OCR MEI C1 — Question 1 9 marks

Exam BoardOCR MEI
ModuleC1 (Core Mathematics 1)
Marks9
TopicQuadratic Functions
1
  1. Express \(x ^ { 2 } - 5 x + 6\) in the form \(( x - a ) ^ { 2 } - b\). Hence state the coordinates of the turning point of the curve \(y = x ^ { 2 } - 5 x + 6\).
  2. Find the coordinates of the intersections of the curve \(y = x ^ { 2 } - 5 x + 6\) with the axes and sketch this curve.
  3. Solve the simultaneous equations \(y = x ^ { 2 } - 5 x + 6\) and \(x + y = 2\). Hence show that the line \(x + y = 2\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 6\) at one of the points where the curve intersects the axes. [4] \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{973ad9eb-33f2-432e-9449-e54c1728008b-1_1292_1401_887_359} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 3 }\).
  4. Draw accurately, on the copy of Fig. 12, the graph of \(y = x ^ { 2 } - 4 x + 1\) for \(- 1 \leqslant x \leqslant 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac { 1 } { x - 3 }\) and \(y = x ^ { 2 } - 4 x + 1\).
  5. Show algebraically that, where the curves intersect, \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\).
  6. Use the fact that \(x = 4\) is a root of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\) to find a quadratic factor of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4\). Hence find the exact values of the other two roots of this equation. [5]
  7. Find algebraically the coordinates of the points of intersection of the curve \(y = 4 x ^ { 2 } + 24 x + 31\) and the line \(x + y = 10\).
  8. Express \(4 x ^ { 2 } + 24 x + 31\) in the form \(a ( x + b ) ^ { 2 } + c\).
  9. For the curve \(y = 4 x ^ { 2 } + 24 x + 31\),
    (A) write down the equation of the line of symmetry,
    (B) write down the minimum \(y\)-value on the curve.
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