OCR C1 (Core Mathematics 1) 2005 June

Question 1
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1 Solve the inequality \(x ^ { 2 } - 6 x - 40 \geqslant 0\).
Question 2
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2
  1. Express \(3 x ^ { 2 } + 12 x + 7\) in the form \(3 ( x + a ) ^ { 2 } + b\).
  2. Hence write down the equation of the line of symmetry of the curve \(y = 3 x ^ { 2 } + 12 x + 7\).
Question 3
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3
  1. Sketch the curve \(y = x ^ { 3 }\).
  2. Describe a transformation that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = - x ^ { 3 }\).
  3. The curve \(y = x ^ { 3 }\) is translated by \(p\) units, parallel to the \(x\)-axis. State the equation of the curve after it has been transformed.
Question 4
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4 Solve the equation \(x ^ { 6 } + 26 x ^ { 3 } - 27 = 0\).
Question 5
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5
  1. Simplify \(2 x ^ { \frac { 2 } { 3 } } \times 3 x ^ { - 1 }\).
  2. Express \(2 ^ { 40 } \times 4 ^ { 30 }\) in the form \(2 ^ { n }\).
  3. Express \(\frac { 26 } { 4 - \sqrt { } 3 }\) in the form \(a + b \sqrt { } 3\).
Question 6
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6 Given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 3 x - 4 )\),
  1. express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\),
  2. find \(\mathrm { f } ^ { \prime } ( x )\),
  3. find \(\mathrm { f } ^ { \prime \prime } ( x )\).
Question 7
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7
  1. Calculate the discriminant of each of the following:
    (a) \(x ^ { 2 } + 6 x + 9\),
    (b) \(x ^ { 2 } - 10 x + 12\),
    (c) \(x ^ { 2 } - 2 x + 5\).
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_391_446_628_397} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_394_449_625_888} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_389_442_630_1384} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_394_446_1119_644} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_396_447_1119_1137} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure} State with reasons which of the diagrams corresponds to the curve
    (a) \(y = x ^ { 2 } + 6 x + 9\),
    (b) \(y = x ^ { 2 } - 10 x + 12\),
    (c) \(y = x ^ { 2 } - 2 x + 5\).
Question 8
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8
  1. Describe completely the curve \(x ^ { 2 } + y ^ { 2 } = 25\).
  2. Find the coordinates of the points of intersection of the curve \(x ^ { 2 } + y ^ { 2 } = 25\) and the line \(2 x + y - 5 = 0\).
Question 9
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9
  1. Find the gradient of the line \(l _ { 1 }\) which has equation \(4 x - 3 y + 5 = 0\).
  2. Find an equation of the line \(l _ { 2 }\), which passes through the point ( 1,2 ) and which is perpendicular to the line \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\). The line \(l _ { 1 }\) crosses the \(x\)-axis at \(P\) and the line \(l _ { 2 }\) crosses the \(y\)-axis at \(Q\).
  3. Find the coordinates of the mid-point of \(P Q\).
  4. Calculate the length of \(P Q\), giving your answer in the form \(\frac { \sqrt { } a } { b }\), where \(a\) and \(b\) are integers.
Question 10
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10
  1. Given that \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find the coordinates of the stationary points on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\).
  3. Determine whether each stationary point is a maximum point or a minimum point.
  4. Given that \(24 x + 3 y + 2 = 0\) is the equation of the tangent to the curve at the point ( \(p , q\) ), find \(p\) and \(q\).