Questions — OCR (4619 questions)

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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 FP1 2006 June Q2
4 marks Easy -1.2
2 The transformation S is a shear parallel to the \(x\)-axis in which the image of the point ( 1,1 ) is the point \(( 0,1 )\).
  1. Draw a diagram showing the image of the unit square under S .
  2. Write down the matrix that represents S .
OCR FP1 2006 June Q3
5 marks Easy -1.2
3 One root of the quadratic equation \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are real, is the complex number 2-3i.
  1. Write down the other root.
  2. Find the values of \(p\) and \(q\).
OCR FP1 2006 June Q4
5 marks Moderate -0.5
4 Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } + r ^ { 2 } \right) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + 1 )$$
OCR FP1 2006 June Q5
8 marks Moderate -0.8
5 The complex numbers \(3 - 2 \mathrm { i }\) and \(2 + \mathrm { i }\) are denoted by \(z\) and \(w\) respectively. Find, giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain these answers,
  1. \(2 z - 3 w\),
  2. \(( \mathrm { i } z ) ^ { 2 }\),
  3. \(\frac { z } { w }\).
OCR FP1 2006 June Q6
7 marks Moderate -0.5
6 In an Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z | = 2 \quad \text { and } \quad \arg z = \frac { 1 } { 3 } \pi$$ respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Hence find, in the form \(x + \mathrm { i } y\), the complex number representing the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
OCR FP1 2006 June Q7
8 marks Standard +0.3
7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\).
  1. Find \(\mathbf { A } ^ { 2 }\) and \(\mathbf { A } ^ { 3 }\).
  2. Hence suggest a suitable form for the matrix \(\mathbf { A } ^ { n }\).
  3. Use induction to prove that your answer to part (ii) is correct.
OCR FP1 2006 June Q8
10 marks Standard +0.3
8 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 2 \\ 1 & a & 0 \\ 1 & 2 & 1 \end{array} \right)\).
  1. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
  2. Hence find the values of \(a\) for which \(\mathbf { M }\) is singular.
  3. State, giving a brief reason in each case, whether the simultaneous equations $$\begin{aligned} a x + 4 y + 2 z & = 3 a \\ x + a y & = 1 \\ x + 2 y + z & = 3 \end{aligned}$$ have any solutions when
    (a) \(a = 3\),
    (b) \(a = 2\).
OCR FP1 2006 June Q9
10 marks Moderate -0.5
9
  1. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \left\{ ( r + 1 ) ^ { 3 } - r ^ { 3 } \right\} = ( n + 1 ) ^ { 3 } - 1$$
  2. Show that \(( r + 1 ) ^ { 3 } - r ^ { 3 } \equiv 3 r ^ { 2 } + 3 r + 1\).
  3. Use the results in parts (i) and (ii) and the standard result for \(\sum _ { r = 1 } ^ { n } r\) to show that $$3 \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 1 )$$
OCR FP1 2006 June Q10
11 marks Standard +0.3
10 The cubic equation \(x ^ { 3 } - 2 x ^ { 2 } + 3 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\). The cubic equation \(x ^ { 3 } + p x ^ { 2 } + 10 x + q = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).
  2. Find the value of \(p\).
  3. Find the value of \(q\).
OCR FP1 2007 June Q1
4 marks Easy -1.2
1 The complex number \(a + \mathrm { i } b\) is denoted by \(z\). Given that \(| z | = 4\) and \(\arg z = \frac { 1 } { 3 } \pi\), find \(a\) and \(b\).
OCR FP1 2007 June Q2
5 marks Standard +0.3
2 Prove by induction that, for \(n \geqslant 1 , \sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }\).
OCR FP1 2007 June Q3
6 marks Moderate -0.5
3 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 1 \right) = n ^ { 3 }$$
OCR FP1 2007 June Q4
6 marks Standard +0.3
4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & 1 \\ 3 & 5 \end{array} \right)\).
  1. Find \(\mathbf { A } ^ { - 1 }\). The matrix \(\mathbf { B } ^ { - 1 }\) is given by \(\mathbf { B } ^ { - 1 } = \left( \begin{array} { r r } 1 & 1 \\ 4 & - 1 \end{array} \right)\).
