Questions — OCR MEI FP1 (195 questions)

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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 \(\frac { 1 } { 1 \times 3 } + \frac { 1 } { 3 \times 5 } + \frac { 1 } { 5 \times 7 } + \ldots + \frac { 1 } { ( 2 n - 1 ) ( 2 n + 1 ) } = \frac { n } { 2 n + 1 }\).
OCR MEI FP1 2014 June Q7
12 marks Standard +0.8
7 A curve has equation \(y = \frac { x ^ { 2 } - 5 } { ( x + 3 ) ( x - 2 ) ( a x - 1 ) }\), where \(a\) is a constant.
  1. Find the coordinates of the points where the curve crosses the \(x\)-axis and the \(y\)-axis.
  2. You are given that the curve has a vertical asymptote at \(x = \frac { 1 } { 2 }\). Write down the value of \(a\) and the equations of the other asymptotes.
  3. Sketch the curve.
  4. Find the set of values of \(x\) for which \(y > 0\).
OCR MEI FP1 2014 June Q8
12 marks Standard +0.8
8 You are given the complex number \(w = 2 + 2 \sqrt { 3 } \mathrm { j }\).
  1. Express \(w\) in modulus-argument form.
  2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$- \frac { \pi } { 2 } \leqslant \arg z \leqslant \frac { \pi } { 3 } \text { and } | z | \leqslant 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(| z - w |\).
OCR MEI FP1 2014 June Q9
12 marks Standard +0.3
9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 1 & 3 & - 1 \\ - 1 & \alpha & - 1 \\ - 2 & - 1 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c c } 3 \alpha - 1 & - 8 & \alpha - 3 \\ 5 & 1 & 2 \\ 2 \alpha + 1 & - 5 & \alpha + 3 \end{array} \right)\) and \(\mathbf { A B } = \left( \begin{array} { c c c } \gamma & 0 & 0 \\ \beta & \gamma & 0 \\ 0 & 0 & \gamma \end{array} \right)\).
  1. Show that \(\beta = 0\).
  2. Find \(\gamma\) in terms of \(\alpha\).
  3. Write down \(\mathbf { A } ^ { - 1 }\) for the case when \(\alpha = 2\). State the value of \(\alpha\) for which \(\mathbf { A } ^ { - 1 }\) does not exist.
  4. Use your answer to part (iii) to solve the following simultaneous equations. $$\begin{aligned} x + 3 y - z & = 25 \\ - x + 2 y - z & = 11 \\ - 2 x - y + 3 z & = - 23 \end{aligned}$$
OCR MEI FP1 2015 June Q1
6 marks Moderate -0.8
1 Given that \(\mathbf { M } \binom { x } { y } = \binom { 1 } { 3 }\), where \(\mathbf { M } = \left( \begin{array} { r r } 4 & - 3 \\ 8 & 21 \end{array} \right)\), find \(x\) and \(y\).
OCR MEI FP1 2015 June Q2
5 marks Moderate -0.8
2 Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 13 = 0\).
Find the modulus and argument of each root.
OCR MEI FP1 2015 June Q3
6 marks Moderate -0.3
3 The equation \(2 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } + \mathrm { r } = 0\) has a root at \(x = 4\). The sum of the roots is 6 and the product of the roots is - 10 . Find \(p , q\) and \(r\).
OCR MEI FP1 2015 June Q4
6 marks Standard +0.3
4 Indicate, on a single Argand diagram
  1. the set of points for which \(\arg ( z - ( - 1 - \mathrm { j } ) ) = \frac { \pi } { 4 }\),
  2. the set of points for which \(| z - ( 1 + 2 j ) | = 2\),
  3. the set of points for which \(| z - ( 1 + 2 j ) | \geqslant 2\) and \(0 \leqslant \arg ( z - ( - 1 - j ) ) \leqslant \frac { \pi } { 4 }\).
OCR MEI FP1 2015 June Q5
7 marks Moderate -0.3
5
  1. Show that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( 2 \mathrm { r } - 1 ) = \mathrm { n } ^ { 2 }\).
  2. Show that \(\frac { \sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( 2 \mathrm { r } - 1 ) } { \sum _ { \mathrm { r } = \mathrm { n } + 1 } ^ { 2 \mathrm { n } } ( 2 \mathrm { r } - 1 ) } = \mathrm { k }\), where \(k\) is a constant to be determined.
OCR MEI FP1 2015 June Q6
6 marks Standard +0.3
6 A sequence is defined by \(u _ { 1 } = 3\) and \(u _ { n + 1 } = 3 u _ { n } - 5\). Prove by induction that \(u _ { n } = \frac { 3 ^ { n - 1 } + 5 } { 2 }\). Section B (36 marks)
OCR MEI FP1 2015 June Q7
12 marks Standard +0.8
7 A curve has equation \(\mathrm { y } = \frac { ( 3 \mathrm { x } + 2 ) ( \mathrm { x } - 3 ) } { ( \mathrm { x } - 2 ) ( \mathrm { x } + 1 ) }\).
