📅 Free live lesson: Tuesday 17th NovemberSave your free place →

Mixed Topic Worksheet Grades 8/9: Day 020

Grades 8/9Calculator

  1. Question 1. ABCDEFGH is a cuboid.
    Work out the length of AG.
    Give your answer to 1 decimal place.

    Show worked answerHide worked answer
    1. M1: AC² = 9² + 4²
    2. M1: AG² = AC² + 7²
    3. A1: AG = 12.1 from AG² = 146
    Answer: 12.1
    3D Pythagoras
    9 cm7 cmADBCFGEH4 cm
    [3 marks]

  2. Question 2. Show that x² - 4x + 9 = 0 has no real solutions
    You must use completing the square.

    Show worked answerHide worked answer
    1. M1: (x - 2)² - 4 + 9 = 0
    2. M1: (x - 2)² = -5
    3. A1: A square cannot be negative so there are no real solutions
    Answer: (x - 2)² = -5 has no real solution because a square cannot be negative
    Solving Quadratics by Completing the Square (a = 1)
    [3 marks]

  3. Question 3. Prove that the sum of the squares of two consecutive odd numbers is always 2 more than a multiple of 8.

    Show worked answerHide worked answer
    1. M1: (2n+1)² + (2n+3)²
    2. M1: = 8n² + 16n + 10
    3. C1: = 8(n² + 2n + 1) + 2, which is 2 more than a multiple of 8
    Answer: 8(n²+2n+1)+2, so 2 more than a multiple of 8
    Proof with Consecutive Numbers
    [3 marks]

  4. Question 4. A Venn diagram of 30 people shows: A only 8, both A and B 6, B only 10, neither the rest. A person not in B is chosen. Find the probability that they are in A.

    Show worked answerHide worked answer
    1. M1: neither = 6, so B' has 8 + 6 = 14
    2. M1: dfrac814
    3. A1: dfrac47
    Answer: dfrac47
    Conditional Probability with Venn Diagrams
    [3 marks]

  5. Question 5. overrightarrowAB=2mathbfa+5mathbfb and overrightarrowBC=6mathbfa+(k+3)mathbfb. A, B and C lie on a straight line. Work out the value of k.

    Show worked answerHide worked answer
    1. M1: 6=3×2, so overrightarrowBC=3overrightarrowAB
    2. M1: k+3=3×5=15
    3. A1: k=12
    Answer: k=12
    Vectors Along a Straight Line
    [3 marks]
📅 Free lesson: Tue 17th NovSave your place →

Mixed Topic Worksheet Grades 8/9: Day 020

Grades 8/9Calculator

  1. Question 1. ABCDEFGH is a cuboid.
    Work out the length of AG.
    Give your answer to 1 decimal place.

    [3 marks]
    9 cm7 cmADBCFGEH4 cm
    Show worked answerHide worked answer
    1. M1: AC² = 9² + 4²
    2. M1: AG² = AC² + 7²
    3. A1: AG = 12.1 from AG² = 146
    Answer: 12.1
    3D Pythagoras

  2. Question 2. Show that x² - 4x + 9 = 0 has no real solutions
    You must use completing the square.

    [3 marks]
    Show worked answerHide worked answer
    1. M1: (x - 2)² - 4 + 9 = 0
    2. M1: (x - 2)² = -5
    3. A1: A square cannot be negative so there are no real solutions
    Answer: (x - 2)² = -5 has no real solution because a square cannot be negative
    Solving Quadratics by Completing the Square (a = 1)

  3. Question 3. Prove that the sum of the squares of two consecutive odd numbers is always 2 more than a multiple of 8.

    [3 marks]
    Show worked answerHide worked answer
    1. M1: (2n+1)² + (2n+3)²
    2. M1: = 8n² + 16n + 10
    3. C1: = 8(n² + 2n + 1) + 2, which is 2 more than a multiple of 8
    Answer: 8(n²+2n+1)+2, so 2 more than a multiple of 8
    Proof with Consecutive Numbers

  4. Question 4. A Venn diagram of 30 people shows: A only 8, both A and B 6, B only 10, neither the rest. A person not in B is chosen. Find the probability that they are in A.

    [3 marks]
    Show worked answerHide worked answer
    1. M1: neither = 6, so B' has 8 + 6 = 14
    2. M1: dfrac814
    3. A1: dfrac47
    Answer: dfrac47
    Conditional Probability with Venn Diagrams

  5. Question 5. overrightarrowAB=2mathbfa+5mathbfb and overrightarrowBC=6mathbfa+(k+3)mathbfb. A, B and C lie on a straight line. Work out the value of k.

    [3 marks]
    Show worked answerHide worked answer
    1. M1: 6=3×2, so overrightarrowBC=3overrightarrowAB
    2. M1: k+3=3×5=15
    3. A1: k=12
    Answer: k=12
    Vectors Along a Straight Line
📅 Free live lesson: Tuesday 17th NovemberSave your free place →

Mixed Topic Worksheet Grades 8/9: Day 020

Grades 8/9Calculator

  1. Question 1. ABCDEFGH is a cuboid.
    Work out the length of AG.
    Give your answer to 1 decimal place.

    Show worked answerHide worked answer
    1. M1: AC² = 9² + 4²
    2. M1: AG² = AC² + 7²
    3. A1: AG = 12.1 from AG² = 146
    Answer: 12.1
    3D Pythagoras
    9 cm7 cmADBCFGEH4 cm
    [3 marks]

  2. Question 2. Show that x² - 4x + 9 = 0 has no real solutions
    You must use completing the square.

    Show worked answerHide worked answer
    1. M1: (x - 2)² - 4 + 9 = 0
    2. M1: (x - 2)² = -5
    3. A1: A square cannot be negative so there are no real solutions
    Answer: (x - 2)² = -5 has no real solution because a square cannot be negative
    Solving Quadratics by Completing the Square (a = 1)
    [3 marks]

  3. Question 3. Prove that the sum of the squares of two consecutive odd numbers is always 2 more than a multiple of 8.

    Show worked answerHide worked answer
    1. M1: (2n+1)² + (2n+3)²
    2. M1: = 8n² + 16n + 10
    3. C1: = 8(n² + 2n + 1) + 2, which is 2 more than a multiple of 8
    Answer: 8(n²+2n+1)+2, so 2 more than a multiple of 8
    Proof with Consecutive Numbers
    [3 marks]

  4. Question 4. A Venn diagram of 30 people shows: A only 8, both A and B 6, B only 10, neither the rest. A person not in B is chosen. Find the probability that they are in A.

    Show worked answerHide worked answer
    1. M1: neither = 6, so B' has 8 + 6 = 14
    2. M1: dfrac814
    3. A1: dfrac47
    Answer: dfrac47
    Conditional Probability with Venn Diagrams
    [3 marks]

  5. Question 5. overrightarrowAB=2mathbfa+5mathbfb and overrightarrowBC=6mathbfa+(k+3)mathbfb. A, B and C lie on a straight line. Work out the value of k.

    Show worked answerHide worked answer
    1. M1: 6=3×2, so overrightarrowBC=3overrightarrowAB
    2. M1: k+3=3×5=15
    3. A1: k=12
    Answer: k=12
    Vectors Along a Straight Line
    [3 marks]