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

Mixed Topic Worksheet Grades 8/9: Day 093

Grades 8/9Non-calculator

  1. Question 1. Write √726 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: √121 × 6
    2. A1: 11√6
    Answer: 11√6
    Simplifying Surds
    [2 marks]

  2. Question 2. Write dfrac2x+1+dfrac3x+4 as a single fraction.
    Give your answer in the form dfracax+b(x+1)(x+4).

    Show worked answerHide worked answer
    1. M1: common denominator (x+1)(x+4)
    2. M1: 2(x+4)+3(x+1)
    3. A1: dfrac5x+11(x+1)(x+4)
    Answer: dfrac5x+11(x+1)(x+4)
    Algebraic Fractions Involving Factorising
    [3 marks]

  3. Question 3. y is directly proportional to the cube of x and inversely proportional to z.
    x is decreased by 10%.
    The value of y does not change.
    Work out the percentage change in z.

    Show worked answerHide worked answer
    1. M1: y = dfrackx³z, so z = dfrackx³y for fixed y
    2. M1: new z is 0.9³ = 0.729 times the old z
    3. A1: z decreases by 27.1%
    Answer: 27.1% decrease
    Direct & Inverse Proportion with Functions
    [3 marks]

  4. Question 4. n is a positive integer.
    Prove that the product of four consecutive integers, plus 1, is a square number.

    Show worked answerHide worked answer
    1. M1: n(n + 3) = n² + 3n and (n + 1)(n + 2) = n² + 3n + 2
    2. M1: (n² + 3n)(n² + 3n + 2) + 1 = (n² + 3n)² + 2(n² + 3n) + 1
    3. A1: (n² + 3n + 1)²
    4. C1: n² + 3n + 1 is an integer, so the result is a square number
    Answer: (n² + 3n + 1)²
    Proof with Consecutive Numbers
    [4 marks]

  5. Question 5. Solve 2x² - 7x - 15 > 0

    Show worked answerHide worked answer
    1. M1 A1: (2x + 3)(x - 5) = 0 so x = -1.5 or x = 5
    2. M1: the outer regions (sketch of the parabola)
    3. A1: x < -1.5 or x > 5
    Answer: x < -1.5 or x > 5
    Solving Quadratic Inequalities
    [4 marks]
📅 Free lesson: Tue 17th NovSave your place →

Mixed Topic Worksheet Grades 8/9: Day 093

Grades 8/9Non-calculator

  1. Question 1. Write √726 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: √121 × 6
    2. A1: 11√6
    Answer: 11√6
    Simplifying Surds

  2. Question 2. Write dfrac2x+1+dfrac3x+4 as a single fraction.
    Give your answer in the form dfracax+b(x+1)(x+4).

    [3 marks]
    Show worked answerHide worked answer
    1. M1: common denominator (x+1)(x+4)
    2. M1: 2(x+4)+3(x+1)
    3. A1: dfrac5x+11(x+1)(x+4)
    Answer: dfrac5x+11(x+1)(x+4)
    Algebraic Fractions Involving Factorising

  3. Question 3. y is directly proportional to the cube of x and inversely proportional to z.
    x is decreased by 10%.
    The value of y does not change.
    Work out the percentage change in z.

    [3 marks]
    Show worked answerHide worked answer
    1. M1: y = dfrackx³z, so z = dfrackx³y for fixed y
    2. M1: new z is 0.9³ = 0.729 times the old z
    3. A1: z decreases by 27.1%
    Answer: 27.1% decrease
    Direct & Inverse Proportion with Functions

  4. Question 4. n is a positive integer.
    Prove that the product of four consecutive integers, plus 1, is a square number.

    [4 marks]
    Show worked answerHide worked answer
    1. M1: n(n + 3) = n² + 3n and (n + 1)(n + 2) = n² + 3n + 2
    2. M1: (n² + 3n)(n² + 3n + 2) + 1 = (n² + 3n)² + 2(n² + 3n) + 1
    3. A1: (n² + 3n + 1)²
    4. C1: n² + 3n + 1 is an integer, so the result is a square number
    Answer: (n² + 3n + 1)²
    Proof with Consecutive Numbers

  5. Question 5. Solve 2x² - 7x - 15 > 0

    [4 marks]
    Show worked answerHide worked answer
    1. M1 A1: (2x + 3)(x - 5) = 0 so x = -1.5 or x = 5
    2. M1: the outer regions (sketch of the parabola)
    3. A1: x < -1.5 or x > 5
    Answer: x < -1.5 or x > 5
    Solving Quadratic Inequalities
📅 Free live lesson: Tuesday 17th NovemberSave your free place →

Mixed Topic Worksheet Grades 8/9: Day 093

Grades 8/9Non-calculator

  1. Question 1. Write √726 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: √121 × 6
    2. A1: 11√6
    Answer: 11√6
    Simplifying Surds
    [2 marks]

  2. Question 2. Write dfrac2x+1+dfrac3x+4 as a single fraction.
    Give your answer in the form dfracax+b(x+1)(x+4).

    Show worked answerHide worked answer
    1. M1: common denominator (x+1)(x+4)
    2. M1: 2(x+4)+3(x+1)
    3. A1: dfrac5x+11(x+1)(x+4)
    Answer: dfrac5x+11(x+1)(x+4)
    Algebraic Fractions Involving Factorising
    [3 marks]

  3. Question 3. y is directly proportional to the cube of x and inversely proportional to z.
    x is decreased by 10%.
    The value of y does not change.
    Work out the percentage change in z.

    Show worked answerHide worked answer
    1. M1: y = dfrackx³z, so z = dfrackx³y for fixed y
    2. M1: new z is 0.9³ = 0.729 times the old z
    3. A1: z decreases by 27.1%
    Answer: 27.1% decrease
    Direct & Inverse Proportion with Functions
    [3 marks]

  4. Question 4. n is a positive integer.
    Prove that the product of four consecutive integers, plus 1, is a square number.

    Show worked answerHide worked answer
    1. M1: n(n + 3) = n² + 3n and (n + 1)(n + 2) = n² + 3n + 2
    2. M1: (n² + 3n)(n² + 3n + 2) + 1 = (n² + 3n)² + 2(n² + 3n) + 1
    3. A1: (n² + 3n + 1)²
    4. C1: n² + 3n + 1 is an integer, so the result is a square number
    Answer: (n² + 3n + 1)²
    Proof with Consecutive Numbers
    [4 marks]

  5. Question 5. Solve 2x² - 7x - 15 > 0

    Show worked answerHide worked answer
    1. M1 A1: (2x + 3)(x - 5) = 0 so x = -1.5 or x = 5
    2. M1: the outer regions (sketch of the parabola)
    3. A1: x < -1.5 or x > 5
    Answer: x < -1.5 or x > 5
    Solving Quadratic Inequalities
    [4 marks]