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 2005 January Q1
1 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 2 & 3
- 2 & 1 \end{array} \right)\).
Find the inverse of \(\mathbf { M }\).
The transformation associated with \(\mathbf { M }\) is applied to a figure of area 2 square units. What is the area of the transformed figure?
OCR MEI FP1 2005 January Q2
2
  1. Show that \(\frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }\).
  2. Hence use the method of differences to find the sum of the series $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$
OCR MEI FP1 2005 January Q3
3
  1. Solve the equation \(\frac { 1 } { x + 2 } = 3 x + 4\).
  2. Solve the inequality \(\frac { 1 } { x + 2 } \leqslant 3 x + 4\).
OCR MEI FP1 2005 January Q4
4 Find \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 2 )\), giving your answer in a factorised form.
OCR MEI FP1 2005 January Q5
5 The roots of the cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + x - 3 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation whose roots are \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\), simplifying your answer as far as you can.
OCR MEI FP1 2005 January Q6
6 Prove by induction that \(\sum _ { r = 1 } ^ { n } r 2 ^ { r - 1 } = 1 + ( n - 1 ) 2 ^ { n }\).
OCR MEI FP1 2005 January Q7
7 A curve has equation \(y = \frac { ( 2 x - 3 ) ( x + 1 ) } { ( x + 4 ) ( x - 2 ) }\).
  1. Write down the values of \(x\) for which \(y = 0\).
  2. Write down the equations of the three asymptotes.
  3. Determine whether the curve approaches the horizontal asymptote from above or from below for
    (A) large positive values of \(x\),
    (B) large negative values of \(x\).
  4. Sketch the curve.
  5. Solve the inequality \(\frac { ( 2 x - 3 ) ( x + 1 ) } { ( x + 4 ) ( x - 2 ) } \leqslant 2\).
OCR MEI FP1 2005 January Q8
8 Two complex numbers are given by \(\alpha = 2 - \mathrm { j }\) and \(\beta = - 1 + 2 \mathrm { j }\).
  1. Find \(\alpha + \beta , \alpha \beta\) and \(\frac { \alpha } { \beta }\) in the form \(a + b \mathrm { j }\), showing your working.
  2. Find the modulus of \(\alpha\), leaving your answer in surd form. Find also the argument of \(\alpha\).
  3. Sketch the locus \(| z - \alpha | = 2\) on an Argand diagram.
  4. On a separate Argand diagram, sketch the locus \(\arg ( z - \beta ) = \frac { 1 } { 4 } \pi\).
OCR MEI FP1 2005 January Q9
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 ( \(\mathbf { v }\) ) is equivalent to a single reflection. What is the equation of the mirror line of this reflection?
OCR MEI FP1 2006 January Q1
1 You are given that \(\mathbf { A } = \left( \begin{array} { l l } 4 & 3
1 & 2 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 2 & - 3
1 & 4 \end{array} \right) , \mathbf { C } = \left( \begin{array} { r r } 1 & - 1
0 & 2
0 & 1 \end{array} \right)\).
  1. Calculate, where possible, \(2 \mathbf { B } , \mathbf { A } + \mathbf { C } , \mathbf { C A }\) and \(\mathbf { A } - \mathbf { B }\).
  2. Show that matrix multiplication is not commutative.
OCR MEI FP1 2006 January Q2
2
  1. Given that \(z = a + b \mathrm { j }\), express \(| z |\) and \(z ^ { * }\) in terms of \(a\) and \(b\).
  2. Prove that \(z z ^ { * } - | z | ^ { 2 } = 0\).
OCR MEI FP1 2006 January Q3
3 Find \(\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r - 1 )\), expressing your answer in a fully factorised form.
OCR MEI FP1 2006 January Q4
4 The matrix equation \(\left( \begin{array} { r r } 6 & - 2
- 3 & 1 \end{array} \right) \binom { x } { y } = \binom { a } { b }\) represents two simultaneous linear equations in \(x\) and \(y\).
  1. Write down the two equations.
  2. Evaluate the determinant of \(\left( \begin{array} { r r } 6 & - 2
    - 3 & 1 \end{array} \right)\). What does this value tell you about the solution of the equations in part (i)?
