Questions — OCR MEI (4333 questions)

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OCR MEI FP1 2011 June Q9
13 marks Moderate -0.3
9 The simultaneous equations $$\begin{aligned} & 2 x - y = 1 \\ & 3 x + k y = b \end{aligned}$$ are represented by the matrix equation \(\mathbf { M } \binom { x } { y } = \binom { 1 } { b }\).
  1. Write down the matrix \(\mathbf { M }\).
  2. State the value of \(k\) for which \(\mathbf { M } ^ { - 1 }\) does not exist and find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\) when \(\mathbf { M } ^ { - 1 }\) exists. Use \(\mathbf { M } ^ { - 1 }\) to solve the simultaneous equations when \(k = 5\) and \(b = 21\).
  3. What can you say about the solutions of the equations when \(k = - \frac { 3 } { 2 }\) ?
  4. The two equations can be interpreted as representing two lines in the \(x - y\) plane. Describe the relationship between these two lines
    (A) when \(k = 5\) and \(b = 21\),
    (B) when \(k = - \frac { 3 } { 2 }\) and \(b = 1\),
    (C) when \(k = - \frac { 3 } { 2 }\) and \(b = \frac { 3 } { 2 }\). RECOGNISING ACHIEVEMENT
OCR MEI FP1 2012 June Q1
5 marks Moderate -0.8
1 You are given that the matrix \(\left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)\) represents a transformation \(A\), and that the matrix \(\left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)\) represents a transformation B .
  1. Describe the transformations A and B .
  2. Find the matrix representing the composite transformation consisting of A followed by B .
  3. What single transformation is represented by this matrix?
OCR MEI FP1 2012 June Q2
7 marks Standard +0.3
2 You are given that \(z _ { 1 }\) and \(z _ { 2 }\) are complex numbers. \(z _ { 1 } = 3 + 3 \sqrt { 3 } \mathrm { j }\), and \(z _ { 2 }\) has modulus 5 and argument \(\frac { \pi } { 3 }\).
  1. Find the modulus and argument of \(z _ { 1 }\), giving your answers exactly.
  2. Express \(z _ { 2 }\) in the form \(a + b \mathrm { j }\), where \(a\) and \(b\) are to be given exactly.
  3. Explain why, when plotted on an Argand diagram, \(z _ { 1 } , z _ { 2 }\) and the origin lie on a straight line.
OCR MEI FP1 2012 June Q3
6 marks Standard +0.8
3 The cubic equation \(3 x ^ { 3 } + 8 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , \frac { \alpha } { 6 }\) and \(\alpha - 7\). Find the values of \(\alpha , p\) and \(q\).
OCR MEI FP1 2012 June Q4
4 marks Moderate -0.8
4 Solve the inequality \(\frac { 3 } { x - 4 } > 1\).
OCR MEI FP1 2012 June Q5
7 marks Standard +0.3
5
  1. Show that \(\frac { 1 } { 2 r + 1 } - \frac { 1 } { 2 r + 3 } \equiv \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\).
  2. Use the method of differences to find \(\sum _ { r = 1 } ^ { 30 } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\), expressing your answer as a fraction.
OCR MEI FP1 2012 June Q6
7 marks Standard +0.3
6 A sequence is defined by \(a _ { 1 } = 1\) and \(a _ { k + 1 } = 3 \left( a _ { k } + 1 \right)\).
  1. Calculate the value of the third term, \(a _ { 3 }\).
  2. Prove by induction that \(a _ { n } = \frac { 5 \times 3 ^ { n - 1 } - 3 } { 2 }\).
OCR MEI FP1 2012 June Q7
14 marks Standard +0.8
7 A curve has equation \(y = \frac { x ^ { 2 } - 25 } { ( x - 3 ) ( x + 4 ) ( 3 x + 2 ) }\).
  1. Write down the coordinates of the points where the curve crosses the axes.
  2. Write down the equations of the asymptotes.
  3. Determine how the curve approaches the horizontal asymptote for large positive values of \(x\), and for large negative values of \(x\).
  4. Sketch the curve.
OCR MEI FP1 2012 June Q8
10 marks Standard +0.3
8
  1. Verify that \(1 + 3 \mathrm { j }\) is a root of the equation \(3 z ^ { 3 } - 2 z ^ { 2 } + 22 z + 40 = 0\), showing your working.
