Questions — OCR C1 (324 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR C1 2007 January Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{82ae6eec-3007-467c-90df-18f2adb9ccb1-2_634_926_1242_612} The graph of \(y = \mathrm { f } ( x )\) for \(- 1 \leqslant x \leqslant 4\) is shown above.
  1. Sketch the graph of \(y = - \mathrm { f } ( x )\) for \(- 1 \leqslant x \leqslant 4\).
  2. The point \(P ( 1,1 )\) on \(y = \mathrm { f } ( x )\) is transformed to the point \(Q\) on \(y = 3 \mathrm { f } ( x )\). State the coordinates of \(Q\).
  3. Describe the transformation which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { f } ( x + 2 )\).
OCR C1 2007 January Q6
6
  1. Express \(2 x ^ { 2 } - 24 x + 80\) in the form \(a ( x - b ) ^ { 2 } + c\).
  2. State the equation of the line of symmetry of the curve \(y = 2 x ^ { 2 } - 24 x + 80\).
  3. State the equation of the tangent to the curve \(y = 2 x ^ { 2 } - 24 x + 80\) at its minimum point.
OCR C1 2007 January Q7
7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases.
  1. \(y = 5 x + 3\)
  2. \(y = \frac { 2 } { x ^ { 2 } }\)
  3. \(y = ( 2 x + 1 ) ( 5 x - 7 )\)
OCR C1 2007 January Q8
8
  1. Find the coordinates of the stationary points of the curve \(y = 27 + 9 x - 3 x ^ { 2 } - x ^ { 3 }\).
  2. Determine, in each case, whether the stationary point is a maximum or minimum point.
  3. Hence state the set of values of \(x\) for which \(27 + 9 x - 3 x ^ { 2 } - x ^ { 3 }\) is an increasing function.
    \(9 \quad A\) is the point \(( 2,7 )\) and \(B\) is the point \(( - 1 , - 2 )\).
OCR C1 2007 January Q10
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
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
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
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
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
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
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
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
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
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 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
1 Solve the inequality \(x ^ { 2 } - 6 x - 40 \geqslant 0\).
OCR C1 2005 June Q2
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
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
4 Solve the equation \(x ^ { 6 } + 26 x ^ { 3 } - 27 = 0\).
OCR C1 2005 June Q5
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
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
  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) \(y = x ^ { 2 } - 2 x + 5\).
OCR C1 2005 June Q8
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
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
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 \(q\).