Questions — OCR MEI (4301 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 2008 January Q12
12 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 4 y = 9\).
  1. Show that the centre of this circle is \(\mathrm { C } ( 4,2 )\) and find the radius of the circle.
  2. Show that the origin lies inside the circle.
  3. Show that AB is a diameter of the circle, where A has coordinates (2, 7) and B has coordinates \(( 6 , - 3 )\).
  4. Find the equation of the tangent to the circle at A . Give your answer in the form \(y = m x + c\).
OCR MEI C1 2009 January Q1
1 State the value of each of the following.
  1. \(2 ^ { - 3 }\)
  2. \(9 ^ { 0 }\)
OCR MEI C1 2009 January Q2
2 Find the equation of the line passing through \(( - 1 , - 9 )\) and \(( 3,11 )\). Give your answer in the form \(y = m x + c\).
OCR MEI C1 2009 January Q3
3 Solve the inequality \(7 - x < 5 x - 2\).
OCR MEI C1 2009 January Q4
4 You are given that \(\mathrm { f } ( x ) = x ^ { 4 } + a x - 6\) and that \(x - 2\) is a factor of \(\mathrm { f } ( x )\).
Find the value of \(a\).
OCR MEI C1 2009 January Q5
5
  1. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( x ^ { 2 } - 3 \right) \left( x ^ { 3 } + 7 x + 1 \right)\).
  2. Find the coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).
OCR MEI C1 2009 January Q6
6 Solve the equation \(\frac { 3 x + 1 } { 2 x } = 4\).
OCR MEI C1 2009 January Q7
7
  1. Express \(125 \sqrt { 5 }\) in the form \(5 ^ { k }\).
  2. Simplify \(\left( 4 a ^ { 3 } b ^ { 5 } \right) ^ { 2 }\).
OCR MEI C1 2009 January Q8
8 Find the range of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 18 = 0\) does not have real roots.
OCR MEI C1 2009 January Q9
9 Rearrange \(y + 5 = x ( y + 2 )\) to make \(y\) the subject of the formula.
OCR MEI C1 2009 January Q10
10
  1. Express \(\sqrt { 75 } + \sqrt { 48 }\) in the form \(a \sqrt { 3 }\).
  2. Express \(\frac { 14 } { 3 - \sqrt { 2 } }\) in the form \(b + c \sqrt { d }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c7fbeb8f-d874-4756-aa53-5471b215902f-3_773_961_354_591} \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{figure} Fig. 11 shows the points A and B , which have coordinates \(( - 1,0 )\) and \(( 11,4 )\) respectively.
OCR MEI C1 2009 January Q12
12
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3 x ^ { 2 } + 6 x + 10\) and the line \(y = 2 - 4 x\).
  2. Write \(3 x ^ { 2 } + 6 x + 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
  3. Hence or otherwise, show that the graph of \(y = 3 x ^ { 2 } + 6 x + 10\) is always above the \(x\)-axis.
OCR MEI C1 2007 June Q1
1 Solve the inequality \(1 - 2 x < 4 + 3 x\).
OCR MEI C1 2007 June Q2
2 Make \(t\) the subject of the formula \(s = \frac { 1 } { 2 } a t ^ { 2 }\).
OCR MEI C1 2007 June Q3
3 The converse of the statement ' \(P \Rightarrow Q\) ' is ' \(Q \Rightarrow P\) '.
Write down the converse of the following statement.
' \(n\) is an odd integer \(\Rightarrow 2 n\) is an even integer.'
Show that this converse is false.
OCR MEI C1 2007 June Q4
4 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + k x + c\). The value of \(\mathrm { f } ( 0 )\) is 6, and \(x - 2\) is a factor of \(\mathrm { f } ( x )\).
Find the values of \(k\) and \(c\).
OCR MEI C1 2007 June Q5
5
  1. Find \(a\), given that \(a ^ { 3 } = 64 x ^ { 12 } y ^ { 3 }\).
  2. Find the value of \(\left( \frac { 1 } { 2 } \right) ^ { - 5 }\).
OCR MEI C1 2007 June Q6
6 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - 2 x ) ^ { 5 }\).
OCR MEI C1 2007 June Q7
7 Solve the equation \(\frac { 4 x + 5 } { 2 x } = - 3\).
OCR MEI C1 2007 June Q8
8
  1. Simplify \(\sqrt { 98 } - \sqrt { 50 }\).
  2. Express \(\frac { 6 \sqrt { 5 } } { 2 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.
OCR MEI C1 2007 June Q9
9
  1. A curve has equation \(y = x ^ { 2 } - 4\). Find the \(x\)-coordinates of the points on the curve where \(y = 21\).
  2. The curve \(y = x ^ { 2 } - 4\) is translated by \(\binom { 2 } { 0 }\). Write down an equation for the translated curve. You need not simplify your answer.
OCR MEI C1 2007 June Q10
10 The triangle shown in Fig. 10 has height \(( x + 1 ) \mathrm { cm }\) and base \(( 2 x - 3 ) \mathrm { cm }\). Its area is \(9 \mathrm {~cm} ^ { 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d8caf0f-7594-42cb-bd40-e6c11e2b6832-3_444_1088_351_715} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Show that \(2 x ^ { 2 } - x - 21 = 0\).
  2. By factorising, solve the equation \(2 x ^ { 2 } - x - 21 = 0\). Hence find the height and base of the triangle. Section B (36 marks)
OCR MEI C1 2007 June Q11
11 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d8caf0f-7594-42cb-bd40-e6c11e2b6832-3_442_1102_1384_717} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} A circle has centre \(C ( 1,3 )\) and passes through the point \(A ( 3,7 )\) as shown in Fig. 11.
  1. Show that the equation of the tangent at A is \(x + 2 y = 17\).
  2. The line with equation \(y = 2 x - 9\) intersects this tangent at the point T . Find the coordinates of T .
  3. The equation of the circle is \(( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). Show that the line with equation \(y = 2 x - 9\) is a tangent to the circle. Give the coordinates of the point where this tangent touches the circle.
OCR MEI C1 2007 June Q12
12
  1. Write \(4 x ^ { 2 } - 24 x + 27\) in the form \(a ( x - b ) ^ { 2 } + c\).
  2. State the coordinates of the minimum point on the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
  3. Solve the equation \(4 x ^ { 2 } - 24 x + 27 = 0\).
  4. Sketch the graph of the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
OCR MEI C1 2007 June Q13
13 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).
  1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
  2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
  3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).