  2. Explain why the equation must have exactly one real root.
  3. Find the other roots of the equation.
OCR MEI FP1 2012 June Q9
12 marks Standard +0.3
9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } - 3 & - 4 & 1 \\ 2 & 1 & k \\ 7 & - 1 & - 1 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r c } - 4 & - 5 & 11 \\ - 19 & - 4 & - 7 \\ - 9 & - 31 & 2 - k \end{array} \right)\) and \(\mathbf { A B } = \left( \begin{array} { c c c } 79 & 0 & - 3 - k \\ - 9 k - 27 & - 31 k - 14 & q \\ p & 0 & 82 + k \end{array} \right)\) where \(p\) and \(q\) are to be determined.
  1. Show that \(p = 0\) and \(q = 15 + 2 k - k ^ { 2 }\). It is now given that \(k = - 3\).
  2. Find \(\mathbf { A B }\) and hence write down the inverse matrix \(\mathbf { A } ^ { - 1 }\).
  3. Use a matrix method to find the values of \(x , y\) and \(z\) that satisfy the equation \(\mathbf { A } \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } 14 \\ - 23 \\ 9 \end{array} \right)\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR MEI FP1 2013 June Q1
5 marks Easy -1.2
1 Find the values of \(A , B , C\) and \(D\) in the identity \(2 x \left( x ^ { 2 } - 5 \right) \equiv ( x - 2 ) \left( A x ^ { 2 } + B x + C \right) + D\).
OCR MEI FP1 2013 June Q2
6 marks Moderate -0.5
2 You are given that \(z = \frac { 3 } { 2 }\) is a root of the cubic equation \(2 z ^ { 3 } + 9 z ^ { 2 } + 2 z - 30 = 0\). Find the other two roots.
OCR MEI FP1 2013 June Q3
6 marks Standard +0.3
3 You are given that \(\mathbf { N } = \left( \begin{array} { r r r } - 9 & - 2 & - 4 \\ 3 & 2 & 2 \\ 5 & 1 & 2 \end{array} \right)\) and \(\mathbf { N } ^ { - 1 } = \left( \begin{array} { r r r } 1 & 0 & 2 \\ 2 & 1 & 3 \\ - \frac { 7 } { 2 } & p & - 6 \end{array} \right)\).
  1. Find the value of \(p\).
  2. Solve the equation \(\mathbf { N } \left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } - 39 \\ 5 \\ 22 \end{array} \right)\).
OCR MEI FP1 2013 June Q4
6 marks Moderate -0.8
4 The complex number \(z _ { 1 }\) is \(3 - 2 \mathrm { j }\) and the complex number \(z _ { 2 }\) has modulus 5 and argument \(\frac { \pi } { 4 }\).
  1. Express \(z _ { 2 }\) in the form \(a + b \mathrm { j }\), giving \(a\) and \(b\) in exact form.
  2. Represent \(z _ { 1 } , z _ { 2 } , z _ { 1 } + z _ { 2 }\) and \(z _ { 1 } - z _ { 2 }\) on a single Argand diagram.
OCR MEI FP1 2013 June Q5
6 marks Standard +0.3
5 You are given that \(\frac { 4 } { ( 4 n - 3 ) ( 4 n + 1 ) } \equiv \frac { 1 } { 4 n - 3 } - \frac { 1 } { 4 n + 1 }\). Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 4 r - 3 ) ( 4 r + 1 ) } = \frac { n } { 4 n + 1 }$$
OCR MEI FP1 2013 June Q6
7 marks Standard +0.8
6 The cubic equation \(x ^ { 3 } - 5 x ^ { 2 } + 3 x - 6 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Find a cubic equation with roots \(\frac { \alpha } { 3 } + 1 , \frac { \beta } { 3 } + 1\) and \(\frac { \gamma } { 3 } + 1\), simplifying your answer as far as possible.
