Questions — OCR MEI C1 (472 questions)

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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 MEI C1 2012 January Q7
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f138ed97-09ca-488e-8651-1217ac2d7b21-2_684_734_1537_662} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 shows the graph of \(y = \mathrm { g } ( x )\). Draw the graphs of the following.
  1. \(y = \mathrm { g } ( x ) + 3\)
  2. \(y = \mathrm { g } ( x + 2 )\)
OCR MEI C1 2012 January Q8
8 Express \(5 x ^ { 2 } + 15 x + 12\) in the form \(a ( x + b ) ^ { 2 } + c\).
Hence state the minimum value of \(y\) on the curve \(y = 5 x ^ { 2 } + 15 x + 12\).
OCR MEI C1 2012 January Q9
9 Complete each of the following by putting the best connecting symbol ( \(\Leftrightarrow , \Leftarrow\) or ⇒ ) in the box. Explain your choice, giving full reasons.
  1. \(n ^ { 3 } + 1\) is an odd integer □ \(n\) is an even integer
  2. \(( x - 3 ) ( x - 2 ) > 0\) □ \(x > 3\) Section B (36 marks)
OCR MEI C1 2012 January Q10
10 Point A has coordinates (4, 7) and point B has coordinates (2, 1).
  1. Find the equation of the line through A and B .
  2. Point C has coordinates ( \(- 1,2\) ). Show that angle \(\mathrm { ABC } = 90 ^ { \circ }\) and calculate the area of triangle ABC .
  3. Find the coordinates of D , the midpoint of AC . Explain also how you can tell, without having to work it out, that \(\mathrm { A } , \mathrm { B }\) and C are all the same distance from D.
OCR MEI C1 2012 January Q11
11 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 23 x + 12\).
  1. Show that \(x = - 3\) is a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) fully.
  2. Sketch the curve \(y = \mathrm { f } ( x )\).
  3. Find the \(x\)-coordinates of the points where the line \(y = 4 x + 12\) intersects \(y = \mathrm { f } ( x )\).
OCR MEI C1 2012 January Q12
12 A circle has equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 20\).
  1. Write down the radius of the circle and the coordinates of its centre.
  2. Find the points of intersection of the circle with the \(y\)-axis and sketch the circle.
  3. Show that, where the line \(y = 2 x + k\) intersects the circle, $$5 x ^ { 2 } + ( 4 k - 4 ) x + k ^ { 2 } - 16 = 0 .$$
  4. Hence find the values of \(k\) for which the line \(y = 2 x + k\) is a tangent to the circle. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR MEI C1 2013 January Q1
1 Find the value of each of the following.
  1. \(\left( \frac { 5 } { 3 } \right) ^ { - 2 }\)
  2. \(81 ^ { \frac { 3 } { 4 } }\)
OCR MEI C1 2013 January Q2
2 Simplify \(\frac { \left( 4 x ^ { 5 } y \right) ^ { 3 } } { \left( 2 x y ^ { 2 } \right) \times \left( 8 x ^ { 10 } y ^ { 4 } \right) }\).
OCR MEI C1 2013 January Q3
3 A circle has diameter \(d\), circumference \(C\), and area \(A\). Starting with the standard formulae for a circle, show that \(C d = k A\), finding the numerical value of \(k\).
OCR MEI C1 2013 January Q4
4 Solve the inequality \(5 x ^ { 2 } - 28 x - 12 \leqslant 0\).
OCR MEI C1 2013 January Q5
5 You are given that \(\mathrm { f } ( x ) = x ^ { 2 } + k x + c\).
Given also that \(\mathrm { f } ( 2 ) = 0\) and \(\mathrm { f } ( - 3 ) = 35\), find the values of the constants \(k\) and \(c\).
OCR MEI C1 2013 January Q6
6 The binomial expansion of \(\left( 2 x + \frac { 5 } { x } \right) ^ { 6 }\) has a term which is a constant. Find this term.
