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Rearranging Harder Formulae

discia.org 10 questions · 47 marks
Name Date
  1. Either paper 3 marks

    A solid cone of radius \(r\), height \(h\) and density \(\rho\) has mass \(M\), where

    \(M = \frac{ 1 }{ 3 }\pi \rho r^2 h\)

    Make \(r\) the subject of the formula.

    Show answer
    \(r = \sqrt{ \frac{ 3M }{ \pi \rho h } }\)
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  2. Either paper 6 marks
    (a)

    Make \(x\) the subject of \(8x + y = 2x + 9\)

    (b)

    Make \(x\) the subject of \(y\left(x - 3\right) = 3x + 8\)

    Show answer
    (a) \(x = \frac{ 9 - y}{ 6 }\)
    (b) \(x = \frac{ 8 + 3y }{y - 3 }\)
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  3. Either paper 4 marks

    Make \(x\) the subject of \(y = \frac{ x - 3 }{x - 5 }\)

    Show answer
    \(x = \frac{ -3 + 5y }{y - 1 }\)
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  4. Either paper 6 marks
    (a)

    Rearrange \(y = \sqrt{ 5x - 14 }\) to make \(x\) the subject.

    (b)

    Rearrange \(y = 2\sqrt{x} + 3\) to make \(x\) the subject.

    Show answer
    (a) \(x = \frac{y^2 + 14 }{ 5 }\)
    (b) \(x = \frac{\left(y - 3\right)^2}{ 4 }\)
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  5. Either paper 6 marks
    (a)

    Rearrange \(y = \frac{ 2 }{ x } + 1\) to make \(x\) the subject.

    (b)

    Rearrange \(y = \frac{ 17 }{x - 2 }\) to make \(x\) the subject.

    Show answer
    (a) \(x = \frac{ 2 }{y - 1 }\)
    (b) \(x = \frac{ 17 }{ y } + 2\)
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  6. Either paper 5 marks

    A car moving in a straight line starts at a speed of \(u\) metres per second and finishes at a speed of \(v\) metres per second. Its acceleration is \(a\) metres per second squared and it travels a distance of \(s\) metres.

    Those four are connected by the formula

    \(v^2 = u^2 + 2as\)

    (a)

    Make \(u\) the subject of the formula.

    (b)

    Work out \(u\) when \(v = 11\), \(a = 8\) and \(s = 7\).

    Show answer
    (a) \(u = \sqrt{v^2 - 2as}\)
    (b) \(3\ \text{m/s}\)
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  7. Either paper 4 marks

    The time period \(T\) of a pendulum of length \(l\) is given by

    \(T = 11\pi\sqrt{ \frac{ l }{ g } }\)

    Make \(g\) the subject of the formula.

    Show answer
    \(g = \frac{ 121\pi^2 l }{ T^2 }\)
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  8. Either paper Higher4 marks

    Two springs are joined end to end. The stiffness of the pair is linked to the stiffness of each one by this formula.

    \(\frac{1}{ k } = \frac{1}{ k_2 } + \frac{1}{ k_1 }\)

    Make \(k_2\) the subject of the formula.

    Show answer
    \(k_2 = \frac{ k k_1 }{ k_1 - k }\)
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  9. Either paper 3 marks

    A solid quarter-cylinder has radius \(r\) and length \(l\). Its volume, \(V\), is given by

    \(V = \frac{ 1 }{ 4 }\pi r^2 l\)

    Find a formula for \(r\).

    Show answer
    \(r = \sqrt{ \frac{ 4V }{ \pi l } }\)
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  10. Either paper 6 marks
    (a)

    Rearrange \(6x + y = 3x - 5\) to make \(x\) the subject.

    (b)

    Rearrange \(y\left(x + 1\right) = 7x + 14\) to make \(x\) the subject.

    Show answer
    (a) \(x = \frac{ -5 - y}{ 3 }\)
    (b) \(x = \frac{ 14 -y }{y - 7 }\)
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discia.org Rearranging Harder Formulae · sheet 1
Answers Rearranging Harder Formulae · sheet 1
  1. \(r = \sqrt{ \frac{ 3M }{ \pi \rho h } }\)
  2. (a) \(x = \frac{ 9 - y}{ 6 }\)
    (b) \(x = \frac{ 8 + 3y }{y - 3 }\)
  3. \(x = \frac{ -3 + 5y }{y - 1 }\)
  4. (a) \(x = \frac{y^2 + 14 }{ 5 }\)
    (b) \(x = \frac{\left(y - 3\right)^2}{ 4 }\)
  5. (a) \(x = \frac{ 2 }{y - 1 }\)
    (b) \(x = \frac{ 17 }{ y } + 2\)
  6. (a) \(u = \sqrt{v^2 - 2as}\)
    (b) \(3\ \text{m/s}\)
  7. \(g = \frac{ 121\pi^2 l }{ T^2 }\)
  8. \(k_2 = \frac{ k k_1 }{ k_1 - k }\)
  9. \(r = \sqrt{ \frac{ 4V }{ \pi l } }\)
  10. (a) \(x = \frac{ -5 - y}{ 3 }\)
    (b) \(x = \frac{ 14 -y }{y - 7 }\)
discia.org Rearranging Harder Formulae · sheet 1