OCR MEI FP1 2006 January Q5
5 The cubic equation \(x ^ { 3 } + 3 x ^ { 2 } - 7 x + 1 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
  2. Find the cubic equation with roots \(2 \alpha , 2 \beta\) and \(2 \gamma\), simplifying your answer as far as possible.
OCR MEI FP1 2006 January Q6
6 Prove by induction that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }\).
OCR MEI FP1 2006 January Q7
7 A curve has equation \(y = \frac { 3 + x ^ { 2 } } { 4 - x ^ { 2 } }\).
  1. Show that \(y\) can never be zero.
  2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
  3. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your description.
  4. Sketch the curve.
  5. Solve the inequality \(\frac { 3 + x ^ { 2 } } { 4 - x ^ { 2 } } \leqslant - 2\).
OCR MEI FP1 2006 January Q8
8 You are given that the complex number \(\alpha = 1 + \mathrm { j }\) satisfies the equation \(z ^ { 3 } + 3 z ^ { 2 } + p z + q = 0\), where \(p\) and \(q\) are real constants.
  1. Find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\) in the form \(a + b \mathrm { j }\). Hence show that \(p = - 8\) and \(q = 10\).
  2. Find the other two roots of the equation.
  3. Represent the three roots on an Argand diagram.
OCR MEI FP1 2006 January Q9
9 A transformation T acts on all points in the plane. The image of a general point P is denoted by \(\mathrm { P } ^ { \prime }\). \(\mathrm { P } ^ { \prime }\) always lies on the line \(y = 2 x\) and has the same \(y\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4048c232-6a4e-4baa-9262-93428f375203-4_821_837_475_612} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Write down the image of the point \(( 10,50 )\) under transformation T .
  2. P has coordinates \(( x , y )\). State the coordinates of \(\mathrm { P } ^ { \prime }\).
  3. All points on a particular line \(l\) are mapped onto the point \(( 3,6 )\). Write down the equation of the line \(l\).
  4. In part (iii), the whole of the line \(l\) was mapped by T onto a single point. There are an infinite number of lines which have this property under T. Describe these lines.
  5. For a different set of lines, the transformation T has the same effect as translation parallel to the \(x\)-axis. Describe this set of lines.
  6. Find the \(2 \times 2\) matrix which represents the transformation.
  7. Show that this matrix is singular. Relate this result to the transformation.
OCR MEI FP1 2007 January Q1
1 Is the following statement true or false? Justify your answer. $$x ^ { 2 } = 4 \text { if and only if } x = 2$$
OCR MEI FP1 2007 January Q2
2
  1. Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 7 = 0\), simplifying your answers as far as possible.
  2. Represent these roots on an Argand diagram.
OCR MEI FP1 2007 January Q3
3 The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 3 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively under the transformation represented by the matrix \(\mathbf { M } = \left( \begin{array} { l l } 2 & 0
0 & \frac { 1 } { 2 } \end{array} \right)\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4a339746-195f-477a-952e-02fbdfd9cce5-2_446_444_1046_808} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Draw a diagram showing the image of the triangle after the transformation, labelling the image of each point clearly.
  2. Describe fully the transformation represented by the matrix \(\mathbf { M }\).
OCR MEI FP1 2007 January Q4
4 Use standard series formulae to find \(\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } + 1 \right)\), factorising your answer as far as possible.
OCR MEI FP1 2007 January Q5
5 The roots of the cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + x - 4 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation whose roots are \(2 \alpha + 1,2 \beta + 1\) and \(2 \gamma + 1\), expressing your answer in a form with integer coefficients.
OCR MEI FP1 2007 January Q6
6 Prove by induction that \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\).
OCR MEI FP1 2007 January Q7
7 A curve has equation \(y = \frac { 5 } { ( x + 2 ) ( 4 - x ) }\).
  1. Write down the value of \(y\) when \(x = 0\).
  2. Write down the equations of the three asymptotes.
  3. Sketch the curve.
  4. Find the values of \(x\) for which \(\frac { 5 } { ( x + 2 ) ( 4 - x ) } = 1\) and hence solve the inequality $$\frac { 5 } { ( x + 2 ) ( 4 - x ) } < 1 .$$