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

Mixed Topic Worksheet Grades 8/9: Day 027

Grades 8/9Non-calculator

  1. Question 1. The histogram shows information about the masses, m kg, of some sacks.
    Work out the total number of sacks.

    Show worked answerHide worked answer
    1. M1: 1 × (20 - 10), 3 × (25 - 20), 2.5 × (35 - 25) and 1.5 × (55 - 35) (at least two correct)
    2. M1: 10 + 15 + 25 + 30
    3. A1: 80
    Answer: 80
    Interpreting Histograms
    Mass, m (kg)Frequency density012341020304050
    [3 marks]

  2. Question 2. A right-angled triangle has hypotenuse 14 cm and another side 7 cm.
    Work out the third side.
    Give your answer as a√b, where a and b are integers and b is as small as possible.

    Show worked answerHide worked answer
    1. M1: 14² - 7²
    2. M1: √49 × 3
    3. A1: 7√3
    Answer: 7√3
    Simplifying Surds
    [3 marks]

  3. Question 3. A, B, C, D lie on a circle. Angle ABC = (5x - 8)^°, angle ADC = (3x + 12)^°
    Work out x. Show your working.

    Show worked answerHide worked answer
    1. M1: (5x - 8) + (3x + 12) = 180
    2. M1: 8x + 4 = 180
    3. A1: x = 22
    Answer: 22
    Circle Theorems
    (5x − 8)°(3x + 12)°ABCDNot drawn accurately
    [3 marks]

  4. Question 4. n is an integer. Prove that (n+3)² - (n-3)² is a multiple of 12.

    Show worked answerHide worked answer
    1. M1: n²+6n+9 - (n²-6n+9)
    2. M1: = 12n
    3. C1: 12n is a multiple of 12
    Answer: 12n, a multiple of 12
    Proof with Consecutive Numbers
    [3 marks]

  5. Question 5. The circle C, centre O, has equation x² + y² = 40
    P is the point (6, -2).
    (a) Show that P lies on C. [1]
    (b) Find an equation of the tangent to C at P.
    Give your answer in the form y = mx + c [3]

    Show worked answerHide worked answer
    1. (a) B1: 6² + (-2)² = 36 + 4 = 40
    2. (b) M1: Gradient of OP = dfrac-26 = -dfrac13
    3. (b) M1: Tangent gradient 3, so y + 2 = 3(x - 6)
    4. (b) A1: y = 3x - 20
    Answer: (a) 36 + 4 = 40; (b) y = 3x - 20
    Finding Equations of Tangents to Circles
    [4 marks]
📅 Free lesson: Tue 17th NovSave your place →

Mixed Topic Worksheet Grades 8/9: Day 027

Grades 8/9Non-calculator

  1. Question 1. The histogram shows information about the masses, m kg, of some sacks.
    Work out the total number of sacks.

    [3 marks]
    Mass, m (kg)Frequency density012341020304050
    Show worked answerHide worked answer
    1. M1: 1 × (20 - 10), 3 × (25 - 20), 2.5 × (35 - 25) and 1.5 × (55 - 35) (at least two correct)
    2. M1: 10 + 15 + 25 + 30
    3. A1: 80
    Answer: 80
    Interpreting Histograms

  2. Question 2. A right-angled triangle has hypotenuse 14 cm and another side 7 cm.
    Work out the third side.
    Give your answer as a√b, where a and b are integers and b is as small as possible.

    [3 marks]
    Show worked answerHide worked answer
    1. M1: 14² - 7²
    2. M1: √49 × 3
    3. A1: 7√3
    Answer: 7√3
    Simplifying Surds

  3. Question 3. A, B, C, D lie on a circle. Angle ABC = (5x - 8)^°, angle ADC = (3x + 12)^°
    Work out x. Show your working.

