1.02c Simultaneous equations: two variables by elimination and substitution

284 questions

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OCR MEI Paper 2 Specimen Q1
5 marks Moderate -0.8
In this question you must show detailed reasoning. Find the coordinates of the points of intersection of the curve \(y = x^2 + x\) and the line \(2x + y = 4\). [5]
WJEC Unit 1 2022 June Q15
8 marks Challenging +1.2
Solve the simultaneous equations $$3\log_u(x^2y) - \log_u(x^2y^2) + \log_u\left(\frac{9}{x^2y^2}\right) = \log_u 36,$$ $$\log_u y - \log_u(x + 3) = 0.$$ [8]
SPS SPS FM 2020 October Q8
10 marks Moderate -0.8
The equation of a circle is \(x^2 + y^2 + 6x - 2y - 10 = 0\).
  1. Find the centre and radius of the circle. [3]
  2. Find the coordinates of any points where the line \(y = 2x - 3\) meets the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [4]
  3. State what can be deduced from the answer to part ii. about the line \(y = 2x - 3\) and the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [1]
  4. The point \(A(-1,5)\) lies on the circumference of the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]
SPS SPS SM 2020 October Q7
11 marks Moderate -0.3
  1. Sketch the curves \(y = \frac{3}{x^2}\) and \(y = x^2 - 2\) on the axes provided below. \includegraphics{figure_1} [3]
  2. In this question you must show detailed reasoning. Find the exact coordinates of the points of interception of the curves \(y = \frac{3}{x^2}\) and \(y = x^2 - 2\). [6]
  3. Hence, solve the inequality \(\frac{3}{x^2} \leq x^2 - 2\), giving your answer in interval notation. [2]
SPS SPS SM 2020 October Q8
10 marks Moderate -0.8
The equation of a circle is \(x^2 + y^2 + 6x - 2y - 10 = 0\).
  1. Find the centre and radius of the circle. [3]
  2. Find the coordinates of any points where the line \(y = 2x - 3\) meets the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [4]
  3. State what can be deduced from the answer to part ii. about the line \(y = 2x - 3\) and the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [1]
  4. The point \(A(-1,5)\) lies on the circumference of the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]
SPS SPS SM 2020 October Q9
6 marks Standard +0.8
In this question you must show detailed reasoning. Solve the following simultaneous equations: $$(\log_3 x)^2 + \log_3(y^2) = 5$$ $$\log_3(\sqrt{3xy^{-1}}) = 2$$ [6]
SPS SPS SM 2025 February Q5
7 marks Standard +0.3
A curve has the following properties: • The gradient of the curve is given by \(\frac{dy}{dx} = -2x\). • The curve passes through the point \((4, -13)\). Determine the coordinates of the points where the curve meets the line \(y = 2x\). [7]
OCR H240/01 2017 Specimen Q1
4 marks Moderate -0.5
Solve the simultaneous equations. \(x^2 + 8x + y^2 = 84\) \(x - y = 10\) [4]
Pre-U Pre-U 9794/2 Specimen Q2
4 marks Standard +0.3
Solve the simultaneous equations $$x - 2y = 5,$$ $$\frac{4}{x} - \frac{2}{y} = 5.$$ [4]