Cambridge 9709 Pure Mathematics 1 — Quadratics Test

Published

October 29, 2026

Instructions

  • Time allowed: 90 minutes
  • Total marks: 50
  • Answer all questions.
  • Give non-exact numerical answers correct to 3 significant figures, unless otherwise stated.
  • Show sufficient working; marks may be lost for missing essential steps.

1. The expression \(9x^2 - 36x + 8\) can be written in the form \(p(x+q)^2 + r\), where \(p\), \(q\) and \(r\) are constants.

  1. Find the values of \(p\), \(q\) and \(r\). [3]
  2. Hence state the minimum value of \(9x^2 - 36x + 8\) and the value of \(x\) at which it occurs. [2]
  3. Hence find the set of values of the constant \(k\) for which \(9x^2 - 36x + 8 = k\) has no real solutions. [1]

2. The equation \(3kx^2 + (k+8)x + 3 = 0\), where \(k\) is a constant, has two distinct real roots.

  1. Show that \(k^2 - 20k + 64 > 0\). [3]
  2. Hence find the set of possible values of \(k\). [3]
  3. State one value of \(k\) for which the equation has no real roots. [1]

3. The curve \(C\) has equation \(y = 2x^2 - 8x + 5\).

  1. Express \(2x^2 - 8x + 5\) in the form \(a(x+b)^2 + c\). [3]
  2. Write down the coordinates of the vertex of \(C\). [1]
  3. Sketch \(C\), showing the vertex and the \(y\)-intercept. [2]
  4. Find the equation of the line of symmetry of \(C\). [2]

4. The line \(y = 2x + 3\) intersects the curve \(y = x^2 - 4x + k\), where \(k\) is a constant.

  1. Show that the \(x\)-coordinates of the points of intersection satisfy \(x^2 - 6x + k - 3 = 0\). [2]
  2. Given that the line is a tangent to the curve, find the value of \(k\). [3]
  3. For this value of \(k\), find the coordinates of the point of contact. [4]

5. A quadratic curve has equation \(y = x^2 - 2mx + m^2 + 3\), where \(m\) is a constant.

  1. Show that the curve has no real \(x\)-intercepts for all values of \(m\). [4]
  2. Find the minimum value of \(y\) in terms of \(m\). [2]
  3. State the coordinates of the vertex in terms of \(m\). [2]

6. The curve \(C\) has equation \(y = x^2 - 6x + 11\).

  1. Express \(x^2 - 6x + 11\) in the form \((x-a)^2 + b\). [2]
  2. Sketch \(C\), showing the coordinates of its vertex and its \(y\)-intercept. [3]
  3. The line \(y = 2x + c\) intersects \(C\) at two distinct points. Show that \(c < 1\). [4]
  4. Given that \(c = -3\), find the coordinates of the two points of intersection. Give your answers in exact form. [3]