Questions — OCR MEI FP1 (190 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 FP1 2015 June Q4
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
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 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
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
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
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
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
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
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
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
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 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
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
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
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}