1\) and \(n = 1\), so \(n^2\) is not greater than \(n\). This counter-example shows she is wrong. (\(n = 0.5\) also works.)

    Mark scheme

    • A valid counter-example, such as \(n = 1\) or \(n = 0.5\) — M1
    • Substitutes and shows the statement is false — Q1
  5. 5 Prove [3 marks]

    Prove that \((2n + 1)^2 - (2n - 1)^2\) is a multiple of 8 for every positive integer \(n\).

    Show answerHide answer

    Model answer

    \((2n + 1)^2 = 4n^2 + 4n + 1\) and \((2n - 1)^2 = 4n^2 - 4n + 1\). The difference is \(8n\), which is a multiple of 8.

    Mark scheme

    • \(4n^2 + 4n + 1\) or \(4n^2 - 4n + 1\) — M1
    • \(8n\) — M1
    • States that \(8n\) is a multiple of 8 — Q1
  6. 6 Write [3 marks]

    Write \(\dfrac{2}{x + 1} - \dfrac{1}{x - 2}\) as a single fraction, in its simplest form.

    Show answerHide answer

    Model answer

    The common denominator is \((x + 1)(x - 2)\), so the fraction is \(\dfrac{2(x - 2) - (x + 1)}{(x + 1)(x - 2)} = \dfrac{x - 5}{(x + 1)(x - 2)}\).

    Mark scheme

    • Common denominator \((x + 1)(x - 2)\) — M1
    • \(2(x - 2) - (x + 1)\) — M1
    • \(\dfrac{x - 5}{(x + 1)(x - 2)}\) — 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: \(4n\)

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