Questions — OCR FP1 (201 questions)

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OCR FP1 2010 June Q10
10 The complex number \(z\), where \(0 < \arg z < \frac { 1 } { 2 } \pi\), is such that \(z ^ { 2 } = 3 + 4 \mathrm { i }\).
  1. Use an algebraic method to find \(z\).
  2. Show that \(z ^ { 3 } = 2 + 11 \mathrm { i }\). The complex number \(w\) is the root of the equation $$w ^ { 6 } - 4 w ^ { 3 } + 125 = 0$$ for which \(- \frac { 1 } { 2 } \pi < \arg w < 0\).
  3. Find \(w\).
OCR FP1 2011 June Q1
1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & a
0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 2 & a
4 & 1 \end{array} \right)\). I denotes the \(2 \times 2\) identity matrix. Find
  1. \(\mathbf { A } + 3 \mathbf { B } - 4 \mathbf { I }\),
  2. AB.
OCR FP1 2011 June Q2
2 Prove by induction that, for \(n \geqslant 1 , \sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }\).
OCR FP1 2011 June Q3
3 By using the determinant of an appropriate matrix, find the values of \(k\) for which the simultaneous equations $$\begin{aligned} & k x + 8 y = 1
& 2 x + k y = 3 \end{aligned}$$ do not have a unique solution.
OCR FP1 2011 June Q4
4 Find \(\sum _ { r = 1 } ^ { 2 n } \left( 3 r ^ { 2 } - \frac { 1 } { 2 } \right)\), expressing your answer in a fully factorised form.
OCR FP1 2011 June Q5
5 The complex number \(1 + \mathrm { i } \sqrt { 3 }\) is denoted by \(a\).
  1. Find \(| a |\) and \(\arg a\).
  2. Sketch on a single Argand diagram the loci given by \(| z - a | = | a |\) and \(\arg ( z - a ) = \frac { 1 } { 2 } \pi\).
OCR FP1 2011 June Q6
6 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r r } a & 1 & 0
1 & 2 & 1
- 1 & 3 & 4 \end{array} \right)\), where \(a \neq 1\). Find \(\mathbf { C } ^ { - 1 }\).
OCR FP1 2011 June Q7
7
  1. Show that \(\frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \equiv \frac { 2 } { r ^ { 2 } - 1 }\).
  2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 2 } ^ { n } \frac { 2 } { r ^ { 2 } - 1 }\).
  3. Find the value of \(\sum _ { r = 1000 } ^ { \infty } \frac { 2 } { r ^ { 2 } - 1 }\).
OCR FP1 2011 June Q8
8 The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { l l } 0 & 3
3 & 0 \end{array} \right)\).
  1. The diagram in the printed answer book shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { X }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
  2. The transformation represented by \(\mathbf { X }\) is equivalent to a transformation A , followed by a transformation B. Give geometrical descriptions of possible transformations A and B and state the matrices that represent them.
OCR FP1 2011 June Q9
9 One root of the quadratic equation \(x ^ { 2 } + a x + b = 0\), where \(a\) and \(b\) are real, is \(16 - 30 \mathrm { i }\).
  1. Write down the other root of the quadratic equation.
  2. Find the values of \(a\) and \(b\).
  3. Use an algebraic method to solve the quartic equation \(y ^ { 4 } + a y ^ { 2 } + b = 0\).
OCR FP1 2011 June Q10
10 The cubic equation \(x ^ { 3 } + 3 x ^ { 2 } + 2 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Use the substitution \(x = \frac { 1 } { \sqrt { u } }\) to show that \(4 u ^ { 3 } + 12 u ^ { 2 } + 9 u - 1 = 0\).
  2. Hence find the values of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } }\) and \(\frac { 1 } { \alpha ^ { 2 } \beta ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } \gamma ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } \alpha ^ { 2 } }\).
OCR FP1 2012 June Q1
1 The complex numbers \(z\) and \(w\) are given by \(z = 6 - \mathrm { i }\) and \(w = 5 + 4 \mathrm { i }\). Giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain them, find
  1. \(z + 3 w\),
  2. \(\frac { Z } { W }\).
OCR FP1 2012 June Q2
2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 1
4 & 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 1 & 0
3 & 2 \end{array} \right)\). Find
  1. \(\mathbf { A B }\),
  2. \(\mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 }\).