  2. Find \(( \mathbf { A B } ) ^ { - 1 }\).
OCR FP1 2007 June Q5
7 marks Standard +0.3
5
  1. Show that $$\frac { 1 } { r } - \frac { 1 } { r + 1 } = \frac { 1 } { r ( r + 1 ) }$$
  2. Hence find an expression, in terms of \(n\), for $$\frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \ldots + \frac { 1 } { n ( n + 1 ) }$$
  3. Hence find the value of \(\sum _ { r = n + 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) }\).
OCR FP1 2007 June Q6
8 marks Standard +0.3
6 The cubic equation \(3 x ^ { 3 } - 9 x ^ { 2 } + 6 x + 2 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. (a) Write down the values of \(\alpha + \beta + \gamma\) and \(\alpha \beta + \beta \gamma + \gamma \alpha\).
    (b) Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
  2. (a) Use the substitution \(x = \frac { 1 } { u }\) to find a cubic equation in \(u\) with integer coefficients.
    (b) Use your answer to part (ii) (a) to find the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\).
OCR FP1 2007 June Q7
8 marks Standard +0.3
7 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 0 \\ 0 & a & 4 \\ 2 & 3 & 1 \end{array} \right)\).
  1. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
  2. In the case when \(a = 2\), state whether \(\mathbf { M }\) is singular or non-singular, justifying your answer.
  3. In the case when \(a = 4\), determine whether the simultaneous equations $$\begin{aligned} a x + 4 y \quad = & 6 \\ a y + 4 z & = 8 \\ 2 x + 3 y + z & = 1 \end{aligned}$$ have any solutions.
OCR FP1 2007 June Q8
8 marks Standard +0.3
8 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 3 | = 3\) and arg \(( z - 1 ) = \frac { 1 } { 4 } \pi\) respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Indicate, by shading, the region of the Argand diagram for which $$| z - 3 | \leqslant 3 \text { and } 0 \leqslant \arg ( z - 1 ) \leqslant \frac { 1 } { 4 } \pi$$
OCR FP1 2007 June Q9
9 marks Moderate -0.3
9
  1. Write down the matrix, \(\mathbf { A }\), that represents an enlargement, centre ( 0,0 ), with scale factor \(\sqrt { 2 }\).
  2. The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \\ - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { B }\).
  3. Given that \(\mathbf { C } = \mathbf { A B }\), show that \(\mathbf { C } = \left( \begin{array} { r r } 1 & 1 \\ - 1 & 1 \end{array} \right)\).
  4. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\).
  5. Write down the determinant of \(\mathbf { C }\) and explain briefly how this value relates to the transformation represented by \(\mathbf { C }\).
OCR FP1 2007 June Q10
11 marks Standard +0.3
10
  1. Use an algebraic method to find the square roots of the complex number \(16 + 30 \mathrm { i }\).
  2. Use your answers to part (i) to solve the equation \(z ^ { 2 } - 2 z - ( 15 + 30 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\).
OCR FP1 2008 June Q1
4 marks Moderate -0.8
1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 4 & 1 \\ 5 & 2 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find
  1. \(\mathbf { A } - 3 \mathbf { I }\),
  2. \(\mathrm { A } ^ { - 1 }\).
OCR FP1 2008 June Q2
7 marks Standard +0.3
2 The complex number \(3 + 4 \mathrm { i }\) is denoted by \(a\).
  1. Find \(| a |\) and \(\arg a\).
  2. Sketch on a single Argand diagram the loci given by
    (a) \(| z - a | = | a |\),
    (b) \(\arg ( z - 3 ) = \arg a\).
OCR FP1 2008 June Q3
6 marks Standard +0.3
3
  1. Show that \(\frac { 1 } { r ! } - \frac { 1 } { ( r + 1 ) ! } = \frac { r } { ( r + 1 ) ! }\).
  2. Hence find an expression, in terms of \(n\), for $$\frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { n } { ( n + 1 ) ! }$$
OCR FP1 2008 June Q4
6 marks Standard +0.8
4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 1 \\ 0 & 1 \end{array} \right)\). Prove by induction that, for \(n \geqslant 1\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & \frac { 1 } { 2 } \left( 3 ^ { n } - 1 \right) \\ 0 & 1 \end{array} \right)$$
OCR FP1 2008 June Q5
6 marks Moderate -0.3
5 Find \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 )\), expressing your answer in a fully factorised form.
OCR FP1 2008 June Q6
7 marks Moderate -0.5
6 The cubic equation \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are real, has roots ( \(3 + \mathrm { i }\) ) and 2 .
  1. Write down the other root of the equation.
  2. Find the values of \(a , b\) and \(c\).