Questions — OCR FP1 (201 questions)

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OCR FP1 2008 January Q6
6 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z | = | z - 4 \mathbf { i } | \quad \text { and } \quad \arg z = \frac { 1 } { 6 } \pi$$ respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Hence find, in the form \(x +\) i \(y\), the complex number represented by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
OCR FP1 2008 January Q7
7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c } a & 3
- 2 & 1 \end{array} \right)\).
  1. Given that \(\mathbf { A }\) is singular, find \(a\).
  2. Given instead that \(\mathbf { A }\) is non-singular, find \(\mathbf { A } ^ { - 1 }\) and hence solve the simultaneous equations $$\begin{aligned} a x + 3 y & = 1
    - 2 x + y & = - 1 \end{aligned}$$
OCR FP1 2008 January Q8
8 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 1\) and \(u _ { n + 1 } = u _ { n } + 2 n + 1\).
  1. Show that \(u _ { 4 } = 16\).
  2. Hence suggest an expression for \(u _ { n }\).
  3. Use induction to prove that your answer to part (ii) is correct.
OCR FP1 2008 January Q9
9
  1. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = ( \alpha + \beta ) ^ { 3 } - 3 \alpha \beta ( \alpha + \beta )\).
  2. The quadratic equation \(x ^ { 2 } - 5 x + 7 = 0\) has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\).
  3. Show that \(\frac { 2 } { r } - \frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }\).
  4. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }$$
  5. Hence write down the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) }\).
  6. Given that \(\sum _ { r = N + 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) ( r + 2 ) } = \frac { 7 } { 10 }\), find the value of \(N\).
OCR FP1 2005 June Q1
1 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 6 r ^ { 2 } + 2 r + 1 \right) = n \left( 2 n ^ { 2 } + 4 n + 3 \right)$$
OCR FP1 2005 June Q2
2 The matrices \(\mathbf { A }\) and \(\mathbf { I }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & 2
1 & 3 \end{array} \right)\) and \(\mathbf { I } = \left( \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right)\) respectively.
  1. Find \(\mathbf { A } ^ { 2 }\) and verify that \(\mathbf { A } ^ { 2 } = 4 \mathbf { A } - \mathbf { I }\).
  2. Hence, or otherwise, show that \(\mathbf { A } ^ { - 1 } = 4 \mathbf { I } - \mathbf { A }\).
OCR FP1 2005 June Q3
3 The complex numbers \(2 + 3 \mathrm { i }\) and \(4 - \mathrm { i }\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + \mathrm { i } y\), showing clearly how you obtain your answers.
  1. \(z + 5 w\),
  2. \(z ^ { * } w\), where \(z ^ { * }\) is the complex conjugate of \(z\),
  3. \(\frac { 1 } { w }\).
OCR FP1 2005 June Q4
4 Use an algebraic method to find the square roots of the complex number 21-20i.
OCR FP1 2005 June Q5
5
  1. Show that $$\frac { r + 1 } { r + 2 } - \frac { r } { r + 1 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$
  2. Hence find an expression, in terms of \(n\), for $$\frac { 1 } { 6 } + \frac { 1 } { 12 } + \frac { 1 } { 20 } + \ldots + \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$
  3. Hence write down the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( r + 1 ) ( r + 2 ) }\).
OCR FP1 2005 June Q6
6 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by $$| z - 2 \mathrm { i } | = 2 \quad \text { and } \quad | z + 1 | = | z + \mathrm { i } |$$ respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Hence write down the complex numbers represented by the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
    \(7 \quad\) The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r r r } a & 1 & 3
    2 & 1 & - 1
    0 & 1 & 2 \end{array} \right)\).
  3. Given that \(\mathbf { B }\) is singular, show that \(a = - \frac { 2 } { 3 }\).
  4. Given instead that \(\mathbf { B }\) is non-singular, find the inverse matrix \(\mathbf { B } ^ { - 1 }\).
  5. Hence, or otherwise, solve the equations $$\begin{aligned} - x + y + 3 z & = 1
    2 x + y - z & = 4
    y + 2 z & = - 1 \end{aligned}$$
OCR FP1 2005 June Q8
8
  1. The quadratic equation \(x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha\) and \(\beta\).
    1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
    2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 4\).
    3. Hence find a quadratic equation which has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
  2. The cubic equation \(x ^ { 3 } - 12 x ^ { 2 } + a x - 48 = 0\) has roots \(p , 2 p\) and \(3 p\).
    1. Find the value of \(p\).
    2. Hence find the value of \(a\).
OCR FP1 2005 June Q9
9
  1. Write down the matrix \(\mathbf { C }\) which represents a stretch, scale factor 2 , in the \(x\)-direction.
  2. The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { l l } 1 & 3
    0 & 1 \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { D }\).
  3. The matrix \(\mathbf { M }\) represents the combined effect of the transformation represented by \(\mathbf { C }\) followed by the transformation represented by \(\mathbf { D }\). Show that $$\mathbf { M } = \left( \begin{array} { l l } 2 & 3
    0 & 1 \end{array} \right)$$
  4. Prove by induction that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right)
    0 & 1 \end{array} \right)\), for all positive integers \(n\).
OCR FP1 2006 June Q1
\(\mathbf { 1 }\) The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 4 & 1
0 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 1 & 1
0 & - 1 \end{array} \right)\).
  1. Find \(\mathbf { A } + 3 \mathbf { B }\).
  2. Show that \(\mathbf { A } - \mathbf { B } = k \mathbf { I }\), where \(\mathbf { I }\) is the identity matrix and \(k\) is a constant whose value should be stated. \end{itemize}
OCR FP1 2006 June Q2
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
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
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
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
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
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
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
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
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
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
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
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 \(n\), $$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 1 \right) = n ^ { 3 }$$