  1. Write down the equations of the three asymptotes and the coordinates of the points where the curve crosses the axes.
  2. Sketch the curve, justifying how it approaches the horizontal asymptote.
  3. Find the set of values of \(x\) for which \(y \geqslant 3\).
OCR MEI FP1 2015 June Q8
12 marks Standard +0.3
8 The complex number \(5 + 4 \mathrm { j }\) is denoted by \(\alpha\).
  1. Find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\), showing your working.
  2. The real numbers \(q\) and \(r\) are such that \(\alpha ^ { 3 } + \mathrm { q } \alpha ^ { 2 } + 11 \alpha + \mathrm { r } = 0\). Find \(q\) and \(r\). Let \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + \mathrm { qz } ^ { 2 } + 11 \mathrm { z } + \mathrm { r }\), where \(q\) and \(r\) are as in part (ii).
  3. Solve the equation \(\mathrm { f } ( z ) = 0\).
  4. Solve the equation \(z ^ { 4 } + q z ^ { 3 } + 11 z ^ { 2 } + r z = z ^ { 3 } + q z ^ { 2 } + 11 z + r\).
OCR MEI FP1 2015 June Q9
12 marks Moderate -0.3
9 The triangle ABC has vertices at \(\mathrm { A } ( 0,0 ) , \mathrm { B } ( 0,2 )\) and \(\mathrm { C } ( 4,1 )\). The matrix \(\left( \begin{array} { r r } 1 & - 2 \\ 3 & 0 \end{array} \right)\) represents a transformation T .
  1. The transformation \(T\) maps triangle \(A B C\) onto triangle \(A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Find the coordinates of \(A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\). Triangle \(A ^ { \prime } B ^ { \prime } C ^ { \prime }\) is now mapped onto triangle \(A ^ { \prime \prime } B ^ { \prime \prime } C ^ { \prime \prime }\) using the matrix \(\mathbf { M } = \left( \begin{array} { l l } 4 & 0 \\ 0 & 2 \end{array} \right)\).
  2. Describe fully the transformation represented by \(\mathbf { M }\).
  3. Triangle \(\mathrm { A } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\) is now mapped back onto ABC by a single transformation. Find the matrix representing this transformation.
  4. Calculate the area of \(A ^ { \prime \prime } B ^ { \prime \prime } C ^ { \prime \prime }\).
OCR MEI FP1 2016 June Q1
4 marks Moderate -0.3
1 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } 8 & - 2 \\ p & 1 \end{array} \right)\), where \(p \neq - 4\).
  1. Find the inverse of \(\mathbf { M }\) in terms of \(p\).
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{578345cb-e7a1-41fd-abf8-a977912965e8-2_1086_885_584_587} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} The triangle shown in Fig. 1 undergoes the transformation represented by the matrix \(\left( \begin{array} { c c } 8 & - 2 \\ 3 & 1 \end{array} \right)\). Find the area of the image of the triangle following this transformation.
OCR MEI FP1 2016 June Q2
6 marks Standard +0.3
2 The complex number \(z _ { 1 }\) is \(2 - 5 \mathrm { j }\) and the complex number \(z _ { 2 }\) is \(( a - 1 ) + ( 2 - b ) \mathrm { j }\), where \(a\) and \(b\) are real.
  1. Express \(\frac { z _ { 1 } { } ^ { * } } { z _ { 1 } }\) in the form \(x + y \mathrm { j }\), giving \(x\) and \(y\) in exact form. You must show clearly how you obtain your
    answer.
  2. Given that \(\frac { z _ { 1 } { } ^ { * } } { z _ { 1 } } = z _ { 2 }\), find the exact values of \(a\) and \(b\).
OCR MEI FP1 2016 June Q3
6 marks Standard +0.3
3 You are given that \(\mathbf { A } = \left( \begin{array} { c c c } \lambda & 6 & - 4 \\ 2 & 5 & - 1 \\ - 1 & 4 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c c } - 19 & 34 & - 14 \\ 5 & - 5 & 5 \\ - 13 & 18 & - 3 \end{array} \right)\) and \(\mathbf { A B } = \mu \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity
matrix.
  1. Find the values of \(\lambda\) and \(\mu\).
  2. Hence find \(\mathbf { B } ^ { - 1 }\).
OCR MEI FP1 2016 June Q4
6 marks Standard +0.3
4
  1. Use standard series to show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 2 r - p ) = \frac { 1 } { 6 } n ( n + 1 ) \left( 3 n ^ { 2 } + ( 3 - 2 p ) n - p \right) ,$$ where \(p\) is a constant.
  2. Given that the coefficients of \(n ^ { 3 }\) and \(n ^ { 4 }\) in the expression for \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 2 r - p )\) are equal, find the value of \(p\).
