Questions — OCR MEI (4333 questions)

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OCR MEI FP1 2009 June Q2
5 marks Moderate -0.8
2 Show that \(z = 3\) is a root of the cubic equation \(z ^ { 3 } + z ^ { 2 } - 7 z - 15 = 0\) and find the other roots.
OCR MEI FP1 2009 June Q3
7 marks Standard +0.3
3
  1. Sketch the graph of \(y = \frac { 2 } { x + 4 }\).
  2. Solve the inequality $$\frac { 2 } { x + 4 } \leqslant x + 3$$ showing your working clearly.
OCR MEI FP1 2009 June Q4
6 marks Standard +0.3
4 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } + p x + q = 0\) are \(2 w , - 6 w\) and \(3 w\). Find the values of the roots and the values of \(p\) and \(q\).
OCR MEI FP1 2009 June Q5
6 marks Standard +0.3
5
  1. Show that \(\frac { 1 } { 5 r - 2 } - \frac { 1 } { 5 r + 3 } \equiv \frac { 5 } { ( 5 r - 2 ) ( 5 r + 3 ) }\) for all integers \(r\).
  2. Hence use the method of differences to show that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 r - 2 ) ( 5 r + 3 ) } = \frac { n } { 3 ( 5 n + 3 ) }\).
OCR MEI FP1 2009 June Q6
7 marks Standard +0.3
6 Prove by induction that \(3 + 10 + 17 + \ldots + ( 7 n - 4 ) = \frac { 1 } { 2 } n ( 7 n - 1 )\) for all positive integers \(n\). Section B (36 marks)
OCR MEI FP1 2009 June Q7
12 marks Standard +0.3
7 A curve has equation \(y = \frac { ( x + 2 ) ( 3 x - 5 ) } { ( 2 x + 1 ) ( x - 1 ) }\).
  1. Write down the coordinates of the points where the curve crosses the axes.
  2. Write down the equations of the three asymptotes.
  3. Determine whether the curve approaches the horizontal asymptote from above or below for
    (A) large positive values of \(x\),
    (B) large negative values of \(x\).
  4. Sketch the curve.
OCR MEI FP1 2009 June Q8
12 marks Standard +0.3
8 Fig. 8 shows an Argand diagram. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-3_421_586_1105_778} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Write down the equation of the locus represented by the perimeter of the circle in the Argand diagram.
  2. Write down the equation of the locus represented by the half-line \(\ell\) in the Argand diagram.
  3. Express the complex number represented by the point P in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
  4. Use inequalities to describe the set of points that fall within the shaded region (excluding its boundaries) in the Argand diagram.
OCR MEI FP1 2009 June Q9
12 marks Moderate -0.8
9 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right) , \mathbf { N } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).
  1. The matrix products \(\mathbf { Q } ( \mathbf { M N } )\) and \(( \mathbf { Q M } ) \mathbf { N }\) are identical. What property of matrix multiplication does this illustrate? Find QMN. \(\mathbf { M } , \mathbf { N }\) and \(\mathbf { Q }\) represent the transformations \(\mathrm { M } , \mathrm { N }\) and Q respectively.
  2. Describe the transformations \(\mathrm { M } , \mathrm { N }\) and Q . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-4_668_908_788_621} \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{figure}
  3. The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 9 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively by the composite transformation N followed by M followed by Q . Draw a diagram showing the image of the triangle after this composite transformation, labelling the image of each point clearly.
OCR MEI FP1 2010 June Q1
4 marks Moderate -0.8
1 Find the values of \(A , B\) and \(C\) in the identity \(4 x ^ { 2 } - 16 x + C \equiv A ( x + B ) ^ { 2 } + 2\).
OCR MEI FP1 2010 June Q2
6 marks Moderate -0.3
2 You are given that \(\mathbf { M } = \left( \begin{array} { r r } 2 & - 5 \\ 3 & 7 \end{array} \right)\). \(\mathbf { M } \binom { x } { y } = \binom { 9 } { - 1 }\) represents two simultaneous equations.
  1. Write down these two equations.
  2. Find \(\mathbf { M } ^ { - 1 }\) and use it to solve the equations.
OCR MEI FP1 2010 June Q3
6 marks Moderate -0.3
3 The cubic equation \(2 z ^ { 3 } - z ^ { 2 } + 4 z + k = 0\), where \(k\) is real, has a root \(z = 1 + 2 \mathrm { j }\).
Write down the other complex root. Hence find the real root and the value of \(k\).
OCR MEI FP1 2010 June Q4
6 marks Standard +0.3
4 The roots of the cubic equation \(x ^ { 3 } - 2 x ^ { 2 } - 8 x + 11 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).
OCR MEI FP1 2010 June Q5
6 marks Standard +0.3
5 Use the result \(\frac { 1 } { 5 r - 1 } - \frac { 1 } { 5 r + 4 } \equiv \frac { 5 } { ( 5 r - 1 ) ( 5 r + 4 ) }\) and the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 r - 1 ) ( 5 r + 4 ) }$$ simplifying your answer.
OCR MEI FP1 2010 June Q6
8 marks Standard +0.8
6 A sequence is defined by \(u _ { 1 } = 2\) and \(u _ { n + 1 } = \frac { u _ { n } } { 1 + u _ { n } }\).
  1. Calculate \(u _ { 3 }\).
  2. Prove by induction that \(u _ { n } = \frac { 2 } { 2 n - 1 }\). Section B (36 marks)
OCR MEI FP1 2010 June Q7
12 marks Standard +0.8
7 Fig. 7 shows an incomplete sketch of \(y = \frac { ( 2 x - 1 ) ( x + 3 ) } { ( x - 3 ) ( x - 2 ) }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e449d411-aaa9-4167-aa9c-c28d31446d52-3_786_1376_450_386} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the coordinates of the points where the curve cuts the axes.
