Questions C1 (1442 questions)

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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
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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 2010 January Q3
Moderate -0.8
3 Find the equation of the normal to the curve \(y = x ^ { 3 } - 4 x ^ { 2 } + 7\) at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 2010 January Q4
Easy -1.3
4 Solve the equations
  1. \(3 ^ { m } = 81\),
  2. \(\left( 36 p ^ { 4 } \right) ^ { \frac { 1 } { 2 } } = 24\),
  3. \(5 ^ { n } \times 5 ^ { n + 4 } = 25\).
OCR C1 2010 January Q5
Standard +0.3
5 Solve the equation \(x - 8 \sqrt { x } + 13 = 0\), giving your answers in the form \(p \pm q \sqrt { r }\), where \(p , q\) and \(r\) are integers.
OCR C1 2010 January Q6
Easy -1.3
6
\includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-2_394_846_1868_648} The diagram shows part of the curve \(y = x ^ { 2 } + 5\). The point \(A\) has coordinates ( 1,6 ). The point \(B\) has coordinates ( \(a , a ^ { 2 } + 5\) ), where \(a\) is a constant greater than 1 . The point \(C\) is on the curve between \(A\) and \(B\).
  1. Find by differentiation the value of the gradient of the curve at the point \(A\).
  2. The line segment joining the points \(A\) and \(B\) has gradient 2.3. Find the value of \(a\).
  3. State a possible value for the gradient of the line segment joining the points \(A\) and \(C\).
OCR C1 2010 January Q7
Easy -1.3
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{918d83c3-1608-4482-9d3d-8af05e65f353-3_618_606_255_397} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{918d83c3-1608-4482-9d3d-8af05e65f353-3_622_622_251_1128} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{918d83c3-1608-4482-9d3d-8af05e65f353-3_620_613_986_395} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{918d83c3-1608-4482-9d3d-8af05e65f353-3_620_611_986_1128} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Each diagram shows a quadratic curve. State which diagram corresponds to the curve
    (a) \(y = ( 3 - x ) ^ { 2 }\),
    (b) \(y = x ^ { 2 } + 9\),
    (c) \(y = ( 3 - x ) ( x + 3 )\).
  2. Give the equation of the curve which does not correspond to any of the equations in part (i).
OCR C1 2010 January Q8
Moderate -0.3
8 A circle has equation \(x ^ { 2 } + y ^ { 2 } + 6 x - 4 y - 4 = 0\).
  1. Find the centre and radius of the circle.
  2. Find the coordinates of the points where the circle meets the line with equation \(y = 3 x + 4\).
OCR C1 2010 January Q9
Moderate -0.8
9 Given that \(\mathrm { f } ( x ) = \frac { 1 } { x } - \sqrt { x } + 3\),
  1. find \(\mathrm { f } ^ { \prime } ( x )\),
  2. find \(\mathrm { f } ^ { \prime \prime } ( 4 )\).
OCR C1 2010 January Q10
Moderate -0.3
10 The quadratic equation \(k x ^ { 2 } - 30 x + 25 k = 0\) has equal roots. Find the possible values of \(k\).
OCR C1 2010 January Q11
Standard +0.3
11 A lawn is to be made in the shape shown below. The units are metres.
\includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-4_412_698_486_726}
  1. The perimeter of the lawn is \(P \mathrm {~m}\). Find \(P\) in terms of \(x\).
  2. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the lawn is given by \(A = 9 x ^ { 2 } + 6 x\). The perimeter of the lawn must be at least 39 m and the area of the lawn must be less than \(99 \mathrm {~m} ^ { 2 }\).
  3. By writing down and solving appropriate inequalities, determine the set of possible values of \(x\).
OCR C1 2011 January Q1
Moderate -0.8
1 The points \(A\) and \(B\) have coordinates \(( 6,1 )\) and \(( - 2,7 )\) respectively.
  1. Find the length of \(A B\).
  2. Find the gradient of the line \(A B\).
  3. Determine whether the line \(4 x - 3 y - 10 = 0\) is perpendicular to \(A B\).
OCR C1 2011 January Q2
Moderate -0.3
2 Given that $$( x - p ) \left( 2 x ^ { 2 } + 9 x + 10 \right) = \left( x ^ { 2 } - 4 \right) ( 2 x + q )$$ for all values of \(x\), find the constants \(p\) and \(q\).
OCR C1 2011 January Q3
Easy -1.8
3 Express each of the following in the form \(8 ^ { p }\) :
  1. \(\sqrt { 8 }\),
  2. \(\frac { 1 } { 64 }\),
  3. \(2 ^ { 6 } \times 2 ^ { 2 }\).