OCR MEI FP1 2016 June Q5
8 marks Standard +0.3
5 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z + 3 - 4 \mathrm { j } | = 5\) and arg \(( z + 3 - 6 \mathrm { j } ) = \frac { 1 } { 2 } \pi\) respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Write down the complex number represented by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
  3. Indicate, by shading on your sketch, the region satisfying $$| z + 3 - 4 \mathrm { j } | \geqslant 5 \text { and } \frac { 1 } { 2 } \pi \leqslant \arg ( z + 3 - 6 \mathrm { j } ) \leqslant \frac { 3 } { 4 } \pi .$$
OCR MEI FP1 2016 June Q6
6 marks Standard +0.8
6 A sequence is defined by \(u _ { 1 } = 8\) and \(u _ { n + 1 } = 3 u _ { n } + 2 n + 5\). Prove by induction that \(u _ { n } = 4 \left( 3 ^ { n } \right) - n - 3\).
OCR MEI FP1 2016 June Q7
13 marks Standard +0.8
7 The function \(\mathrm { f } ( z ) = 2 z ^ { 4 } - 9 z ^ { 3 } + A z ^ { 2 } + B z - 26\) has real coefficients. The equation \(\mathrm { f } ( z ) = 0\) has two real roots, \(\alpha\) and \(\beta\), where \(\alpha > \beta\), and two complex roots, \(\gamma\) and \(\delta\), where \(\gamma = 3 + 2 \mathrm { j }\).
  1. Show that \(\alpha + \beta = - \frac { 3 } { 2 }\) and find the value of \(\alpha \beta\).
  2. Hence find the two real roots \(\alpha\) and \(\beta\).
  3. Find the values of \(A\) and \(B\).
  4. Write down the roots of the equation \(\mathrm { f } \left( \frac { w } { \mathrm { j } } \right) = 0\).
OCR MEI FP1 2016 June Q8
12 marks Standard +0.8
8 A curve has equation \(y = \frac { 3 x ^ { 2 } - 9 } { x ^ { 2 } + 3 x - 4 }\).
  1. Find the equations of the two vertical asymptotes and the one horizontal asymptote of this curve.
  2. State, with justification, how the curve approaches the horizontal asymptote for large positive and large negative values of \(x\).
  3. Sketch the curve.
  4. Solve the inequality \(\frac { 3 x ^ { 2 } - 9 } { x ^ { 2 } + 3 x - 4 } \geqslant 0\).
OCR MEI FP1 2016 June Q9
11 marks Challenging +1.2
9 You are given that \(\frac { 3 } { 4 ( 2 r - 1 ) } - \frac { 1 } { 2 r + 1 } + \frac { 1 } { 4 ( 2 r + 3 ) } = \frac { 2 r + 5 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) }\).
  1. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 2 r + 5 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { 2 } { 3 } - \frac { 3 } { 4 ( 2 n + 1 ) } + \frac { 1 } { 4 ( 2 n + 3 ) } .$$
  2. Write down the limit to which \(\sum _ { r = 1 } ^ { n } \frac { 2 r + 5 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) }\) converges as \(n\) tends to infinity.
  3. Find the sum of the finite series $$\frac { 45 } { 39 \times 41 \times 43 } + \frac { 47 } { 41 \times 43 \times 45 } + \frac { 49 } { 43 \times 45 \times 47 } + \ldots + \frac { 105 } { 99 \times 101 \times 103 } ,$$ giving your answer to 3 significant figures. \section*{END OF QUESTION PAPER}
OCR MEI FP1 Q9
Standard +0.3
9 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6 \\ 0.6 & - 0.8 \end{array} \right)\).
  1. Calculate \(\mathbf { M } ^ { 2 }\). You are now given that the matrix \(M\) represents a reflection in a line through the origin.
  2. Explain how your answer to part (i) relates to this information.
  3. By investigating the invariant points of the reflection, find the equation of the mirror line.
  4. Describe fully the transformation represented by the matrix \(\mathbf { P } = \left( \begin{array} { c c } 0.8 & - 0.6 \\ 0.6 & 0.8 \end{array} \right)\).
  5. A composite transformation is formed by the transformation represented by \(\mathbf { P }\) followed by the transformation represented by \(\mathbf { M }\). Find the single matrix that represents this composite transformation.
  6. The composite transformation described in part (v) is equivalent to a single reflection. What is the equation of the mirror line of this reflection? \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education} \section*{MEI STRUCTURED MATHEMATICS
    4755
    \textbackslash section*\{Further Concepts For Advanced Mathematics (FP1)\}}
    Tuesday 7 JUNE 2005Afternoon1 hour 30 minutes
    Additional materials:
    Answer booklet
    Graph paper
    MEI Examination Formulae and Tables (MF2)
    TIME 1 hour 30 minutes
    • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
    • Answer all the questions.
    • You are permitted to use a graphical calculator in this paper.
    • The number of marks is given in brackets [ ] at the end of each question or part question.
    • You are advised that an answer may receive no marks unless you show sufficient defail of the working to indicate that a correct method is being used.
    • Final answers should be given to a degree of accuracy appropriate to the context.
    • The total number of marks for this paper is 72.