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

Mixed Topic Worksheet Grades 8/9: Day 083

Grades 8/9Non-calculator

  1. Question 1. Write √384 in the form a√b, where a and b are integers and b is as small as possible.

    Show worked answerHide worked answer
    1. M1: √64 × 6
    2. A1: 8√6
    Answer: 8√6
    Simplifying Surds
    [2 marks]

  2. Question 2. Prove algebraically that when the smaller of two consecutive positive integers is subtracted from their product, the result is a square number.

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    1. M1: n(n + 1) - n
    2. M1: n² + n - n
    3. C1: n², which is a square number
    Answer: n², a square number
    Proof with Consecutive Numbers
    [3 marks]

  3. Question 3. ABCD is a cyclic quadrilateral. Angle A = (2x + 10)° and angle C = (3x - 10)°. Work out angle C.

    Show worked answerHide worked answer
    1. M1: opposite angles of a cyclic quadrilateral add to 180°: 5x = 180
    2. M1 A1: x = 36, so angle C = 98°
    Answer: 98°
    Circle Theorems
    [3 marks]

  4. Question 4. Solve x² + y² = 25 and y = x + 1.

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    1. M1: x² + (x + 1)² = 25, so x² + x - 12 = 0
    2. M1 A1 A1: x = 3, y = 4 or x = -4, y = -3
    Answer: x = 3, y = 4 or x = -4, y = -3
    Solving Quadratic Simultaneous Equations
    [4 marks]

  5. Question 5. The point P(5, -3) lies on the circle x² + y² = 34. The tangent to the circle at P meets the x-axis at Q. Work out the coordinates of Q

    Show worked answerHide worked answer
    1. M1: gradient of OP = -dfrac35, so gradient of tangent = dfrac53
    2. M1: y + 3 = dfrac53(x - 5) and put y = 0
    3. A2: x = 6.8, so Q(6.8, 0)
    Answer: Q(6.8, 0)
    Finding Equations of Tangents to Circles
    [4 marks]
📅 Free lesson: Tue 17th NovSave your place →

Mixed Topic Worksheet Grades 8/9: Day 083

Grades 8/9Non-calculator

  1. Question 1. Write √384 in the form a√b, where a and b are integers and b is as small as possible.

    [2 marks]
    Show worked answerHide worked answer
    1. M1: √64 × 6
    2. A1: 8√6
    Answer: 8√6
    Simplifying Surds

  2. Question 2. Prove algebraically that when the smaller of two consecutive positive integers is subtracted from their product, the result is a square number.

    [3 marks]
    Show worked answerHide worked answer
    1. M1: n(n + 1) - n
    2. M1: n² + n - n
    3. C1: n², which is a square number
    Answer: n², a square number
    Proof with Consecutive Numbers

  3. Question 3. ABCD is a cyclic quadrilateral. Angle A = (2x + 10)° and angle C = (3x - 10)°. Work out angle C.

    [3 marks]
    Show worked answerHide worked answer
    1. M1: opposite angles of a cyclic quadrilateral add to 180°: 5x = 180
    2. M1 A1: x = 36, so angle C = 98°
    Answer: 98°
    Circle Theorems

  4. Question 4. Solve x² + y² = 25 and y = x + 1.

    [4 marks]
    Show worked answerHide worked answer
    1. M1: x² + (x + 1)² = 25, so x² + x - 12 = 0
    2. M1 A1 A1: x = 3, y = 4 or x = -4, y = -3
    Answer: x = 3, y = 4 or x = -4, y = -3
    Solving Quadratic Simultaneous Equations

  5. Question 5. The point P(5, -3) lies on the circle x² + y² = 34. The tangent to the circle at P meets the x-axis at Q. Work out the coordinates of Q

    [4 marks]
    Show worked answerHide worked answer
    1. M1: gradient of OP = -dfrac35, so gradient of tangent = dfrac53
    2. M1: y + 3 = dfrac53(x - 5) and put y = 0
    3. A2: x = 6.8, so Q(6.8, 0)
    Answer: Q(6.8, 0)
    Finding Equations of Tangents to Circles
📅 Free live lesson: Tuesday 17th NovemberSave your free place →

Mixed Topic Worksheet Grades 8/9: Day 083

Grades 8/9Non-calculator

  1. Question 1. Write √384 in the form a√b, where a and b are integers and b is as small as possible.

    Show worked answerHide worked answer
    1. M1: √64 × 6
    2. A1: 8√6
    Answer: 8√6
    Simplifying Surds
    [2 marks]

  2. Question 2. Prove algebraically that when the smaller of two consecutive positive integers is subtracted from their product, the result is a square number.

    Show worked answerHide worked answer
    1. M1: n(n + 1) - n
    2. M1: n² + n - n
    3. C1: n², which is a square number
    Answer: n², a square number
    Proof with Consecutive Numbers
    [3 marks]

  3. Question 3. ABCD is a cyclic quadrilateral. Angle A = (2x + 10)° and angle C = (3x - 10)°. Work out angle C.

    Show worked answerHide worked answer
    1. M1: opposite angles of a cyclic quadrilateral add to 180°: 5x = 180
    2. M1 A1: x = 36, so angle C = 98°
    Answer: 98°
    Circle Theorems
    [3 marks]

  4. Question 4. Solve x² + y² = 25 and y = x + 1.

    Show worked answerHide worked answer
    1. M1: x² + (x + 1)² = 25, so x² + x - 12 = 0
    2. M1 A1 A1: x = 3, y = 4 or x = -4, y = -3
    Answer: x = 3, y = 4 or x = -4, y = -3
    Solving Quadratic Simultaneous Equations
    [4 marks]

  5. Question 5. The point P(5, -3) lies on the circle x² + y² = 34. The tangent to the circle at P meets the x-axis at Q. Work out the coordinates of Q

    Show worked answerHide worked answer
    1. M1: gradient of OP = -dfrac35, so gradient of tangent = dfrac53
    2. M1: y + 3 = dfrac53(x - 5) and put y = 0
    3. A2: x = 6.8, so Q(6.8, 0)
    Answer: Q(6.8, 0)
    Finding Equations of Tangents to Circles
    [4 marks]