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Quadratic Simultaneous Equations

discia.org 10 questions · 46 marks
Name Date
  1. Either paper 4 marks

    Solve these simultaneous equations.

    \(y = x - 3\)

    \(xy = 28\)

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    \(x = -4,\ y = -7\) or \(x = 7,\ y = 4\)
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  2. Either paper 5 marks

    Solve the simultaneous equations

    \(x^2 + y^2 = 65\)

    \(y = -2x - 10\)

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    \(x = -7,\ y = 4\) or \(x = -1,\ y = -8\)
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  3. Either paper 5 marks

    Solve these simultaneous equations.

    \(y = x^2 + 5x - 8\)

    \(y = 2x - 4\)

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    \(x = -4,\ y = -12\) or \(x = 1,\ y = -2\)
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  4. Either paper 5 marks

    The sum of two numbers is \(8\).

    The sum of the squares of the two numbers is \(40\).

    Work out the two numbers.

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    \(2\) and \(6\)
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  5. Either paper 4 marks

    A rectangle has a perimeter of \(48\) cm and an area of \(119\) cm².

    Find the length and the width of the rectangle.

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    The length is \(17\) cm and the width is \(7\) cm.
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  6. Either paper 4 marks

    Here are the equations of a curve and a straight line.

    \(y = x^2 + 6x - 20\)

    \(y = 4x - 5\)

    They meet at two points. One of them is \((-5, -25)\).

    Calculate the coordinates of the other point.

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    \((3, 7)\)
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  7. Either paper 5 marks

    The line \(y = 7x - 17\) and the curve \(y = x^2 +x - 8\) meet at exactly one point.

    Find the coordinates of that point.

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    \((3, 4)\)
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  8. Either paper 6 marks

    The line \(y = 3x + 7\) crosses the curve \(y = x^2 - 3\) at two points, \(A\) and \(B\).

    Find the length of \(AB\).

    Give your answer in the form \(a\sqrt{ b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.

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    \(7\sqrt{ 10 }\)
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  9. Either paper 4 marks

    A circle has equation \(x^2 + y^2 = 64\).

    A straight line has equation \(y = -3x + 31\).

    Show that the line and the circle do not meet.

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    Substituting the line into the circle gives \(10x^2 - 186x + 897 = 0\), whose discriminant is \(-1284\). It is negative, so there is no value of \(x\) that satisfies both equations, and the line and the circle never meet.
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  10. Either paper 4 marks

    The line \(y = -2x + c\) is a tangent to the curve \(y = x^2 - 12x + 8\).

    Work out the value of \(c\).

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    \(c = -17\)
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discia.org Quadratic Simultaneous Equations · sheet 1
Answers Quadratic Simultaneous Equations · sheet 1
  1. \(x = -4,\ y = -7\) or \(x = 7,\ y = 4\)
  2. \(x = -7,\ y = 4\) or \(x = -1,\ y = -8\)
  3. \(x = -4,\ y = -12\) or \(x = 1,\ y = -2\)
  4. \(2\) and \(6\)
  5. The length is \(17\) cm and the width is \(7\) cm.
  6. \((3, 7)\)
  7. \((3, 4)\)
  8. \(7\sqrt{ 10 }\)
  9. Substituting the line into the circle gives \(10x^2 - 186x + 897 = 0\), whose discriminant is \(-1284\). It is negative, so there is no value of \(x\) that satisfies both equations, and the line and the circle never meet.
  10. \(c = -17\)
discia.org Quadratic Simultaneous Equations · sheet 1