Line-curve intersection points

A question is this type if and only if it asks to find the coordinates where a line intersects a quadratic curve by solving simultaneous equations.

5 questions · Moderate -0.6

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OCR MEI C1 Q12
12 marks Moderate -0.3
  1. Show that the graph of \(y = x^2 - 3x + 11\) is above the \(x\)-axis for all values of \(x\). [3]
  2. Find the set of values of \(x\) for which the graph of \(y = 2x^2 + x - 10\) is above the \(x\)-axis. [4]
  3. Find algebraically the coordinates of the points of intersection of the graphs of $$y = x^2 - 3x + 11 \quad\text{and}\quad y = 2x^2 + x - 10.$$ [5]
OCR MEI C1 2010 June Q10
12 marks Moderate -0.3
  1. Solve, by factorising, the equation \(2x^2 - x - 3 = 0\). [3]
  2. Sketch the graph of \(y = 2x^2 - x - 3\). [3]
  3. Show that the equation \(x^2 - 5x + 10 = 0\) has no real roots. [2]
  4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2x^2 - x - 3\) and \(y = x^2 - 5x + 10\). Give your answer in the form \(a \pm \sqrt{b}\). [4]
OCR MEI C1 2011 June Q11
11 marks Moderate -0.8
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 4x^2 + 24x + 31\) and the line \(x + y = 10\). [5]
  2. Express \(4x^2 + 24x + 31\) in the form \(a(x + b)^2 + c\). [4]
  3. For the curve \(y = 4x^2 + 24x + 31\),
    1. write down the equation of the line of symmetry, [1]
    2. write down the minimum \(y\)-value on the curve. [1]
OCR C1 Q7
11 marks Moderate -0.8
The curve \(C\) has the equation \(y = x^2 + 2x + 4\).
  1. Express \(x^2 + 2x + 4\) in the form \((x + p)^2 + q\) and hence state the coordinates of the minimum point of \(C\). [4]
The straight line \(l\) has the equation \(x + y = 8\).
  1. Sketch \(l\) and \(C\) on the same set of axes. [3]
  2. Find the coordinates of the points where \(l\) and \(C\) intersect. [4]
OCR MEI C1 Q4
11 marks Moderate -0.8
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 4x^2 + 24x + 31\) and the line \(x + y = 10\). [5]
  2. Express \(4x^2 + 24x + 31\) in the form \(a(x + b)^2 + c\). [4]
  3. For the curve \(y = 4x^2 + 24x + 31\),
    1. write down the equation of the line of symmetry, [1]
    2. write down the minimum \(y\)-value on the curve. [1]