  2. Write down the equations of the three asymptotes.
  3. Determine whether the curve approaches the horizontal asymptote from above or below for large positive values of \(x\), justifying your answer. Copy and complete the sketch.
  4. Solve the inequality \(\frac { ( 2 x - 1 ) ( x + 3 ) } { ( x - 3 ) ( x - 2 ) } < 2\).
OCR MEI FP1 2010 June Q8
10 marks Moderate -0.3
8 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = \sqrt { 3 } + \mathrm { j }\) and \(\beta = 3 \mathrm { j }\).
  1. Find the modulus and argument of \(\alpha\) and \(\beta\).
  2. Find \(\alpha \beta\) and \(\frac { \beta } { \alpha }\), giving your answers in the form \(a + b \mathrm { j }\), showing your working.
  3. Plot \(\alpha , \beta , \alpha \beta\) and \(\frac { \beta } { \alpha }\) on a single Argand diagram.
OCR MEI FP1 2010 June Q9
14 marks Moderate -0.3
9 The matrices \(\mathbf { P } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\) represent transformations \(P\) and \(Q\) respectively.
  1. Describe fully the transformations P and Q . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e449d411-aaa9-4167-aa9c-c28d31446d52-4_625_849_470_648} \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{figure} Fig. 9 shows triangle T with vertices \(\mathrm { A } ( 2,0 ) , \mathrm { B } ( 1,2 )\) and \(\mathrm { C } ( 3,1 )\).
    Triangle T is transformed first by transformation P , then by transformation Q .
  2. Find the single matrix that represents this composite transformation.
  3. This composite transformation maps triangle T onto triangle \(\mathrm { T } ^ { \prime }\), with vertices \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). Calculate the coordinates of \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). T' is reflected in the line \(y = - x\) to give a new triangle, T".
  4. Find the matrix \(\mathbf { R }\) that represents reflection in the line \(y = - x\).
  5. A single transformation maps \(\mathrm { T } ^ { \prime \prime }\) onto the original triangle, T . Find the matrix representing this transformation.
OCR MEI FP1 2011 June Q1
5 marks Moderate -0.8
1
  1. Write down the matrix for a rotation of \(90 ^ { \circ }\) anticlockwise about the origin.
  2. Write down the matrix for a reflection in the line \(y = x\).
  3. Find the matrix for the composite transformation of rotation of \(90 ^ { \circ }\) anticlockwise about the origin, followed by a reflection in the line \(y = x\).
  4. What single transformation is equivalent to this composite transformation?
OCR MEI FP1 2011 June Q2
8 marks Moderate -0.8
2 You are given that \(z = 3 - 2 \mathrm { j }\) and \(w = - 4 + \mathrm { j }\).
  1. Express \(\frac { z + w } { w }\) in the form \(a + b \mathrm { j }\).
  2. Express \(w\) in modulus-argument form.
  3. Show \(w\) on an Argand diagram, indicating its modulus and argument.
OCR MEI FP1 2011 June Q3
5 marks Standard +0.8
3 The equation \(x ^ { 3 } + p x ^ { 2 } + q x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\), where $$\begin{gathered} \alpha + \beta + \gamma = 4 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 6 \end{gathered}$$ Find \(p\) and \(q\).
OCR MEI FP1 2011 June Q4
6 marks Standard +0.3
4 Solve the inequality \(\frac { 5 x } { x ^ { 2 } + 4 } < x\).
OCR MEI FP1 2011 June Q5
5 marks Standard +0.3
5 Given that \(\frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) } \equiv \frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 }\), find \(\sum _ { r = 1 } ^ { 20 } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) }\), giving your answer as an exact fraction.
OCR MEI FP1 2011 June Q6
7 marks Standard +0.3
6 Prove by induction that \(1 + 8 + 27 + \ldots + n ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }\). Section B (36 marks)
OCR MEI FP1 2011 June Q7
12 marks Standard +0.3
7 A curve has equation \(y = \frac { ( x + 9 ) ( 3 x - 8 ) } { x ^ { 2 } - 4 }\).
  1. Write down the coordinates of the points where the curve crosses the axes.
  2. Write down the equations of the three asymptotes.
  3. Determine whether the curve approaches the horizontal asymptote from above or below for
    (A) large positive values of \(x\),
    (B) large negative values of \(x\).
  4. Sketch the curve.
OCR MEI FP1 2011 June Q8
11 marks Standard +0.3
8 A polynomial \(\mathrm { P } ( z )\) has real coefficients. Two of the roots of \(\mathrm { P } ( z ) = 0\) are \(2 - \mathrm { j }\) and \(- 1 + 2 \mathrm { j }\).
  1. Explain why \(\mathrm { P } ( z )\) cannot be a cubic. You are given that \(\mathrm { P } ( z )\) is a quartic.
  2. Write down the other roots of \(\mathrm { P } ( z ) = 0\) and hence find \(\mathrm { P } ( z )\) in the form \(z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d\).
  3. Show the roots of \(\mathrm { P } ( z ) = 0\) on an Argand diagram and give, in terms of \(z\), the equation of the circle they lie on.