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Linear/quadratic simultaneous equations

3 exam-style questions. Answers and the mark for each step are on the last page.

  1. Question 1Calculator · 3 marks

    (a) Solve y = 2x + 1 and y = x2x^{2} −- 2 simultaneously. Give both ordered pairs (x, y). (3)

  2. Question 2Non-calculator · 4 marks

    Solve the simultaneous equations x2+y2=20x^2 + y^2 = 20 and y=2xy = 2x.

    (a) Find both solutions, giving each as (x, y). (3)

    (b) Explain what the two solutions represent on a graph. (1)

  3. Question 3Non-calculator · 3 marks

    (a) The line y = kx −- 2 meets the curve y = x2x^{2} −- 4x + 7 at exactly one point. Find both possible values of k. (3)

Answers and marks

Question 1

(a) {(−1, −1), (3, 7)}

  • M1 Establishing x2−2=2x+1x^{2}-2=2x+1 or an equivalent valid method.
  • M1 Establishing (x−3)(x+1)=0(x-3)(x+1)=0 or an equivalent valid method.
  • A1 Correct answer: {(−1, −1), (3, 7)}

Question 2

(a) (2, 4) and (−2, −4)

  • M1 Substitute y = 2x into the first equation.
  • M1 Solve for x.
  • A1 Correct answer: (2, 4) and (−2, −4)

(b) They are the two points where the line y = 2x crosses the circle of radius √20 centred at the origin.

  • C1 Correct conclusion with supporting reasoning: They are the two points where the line y = 2x crosses the circle of radius √20 centred at the origin.

Question 3

(a) −10,2-10, 2

  • P1 Establishing x2−(k+4)x+9=0x^{2}-(k+4)x+9=0 or an equivalent valid method.
  • P1 Establishing (k+4)2−36=0(k+4)^{2}-36=0 or an equivalent valid method.
  • A1 Correct answer: −10,2-10, 2

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Linear/quadratic simultaneous equations

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