OCR C1 2011 January Q4
Moderate -0.5
4 By using the substitution \(u = ( 3 x - 2 ) ^ { 2 }\), find the roots of the equation $$( 3 x - 2 ) ^ { 4 } - 5 ( 3 x - 2 ) ^ { 2 } + 4 = 0$$
OCR C1 2011 January Q5
Easy -1.2
5
  1. Sketch the curve \(y = - x ^ { 3 }\).
  2. The curve \(y = - x ^ { 3 }\) is translated by 3 units in the positive \(x\)-direction. Find the equation of the curve after it has been translated.
  3. Describe a transformation that transforms the curve \(y = - x ^ { 3 }\) to the curve \(y = - 5 x ^ { 3 }\).
OCR C1 2011 January Q6
Moderate -0.8
6 Given that \(y = \frac { 5 } { x ^ { 2 } } - \frac { 1 } { 4 x } + x\), find
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
OCR C1 2011 January Q7
Moderate -0.3
7
  1. Express \(4 x ^ { 2 } + 12 x - 3\) in the form \(p ( x + q ) ^ { 2 } + r\).
  2. Solve the equation \(4 x ^ { 2 } + 12 x - 3 = 0\), giving your answers in simplified surd form.
  3. The quadratic equation \(4 x ^ { 2 } + 12 x - k = 0\) has equal roots. Find the value of \(k\).
OCR C1 2011 January Q8
Moderate -0.8
8
  1. Find the equation of the tangent to the curve \(y = 7 + 6 x - x ^ { 2 }\) at the point \(P\) where \(x = 5\), giving your answer in the form \(a x + b y + c = 0\).
  2. This tangent meets the \(x\)-axis at \(Q\). Find the coordinates of the mid-point of \(P Q\).
  3. Find the equation of the line of symmetry of the curve \(y = 7 + 6 x - x ^ { 2 }\).
  4. State the set of values of \(x\) for which \(7 + 6 x - x ^ { 2 }\) is an increasing function.
OCR C1 2011 January Q9
Standard +0.3
9 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 2 y - 3 = 0\).
  1. Find the coordinates of \(C\) and the radius of the circle.
  2. Find the values of \(k\) for which the line \(y = k\) is a tangent to the circle, giving your answers in simplified surd form.
  3. The points \(S\) and \(T\) lie on the circumference of the circle. \(M\) is the mid-point of the chord \(S T\). Given that the length of \(C M\) is 2 , calculate the length of the chord \(S T\).
  4. Find the coordinates of the point where the circle meets the line \(x - 2 y - 12 = 0\).
OCR C1 2012 January Q1
Easy -1.2
1 Express \(\frac { 15 + \sqrt { 3 } } { 3 - \sqrt { 3 } }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
OCR C1 2012 January Q2
Easy -1.2
2
\includegraphics[max width=\textwidth, alt={}, center]{559e9f1a-340e-4478-adaa-a7361dd70fe8-2_325_479_468_794} The graph of \(y = \mathrm { f } ( x )\) for \(- 2 \leqslant x \leqslant 2\) is shown above.
  1. Sketch the graph of \(y = \mathrm { f } ( - x )\) for \(- 2 \leqslant x \leqslant 2\).
  2. Sketch the graph of \(y = \mathrm { f } ( x ) + 2\) for \(- 2 \leqslant x \leqslant 2\).
OCR C1 2012 January Q3
Moderate -0.8
3 Given that $$5 x ^ { 2 } + p x - 8 = q ( x - 1 ) ^ { 2 } + r$$ for all values of \(x\), find the values of the constants \(p , q\) and \(r\).
OCR C1 2012 January Q4
Easy -1.8
4 Evaluate
  1. \(3 ^ { - 2 }\),
  2. \(16 ^ { \frac { 3 } { 4 } }\),
  3. \(\frac { \sqrt { 200 } } { \sqrt { 8 } }\).
OCR C1 2012 January Q5
Moderate -0.3
5 Find the real roots of the equation \(\frac { 3 } { y ^ { 4 } } - \frac { 10 } { y ^ { 2 } } - 8 = 0\).
OCR C1 2012 January Q6
Moderate -0.8
6 Given that \(\mathrm { f } ( x ) = \frac { 4 } { x } - 3 x + 2\),
  1. find \(\mathrm { f } ^ { \prime } ( x )\),
  2. find \(\mathrm { f } ^ { \prime \prime } \left( \frac { 1 } { 2 } \right)\).
OCR C1 2012 January Q7
Moderate -0.3
7 A curve has equation \(y = ( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\).
  1. Find the coordinates of the minimum point, justifying that it is a minimum.
  2. Calculate the discriminant of \(x ^ { 2 } - 3 x + 5\).
  3. Explain why \(( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\) is always positive for \(x > - 2\).