Questions C1 (1562 questions)

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OCR C1 2007 January Q10
12 marks Moderate -0.3
10 A circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 4 y - 8 = 0\).
  1. Find the centre and radius of the circle.
  2. The circle passes through the point \(( - 3 , k )\), where \(k < 0\). Find the value of \(k\).
  3. Find the coordinates of the points where the circle meets the line with equation \(x + y = 6\).
OCR C1 2008 January Q1
3 marks Easy -1.2
1 Express \(\frac { 4 } { 3 - \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers.
OCR C1 2008 January Q2
3 marks Easy -1.3
2
  1. Write down the equation of the circle with centre \(( 0,0 )\) and radius 7 .
  2. A circle with centre \(( 3,5 )\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 10 y - 30 = 0\). Find the radius of the circle.
OCR C1 2008 January Q3
4 marks Moderate -0.8
3 Given that \(3 x ^ { 2 } + b x + 10 = a ( x + 3 ) ^ { 2 } + c\) for all values of \(x\), find the values of the constants \(a , b\) and \(c\).
OCR C1 2008 January Q4
6 marks Easy -1.2
4 Solve the equations
  1. \(10 ^ { p } = 0.1\),
  2. \(\left( 25 k ^ { 2 } \right) ^ { \frac { 1 } { 2 } } = 15\),
  3. \(t ^ { - \frac { 1 } { 3 } } = \frac { 1 } { 2 }\).
OCR C1 2008 January Q5
7 marks Easy -1.2
5
  1. Sketch the curve \(y = x ^ { 3 } + 2\).
  2. Sketch the curve \(y = 2 \sqrt { x }\).
  3. Describe a transformation that transforms the curve \(y = 2 \sqrt { x }\) to the curve \(y = 3 \sqrt { x }\).
OCR C1 2008 January Q6
8 marks Moderate -0.3
6
  1. Solve the equation \(x ^ { 2 } + 8 x + 10 = 0\), giving your answers in simplified surd form.
  2. Sketch the curve \(y = x ^ { 2 } + 8 x + 10\), giving the coordinates of the point where the curve crosses the \(y\)-axis.
  3. Solve the inequality \(x ^ { 2 } + 8 x + 10 \geqslant 0\).
OCR C1 2008 January Q7
8 marks Moderate -0.8
7
  1. Find the gradient of the line \(l\) which has equation \(x + 2 y = 4\).
  2. Find the equation of the line parallel to \(l\) which passes through the point ( 6,5 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Solve the simultaneous equations $$y = x ^ { 2 } + x + 1 \quad \text { and } \quad x + 2 y = 4$$
OCR C1 2008 January Q8
11 marks Moderate -0.8
8
  1. Find the coordinates of the stationary points on the curve \(y = x ^ { 3 } + x ^ { 2 } - x + 3\).
  2. Determine whether each stationary point is a maximum point or a minimum point.
  3. For what values of \(x\) does \(x ^ { 3 } + x ^ { 2 } - x + 3\) decrease as \(x\) increases?
OCR C1 2008 January Q9
12 marks Moderate -0.3
9 The points \(A\) and \(B\) have coordinates \(( - 5 , - 2 )\) and \(( 3,1 )\) respectively.
  1. Find the equation of the line \(A B\), giving your answer in the form \(a x + b y + c = 0\).
  2. Find the coordinates of the mid-point of \(A B\). The point \(C\) has coordinates (-3,4).
  3. Calculate the length of \(A C\), giving your answer in simplified surd form.
  4. Determine whether the line \(A C\) is perpendicular to the line \(B C\), showing all your working.
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 \(k\)