OCR MEI C1 2013 January Q7
7
  1. Express \(\sqrt { 48 } + \sqrt { 75 }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers.
  2. Simplify \(\frac { 7 + 2 \sqrt { 5 } } { 7 + \sqrt { 5 } }\), expressing your answer in the form \(\frac { a + b \sqrt { 5 } } { c }\), where \(a , b\) and \(c\) are integers.
OCR MEI C1 2013 January Q8
8 Rearrange the equation \(5 c + 9 t = a ( 2 c + t )\) to make \(c\) the subject.
OCR MEI C1 2013 January Q9
9 You are given that \(\mathrm { f } ( x ) = ( x + 2 ) ^ { 2 } ( x - 3 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. State the values of \(x\) which satisfy \(\mathrm { f } ( x + 3 ) = 0\).
OCR MEI C1 2013 January Q10
10
  1. Points A and B have coordinates \(( - 2,1 )\) and \(( 3,4 )\) respectively. Find the equation of the perpendicular bisector of AB and show that it may be written as \(5 x + 3 y = 10\).
  2. Points C and D have coordinates \(( - 5,4 )\) and \(( 3,6 )\) respectively. The line through C and D has equation \(4 y = x + 21\). The point E is the intersection of CD and the perpendicular bisector of AB . Find the coordinates of point E .
  3. Find the equation of the circle with centre E which passes through A and B . Show also that CD is a diameter of this circle.
OCR MEI C1 2013 January Q11
11
  1. Express \(x ^ { 2 } - 5 x + 6\) in the form \(( x - a ) ^ { 2 } - b\). Hence state the coordinates of the turning point of the curve \(y = x ^ { 2 } - 5 x + 6\).
  2. Find the coordinates of the intersections of the curve \(y = x ^ { 2 } - 5 x + 6\) with the axes and sketch this curve.
  3. Solve the simultaneous equations \(y = x ^ { 2 } - 5 x + 6\) and \(x + y = 2\). Hence show that the line \(x + y = 2\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 6\) at one of the points where the curve intersects the axes.
OCR MEI C1 2013 January Q12
12 You are given that \(\mathrm { f } ( x ) = x ^ { 4 } - x ^ { 3 } + x ^ { 2 } + 9 x - 10\).
  1. Show that \(x = 1\) is a root of \(\mathrm { f } ( x ) = 0\) and hence express \(\mathrm { f } ( x )\) as a product of a linear factor and a cubic factor.
  2. Hence or otherwise find another root of \(\mathrm { f } ( x ) = 0\).
  3. Factorise \(\mathrm { f } ( x )\), showing that it has only two linear factors. Show also that \(\mathrm { f } ( x ) = 0\) has only two real roots. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR MEI C1 2009 June Q1
1 A line has gradient - 4 and passes through the point (2,6). Find the coordinates of its points of intersection with the axes.
OCR MEI C1 2009 June Q2
2 Make \(a\) the subject of the formula \(s = u t + \frac { 1 } { 2 } a t ^ { 2 }\).
OCR MEI C1 2009 June Q3
3 When \(x ^ { 3 } - k x + 4\) is divided by \(x - 3\), the remainder is 1 . Use the remainder theorem to find the value of \(k\).
OCR MEI C1 2009 June Q4
4 Solve the inequality \(x ( x - 6 ) > 0\).
OCR MEI C1 2009 June Q5
5
  1. Calculate \({ } ^ { 5 } \mathrm { C } _ { 3 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 + 2 x ) ^ { 5 }\).
OCR MEI C1 2009 June Q6
6 Prove that, when \(n\) is an integer, \(n ^ { 3 } - n\) is always even.
OCR MEI C1 2009 June Q7
7 Find the value of each of the following.
  1. \(5 ^ { 2 } \times 5 ^ { - 2 }\)
  2. \(100 ^ { \frac { 3 } { 2 } }\)