Questions — OCR (4907 questions)

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OCR C1 2008 January Q10
10 marks Standard +0.3
10 Given that \(\mathrm { f } ( x ) = 8 x ^ { 3 } + \frac { 1 } { x ^ { 3 } }\),
  1. find \(\mathrm { f } ^ { \prime \prime } ( x )\),
  2. solve the equation \(\mathrm { f } ( x ) = - 9\).
OCR C1 2005 June Q1
4 marks Moderate -0.8
1 Solve the inequality \(x ^ { 2 } - 6 x - 40 \geqslant 0\).
OCR C1 2005 June Q2
5 marks Moderate -0.8
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 \(y = 3 x ^ { 2 } + 12 x + 7\).
OCR C1 2005 June Q3
5 marks Easy -1.2
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.
OCR C1 2005 June Q4
5 marks Standard +0.3
4 Solve the equation \(x ^ { 6 } + 26 x ^ { 3 } - 27 = 0\).
OCR C1 2005 June Q5
7 marks Easy -1.3
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 \(a + b \sqrt { } 3\).
OCR C1 2005 June Q6
7 marks Easy -1.2
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 \(\mathrm { f } ^ { \prime \prime } ( x )\).
OCR C1 2005 June Q7
7 marks Easy -1.2
7
  1. Calculate the discriminant of each of the following:
    1. \(x ^ { 2 } + 6 x + 9\),
    2. \(x ^ { 2 } - 10 x + 12\),
    3. \(x ^ { 2 } - 2 x + 5\).
    4. \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) \(y = x ^ { 2 } - 2 x + 5\).
OCR C1 2005 June Q8
8 marks Moderate -0.8
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 \(2 x + y - 5 = 0\).
OCR C1 2005 June Q9
11 marks Moderate -0.3
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.
OCR C1 2005 June Q10
13 marks Moderate -0.8
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 \(q\).
OCR C1 2007 June Q1
3 marks Easy -1.8
1 Simplify \(( 2 x + 5 ) ^ { 2 } - ( x - 3 ) ^ { 2 }\), giving your answer in the form \(a x ^ { 2 } + b x + c\).
OCR C1 2007 June Q2
5 marks Easy -1.3
2
  1. On separate diagrams, sketch the graphs of
    1. \(\mathrm { y } = \frac { 1 } { \mathrm { x } }\),
    2. \(y = x ^ { 4 }\).
  2. Describe a transformation that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = 8 x ^ { 3 }\).
OCR C1 2007 June Q3
5 marks Easy -1.8
3 Simplify the following, expressing each answer in the form \(a \sqrt { 5 }\).
  1. \(3 \sqrt { 10 } \times \sqrt { 2 }\)
  2. \(\sqrt { 500 } + \sqrt { 125 }\)
OCR C1 2007 June Q4
5 marks Moderate -0.8
4
  1. Find the discriminant of \(k x ^ { 2 } - 4 x + k\) in terms of \(k\).
  2. The quadratic equation \(k x ^ { 2 } - 4 x + k = 0\) has equal roots. Find the possible values of \(k\)
OCR C1 2007 June Q5
6 marks Moderate -0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{581ef815-59f0-434e-a7ec-9128e74c0323-2_256_1113_1366_516} The diagram shows a rectangular enclosure, with a wall forming one side. A rope, of length 20 metres, is used to form the remaining three sides. The width of the enclosure is x metres.
  1. Show that the enclosed area, \(\mathrm { Am } ^ { 2 }\), is given by $$A = 20 x - 2 x ^ { 2 } .$$
  2. Use differentiation to find the maximum value of A .
OCR C1 2007 June Q6
6 marks Moderate -0.8
6 By using the substitution \(y = ( x + 2 ) ^ { 2 }\), find the real roots of the equation $$( x + 2 ) ^ { 4 } + 5 ( x + 2 ) ^ { 2 } - 6 = 0$$
OCR C1 2007 June Q7
9 marks Easy -1.2
7
  1. Given that \(f ( x ) = x + \frac { 3 } { x }\), find \(f ^ { \prime } ( x )\).
  2. Find the gradient of the curve \(\mathrm { y } = \mathrm { x } ^ { \frac { 5 } { 2 } }\) at the point where \(\mathrm { x } = 4\).
OCR C1 2007 June Q8
9 marks Moderate -0.8
8
  1. Express \(x ^ { 2 } + 8 x + 15\) in the form \(( x + a ) ^ { 2 } - b\).
  2. Hence state the coordinates of the vertex of the curve \(y = x ^ { 2 } + 8 x + 15\).
  3. Solve the inequality \(x ^ { 2 } + 8 x + 15 > 0\).
OCR C1 2007 June Q9
12 marks Moderate -0.3
9 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 4 .
  1. Find the centre of the circle and the value of k . The points \(\mathrm { A } ( 3 , \mathrm { a } )\) and \(\mathrm { B } ( - 1,0 )\) lie on the circumference of the circle, with \(\mathrm { a } > 0\).
  2. Calculate the length of \(A B\), giving your answer in simplified surd form.
  3. Find an equation for the line \(A B\).
OCR C1 2007 June Q10
12 marks Standard +0.3
10
  1. Solve the equation \(3 x ^ { 2 } - 14 x - 5 = 0\). A curve has equation \(\mathrm { y } = 3 \mathrm { x } ^ { 2 } - 14 \mathrm { x } - 5\).
  2. Sketch the curve, indicating the coordinates of all intercepts with the axes.
  3. Find the value of C for which the line \(\mathrm { y } = 4 \mathrm { x } + \mathrm { C }\) is a tangent to the curve.
OCR C1 2008 June Q1
4 marks Easy -1.8
1 Express each of the following in the form \(4 ^ { n }\) :
  1. \(\frac { 1 } { 16 }\),
  2. 64 ,
  3. 8 .
OCR C1 2008 June Q2
3 marks Easy -1.8
2
  1. The curve \(y = x ^ { 2 }\) is translated 2 units in the positive \(x\)-direction. Find the equation of the curve after it has been translated.
  2. The curve \(y = x ^ { 3 } - 4\) is reflected in the \(x\)-axis. Find the equation of the curve after it has been reflected.
OCR C1 2008 June Q3
4 marks Easy -1.8
3 Express each of the following in the form \(k \sqrt { 2 }\), where \(k\) is an integer:
  1. \(\sqrt { 200 }\),
  2. \(\frac { 12 } { \sqrt { 2 } }\),
  3. \(5 \sqrt { 8 } - 3 \sqrt { 2 }\).
OCR C1 2008 June Q4
5 marks Standard +0.3
4 Solve the equation \(2 x - 7 x ^ { \frac { 1 } { 2 } } + 3 = 0\).