OCR MEI FP1 2013 June Q7
12 marks Challenging +1.2
7 Fig. 7 shows an incomplete sketch of \(y = \frac { c x ^ { 2 } } { ( b x - 1 ) ( x + a ) }\) where \(a , b\) and \(c\) are integers. The asymptotes of the curve are also shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{597abea9-6d00-416e-9203-d5bce9bd1af1-3_928_996_493_535} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Determine the values of \(a , b\) and \(c\). Use these values of \(a , b\) and \(c\) throughout the rest of the question.
  2. Determine how the curve approaches the horizontal asymptote for large positive values of \(x\), and for large negative values of \(x\), justifying your answer. On the copy of Fig. 7, sketch the rest of the curve.
  3. Find the \(x\) coordinates of the points on the curve where \(y = 1\). Write down the solution to the inequality \(\frac { c x ^ { 2 } } { ( b x - 1 ) ( x + a ) } < 1\).
OCR MEI FP1 2013 June Q8
12 marks Standard +0.3
8
  1. Use standard series formulae to show that $$\sum _ { r = 1 } ^ { n } [ r ( r - 1 ) - 1 ] = \frac { 1 } { 3 } n ( n + 2 ) ( n - 2 )$$
  2. Prove (*) by mathematical induction.
OCR MEI FP1 2013 June Q9
12 marks Standard +0.3
9
  1. Describe fully the transformation Q , represented by the matrix \(\mathbf { Q }\), where \(\mathbf { Q } = \left( \begin{array} { r l } 0 & 1 \\ - 1 & 0 \end{array} \right)\). The transformation M is represented by the matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { r r } 0 & - 1 \\ 0 & 1 \end{array} \right)\).
  2. M maps all points on the line \(y = 2\) onto a single point, P. Find the coordinates of P.
  3. M maps all points on the plane onto a single line, \(l\). Find the equation of \(l\).
  4. M maps all points on the line \(n\) onto the point ( - 6 , 6). Find the equation of \(n\).
  5. Show that \(\mathbf { M }\) is singular. Relate this to the transformation it represents.
  6. R is the composite transformation M followed by Q . R maps all points on the plane onto the line \(q\). Find the equation of \(q\).
OCR MEI FP1 2014 June Q1
5 marks Moderate -0.8
1 Use standard series formulae to find \(\sum _ { r = 1 } ^ { n } r ( r - 2 )\), factorising your answer as far as possible.
OCR MEI FP1 2014 June Q2
5 marks Moderate -0.5
2 Fig. 2 shows the unit square, OABC , and its image, \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\), after undergoing a transformation. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3df020b0-fb7b-454b-b354-36cc2b8df5f6-2_595_739_571_664} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Write down the matrix \(\mathbf { T }\) representing this transformation. The quadrilateral \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) is reflected in the \(x\)-axis to give a new quadrilateral, \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\).
  2. Write down the matrix representing reflection in the \(x\)-axis.
  3. Find the single matrix that will transform OABC onto \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\).
OCR MEI FP1 2014 June Q3
7 marks Standard +0.3
3 You are given that \(z = 2 + 3 \mathrm { j }\) is a root of the quartic equation \(z ^ { 4 } - 5 z ^ { 3 } + 15 z ^ { 2 } - 5 z - 26 = 0\). Find the other roots.
OCR MEI FP1 2014 June Q4
5 marks Standard +0.3
4 Use the identity \(\frac { 1 } { 2 r + 3 } - \frac { 1 } { 2 r + 5 } \equiv \frac { 2 } { ( 2 r + 3 ) ( 2 r + 5 ) }\) and the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 3 ) ( 2 r + 5 ) }\), expressing your answer as a single fraction.
OCR MEI FP1 2014 June Q5
7 marks Standard +0.8
5 The roots of the cubic equation \(3 x ^ { 3 } - 9 x ^ { 2 } + x - 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(3 \alpha - 1,3 \beta - 1\) and \(3 \gamma - 1\), expressing your answer in a form with integer coefficients.
OCR MEI FP1 2014 June Q6
7 marks Standard +0.3
6 Prove by induction that \(\frac { 1 } { 1 \times 3 } + \frac { 1 } { 3 \times 5 } + \frac { 1 } { 5 \times 7 } + \ldots + \frac { 1 } { ( 2 n - 1 ) ( 2 n + 1 ) } = \frac { n } { 2 n + 1 }\).