    [3 marks]
    (5x − 8)°(3x + 12)°ABCDNot drawn accurately
    Show worked answerHide worked answer
    1. M1: (5x - 8) + (3x + 12) = 180
    2. M1: 8x + 4 = 180
    3. A1: x = 22
    Answer: 22
    Circle Theorems

  4. Question 4. n is an integer. Prove that (n+3)² - (n-3)² is a multiple of 12.

    [3 marks]
    Show worked answerHide worked answer
    1. M1: n²+6n+9 - (n²-6n+9)
    2. M1: = 12n
    3. C1: 12n is a multiple of 12
    Answer: 12n, a multiple of 12
    Proof with Consecutive Numbers

  5. Question 5. The circle C, centre O, has equation x² + y² = 40
    P is the point (6, -2).
    (a) Show that P lies on C. [1]
    (b) Find an equation of the tangent to C at P.
    Give your answer in the form y = mx + c [3]

    [4 marks]
    Show worked answerHide worked answer
    1. (a) B1: 6² + (-2)² = 36 + 4 = 40
    2. (b) M1: Gradient of OP = dfrac-26 = -dfrac13
    3. (b) M1: Tangent gradient 3, so y + 2 = 3(x - 6)
    4. (b) A1: y = 3x - 20
    Answer: (a) 36 + 4 = 40; (b) y = 3x - 20
    Finding Equations of Tangents to Circles
📅 Free live lesson: Tuesday 17th NovemberSave your free place →

Mixed Topic Worksheet Grades 8/9: Day 027

Grades 8/9Non-calculator

  1. Question 1. The histogram shows information about the masses, m kg, of some sacks.
    Work out the total number of sacks.

    Show worked answerHide worked answer
    1. M1: 1 × (20 - 10), 3 × (25 - 20), 2.5 × (35 - 25) and 1.5 × (55 - 35) (at least two correct)
    2. M1: 10 + 15 + 25 + 30
    3. A1: 80
    Answer: 80
    Interpreting Histograms
    Mass, m (kg)Frequency density012341020304050
    [3 marks]

  2. Question 2. A right-angled triangle has hypotenuse 14 cm and another side 7 cm.
    Work out the third side.
    Give your answer as a√b, where a and b are integers and b is as small as possible.

    Show worked answerHide worked answer
    1. M1: 14² - 7²
    2. M1: √49 × 3
    3. A1: 7√3
    Answer: 7√3
    Simplifying Surds
    [3 marks]

  3. Question 3. A, B, C, D lie on a circle. Angle ABC = (5x - 8)^°, angle ADC = (3x + 12)^°
    Work out x. Show your working.

    Show worked answerHide worked answer
    1. M1: (5x - 8) + (3x + 12) = 180
    2. M1: 8x + 4 = 180
    3. A1: x = 22
    Answer: 22
    Circle Theorems
    (5x − 8)°(3x + 12)°ABCDNot drawn accurately
    [3 marks]

  4. Question 4. n is an integer. Prove that (n+3)² - (n-3)² is a multiple of 12.

    Show worked answerHide worked answer
    1. M1: n²+6n+9 - (n²-6n+9)
    2. M1: = 12n
    3. C1: 12n is a multiple of 12
    Answer: 12n, a multiple of 12
    Proof with Consecutive Numbers
    [3 marks]

  5. Question 5. The circle C, centre O, has equation x² + y² = 40
    P is the point (6, -2).
    (a) Show that P lies on C. [1]
    (b) Find an equation of the tangent to C at P.
    Give your answer in the form y = mx + c [3]

    Show worked answerHide worked answer
    1. (a) B1: 6² + (-2)² = 36 + 4 = 40
    2. (b) M1: Gradient of OP = dfrac-26 = -dfrac13
    3. (b) M1: Tangent gradient 3, so y + 2 = 3(x - 6)
    4. (b) A1: y = 3x - 20
    Answer: (a) 36 + 4 = 40; (b) y = 3x - 20
    Finding Equations of Tangents to Circles
    [4 marks]