OCR FP1 2012 June Q3
3 One root of the quadratic equation \(x ^ { 2 } + a x + b = 0\), where \(a\) and \(b\) are real, is the complex number \(4 - 3 \mathrm { i }\). Find the values of \(a\) and \(b\).
OCR FP1 2012 June Q4
4 Find \(\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 2 \right)\), expressing your answer in a fully factorised form.
OCR FP1 2012 June Q5
5 Prove by induction that, for \(n \geqslant 1 , \sum _ { r = 1 } ^ { n } 4 \times 3 ^ { r } = 6 \left( 3 ^ { n } - 1 \right)\).
OCR FP1 2012 June Q6
6 The quadratic equation \(2 x ^ { 2 } + x + 5 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Use the substitution \(x = \frac { 1 } { u + 1 }\) to obtain a quadratic equation in \(u\) with integer coefficients.
  2. Hence, or otherwise, find the value of \(\left( \frac { 1 } { \alpha } - 1 \right) \left( \frac { 1 } { \beta } - 1 \right)\).
OCR FP1 2012 June Q7
7 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 3 - 4 \mathrm { i } | = 4\) and \(| z | = | z - 8 \mathrm { i } |\) respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Hence find the complex numbers represented by the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
  3. Indicate, by shading, the region of the Argand diagram for which $$| z - 3 - 4 i | \leqslant 4 \text { and } | z | \geqslant | z - 8 i | .$$
OCR FP1 2012 June Q8
8
  1. Show that \(\frac { 1 } { r } - \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 2 ) }\).
  2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 2 ) }\).
  3. Given that \(\sum _ { r = N + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) } = \frac { 11 } { 30 }\), find the value of \(N\).
OCR FP1 2012 June Q9
9
  1. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { l l } 1 & 2
    0 & 1 \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { X }\).
  2. The matrix \(\mathbf { Z }\) is given by \(\mathbf { Z } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { 1 } { 2 } ( 2 + \sqrt { 3 } )
    - \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } ( 1 - 2 \sqrt { 3 } ) \end{array} \right)\). The transformation represented by \(\mathbf { Z }\) is equivalent to the transformation represented by \(\mathbf { X }\), followed by another transformation represented by the matrix \(\mathbf { Y }\). Find \(\mathbf { Y }\).
  3. Describe fully the geometrical transformation represented by \(\mathbf { Y }\).
OCR FP1 2012 June Q10
10 The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { r r r } a & 2 & - 1
2 & a & 1
1 & 1 & a \end{array} \right)\).
  1. Find the determinant of \(\mathbf { D }\) in terms of \(a\).
  2. Three simultaneous equations are shown below. $$\begin{array} { r } a x + 2 y - z = 0
    2 x + a y + z = a
    x + y + a z = a \end{array}$$ For each of the following values of \(a\), determine whether or not there is a unique solution. If the solution is not unique, determine whether the equations are consistent or inconsistent.
    (a) \(\quad a = 3\)
    (b) \(a = 2\)
    (c) \(\quad a = 0\) \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR FP1 2014 June Q1
1 Find the determinant of the matrix \(\left( \begin{array} { r r r } a & 4 & - 1
3 & a & 2
a & 1 & 1 \end{array} \right)\).
OCR FP1 2014 June Q2
2 The complex number \(7 + 3 \mathrm { i }\) is denoted by \(z\). Find
  1. \(| z |\) and \(\arg z\),
  2. \(\frac { z } { 4 - \mathrm { i } }\), showing clearly how you obtain your answer.
OCR FP1 2014 June Q3
3 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r l } 2 & 1
- 4 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l } 3 & 1
2 & 3 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find
  1. \(4 \mathbf { A } - \mathbf { B } + 2 \mathbf { I }\),
  2. \(\mathrm { A } ^ { - 1 }\),
  3. \(\left( \mathbf { A B } ^ { - 1 } \right) ^ { - 1 }\).
OCR FP1 2014 June Q4
4
  1. Find the matrix that represents a shear with the \(y\)-axis invariant, the image of the point \(( 1,0 )\) being the point \(( 1,4 )\).
  2. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 }
    - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)\).
    1. Describe fully the geometrical transformation represented by \(\mathbf { X }\).
    2. Find the value of the determinant of \(\mathbf { X }\) and describe briefly how this value relates to the transformation represented by \(\mathbf { X }\).