1.02p Interpret algebraic solutions: graphically

28 questions

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Edexcel C1 Q10
13 marks Moderate -0.3
  1. On the same axes sketch the graphs of the curves with equations
    1. \(y = x^2(x - 2)\), [3]
    2. \(y = x(6 - x)\), [3]
    and indicate on your sketches the coordinates of all the points where the curves cross the \(x\)-axis.
  2. Use algebra to find the coordinates of the points where the graphs intersect. [7]
AQA Paper 3 2022 June Q10
13 marks Standard +0.3
The function f is defined by $$f(x) = \frac{x^2 + 10}{2x + 5}$$ where f has its maximum possible domain. The curve \(y = f(x)\) intersects the line \(y = x\) at the points P and Q as shown below. \includegraphics{figure_10}
  1. State the value of \(x\) which is not in the domain of f. [1 mark]
  2. Explain how you know that the function f is many-to-one. [2 marks]
    1. Show that the \(x\)-coordinates of P and Q satisfy the equation $$x^2 + 5x - 10 = 0$$ [2 marks]
    2. Hence, find the exact \(x\)-coordinate of P and the exact \(x\)-coordinate of Q. [1 mark]
  4. Show that P and Q are stationary points of the curve. Fully justify your answer. [5 marks]
  5. Using set notation, state the range of f. [2 marks]
Pre-U Pre-U 9795/1 2013 November Q3
7 marks Standard +0.3
The curve \(C\) has equation \(y = \frac{2x}{x^2 + 1}\).
  1. Write down the equation of the asymptote of \(C\) and the coordinates of any points where \(C\) meets the coordinate axes. [2]
  2. Show that the curve meets the line \(y = k\) if and only if \(-1 \leqslant k \leqslant 1\). Deduce the coordinates of the turning points of the curve. [5]
[Note: You are NOT required to sketch \(C\).]