Questions — OCR (4628 questions)

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OCR FP1 2011 January Q10
11 marks Moderate -0.5
10
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OCR FP1 2012 January Q1
4 marks Easy -1.2
1 The complex number \(a + 5 \mathrm { i }\), where \(a\) is positive, is denoted by \(z\). Given that \(| z | = 13\), find the value of \(a\) and hence find \(\arg z\).
OCR FP1 2012 January Q2
5 marks Moderate -0.5
2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 3 & 4 \\ 2 & - 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 4 & 6 \\ 3 & - 5 \end{array} \right)\), and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Given that \(p \mathbf { A } + q \mathbf { B } = \mathbf { I }\), find the values of the constants \(p\) and \(q\).
OCR FP1 2012 January Q3
6 marks Standard +0.3
3 Use an algebraic method to find the square roots of \(3 + ( 6 \sqrt { 2 } )\) i. Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.
OCR FP1 2012 January Q4
6 marks Moderate -0.8
4 Find \(\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right)\), expressing your answer in a fully factorised form.
OCR FP1 2012 January Q5
6 marks Moderate -0.8
5
  1. Find the matrix that represents a reflection in the line \(y = - x\).
  2. The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 4 \end{array} \right)\).
    1. Describe fully the geometrical transformation represented by \(\mathbf { C }\).
    2. State the value of the determinant of \(\mathbf { C }\) and describe briefly how this value relates to the transformation represented by \(\mathbf { C }\).
OCR FP1 2012 January Q6
6 marks Moderate -0.3
6 Sketch, on a single Argand diagram, the loci given by \(| z - \sqrt { 3 } - \mathrm { i } | = 2\) and \(\arg z = \frac { 1 } { 6 } \pi\).
OCR FP1 2012 January Q7
9 marks Standard +0.8
7 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 2 & 1 \end{array} \right)\).
  1. Show that \(\mathbf { M } ^ { 4 } = \left( \begin{array} { l l } 81 & 0 \\ 80 & 1 \end{array} \right)\).
  2. Hence suggest a suitable form for the matrix \(\mathbf { M } ^ { n }\), where \(n\) is a positive integer.
  3. Use induction to prove that your answer to part (ii) is correct.
OCR FP1 2012 January Q8
8 marks Standard +0.3
8
  1. Show that \(\frac { r } { r + 1 } - \frac { r - 1 } { r } \equiv \frac { 1 } { r ( r + 1 ) }\).
  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 \(\sum _ { r = n + 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) }\).
OCR FP1 2012 January Q9
10 marks Standard +0.3
\(\mathbf { 9 }\) The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r r } a & 2 & 9 \\ 2 & a & 3 \\ 1 & 0 & - 1 \end{array} \right)\).
  1. Find the determinant of \(\mathbf { X }\) in terms of \(a\).
  2. Hence find the values of \(a\) for which \(\mathbf { X }\) is singular.
  3. Given that \(\mathbf { X }\) is non-singular, find \(\mathbf { X } ^ { - 1 }\) in terms of \(a\).
OCR FP1 2012 January Q10
12 marks Standard +0.3
10 The cubic equation \(3 x ^ { 3 } - 9 x ^ { 2 } + 6 x + 2 = 0\) 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 \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\) has roots \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).
  2. Show that \(c = - \frac { 4 } { 9 }\) and find the values of \(a\) and \(b\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR FP1 2013 January Q1
5 marks Moderate -0.8
1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } a & 1 \\ 1 & 4 \end{array} \right)\), where \(a \neq \frac { 1 } { 4 }\), and \(\mathbf { I }\) denotes the \(2 \times 2\) identity matrix. Find
  1. \(2 \mathbf { A } - 3 \mathbf { I }\),
  2. \(\mathrm { A } ^ { - 1 }\).
OCR FP1 2013 January Q2
6 marks Moderate -0.5
2 Find \(\sum _ { r = 1 } ^ { n } ( r - 1 ) ( r + 1 )\), giving your answer in a fully factorised form.
OCR FP1 2013 January Q3
7 marks Moderate -0.3
3 The complex number \(2 - \mathrm { i }\) is denoted by \(z\).
  1. Find \(| z |\) and \(\arg z\).
  2. Given that \(a z + b z ^ { * } = 4 - 8 \mathrm { i }\), find the values of the real constants \(a\) and \(b\).
OCR FP1 2013 January Q4
4 marks Standard +0.3
4 The quadratic equation \(x ^ { 2 } + x + k = 0\) has roots \(\alpha\) and \(\beta\).
  1. Use the substitution \(x = 2 u + 1\) to obtain a quadratic equation in \(u\).
  2. Hence, or otherwise, find the value of \(\left( \frac { \alpha - 1 } { 2 } \right) \left( \frac { \beta - 1 } { 2 } \right)\) in terms of \(k\).
OCR FP1 2013 January Q5
6 marks Standard +0.3
5 By using the determinant of an appropriate matrix, find the values of \(\lambda\) for which the simultaneous equations $$\begin{array} { r } 3 x + 2 y + 4 z = 5 \\ \lambda y + z = 1 \\ x + \lambda y + \lambda z = 4 \end{array}$$ do not have a unique solution for \(x , y\) and \(z\). \includegraphics[max width=\textwidth, alt={}, center]{f074de40-08b6-47a6-a0d2-d3cbe628cacc-3_556_759_233_653} The diagram shows the unit square \(O A B C\), and its image \(O A B ^ { \prime } C ^ { \prime }\) after a transformation. The points have the following coordinates: \(A ( 1,0 ) , B ( 1,1 ) , C ( 0,1 ) , B ^ { \prime } ( 3,2 )\) and \(C ^ { \prime } ( 2,2 )\).
  1. Write down the matrix, \(\mathbf { X }\), for this transformation.
  2. The transformation represented by \(\mathbf { X }\) is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a pair of possible transformations P and Q and state the matrices that represent them.
  3. Find the matrix that represents transformation Q followed by transformation P .
OCR FP1 2013 January Q7
7 marks Standard +0.3
7
  1. Sketch on a single Argand diagram the loci given by
    (a) \(| z | = 2\),
    (b) \(\quad \arg ( z - 3 - \mathrm { i } ) = \pi\).
  2. Indicate, by shading, the region of the Argand diagram for which $$| z | \leqslant 2 \text { and } 0 \leqslant \arg ( z - 3 - i ) \leqslant \pi .$$
OCR FP1 2013 January Q8
9 marks Standard +0.8
8
  1. Show that \(\frac { 1 } { r } - \frac { 3 } { r + 1 } + \frac { 2 } { r + 2 } \equiv \frac { 2 - r } { r ( r + 1 ) ( r + 2 ) }\).
  2. Hence show that \(\sum _ { r = 1 } ^ { n } \frac { 2 - r } { r ( r + 1 ) ( r + 2 ) } = \frac { n } { ( n + 1 ) ( n + 2 ) }\).
  3. Find the value of \(\sum _ { r = 2 } ^ { \infty } \frac { 2 - r } { r ( r + 1 ) ( r + 2 ) }\).
OCR FP1 2013 January Q9
8 marks Standard +0.8
9
  1. Show that \(( \alpha \beta + \beta \gamma + \gamma \alpha ) ^ { 2 } \equiv \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } + 2 \alpha \beta \gamma ( \alpha + \beta + \gamma )\).
  2. It is given that \(\alpha , \beta\) and \(\gamma\) are the roots of the cubic equation \(x ^ { 3 } + p x ^ { 2 } - 4 x + 3 = 0\), where \(p\) is a constant. Find the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } }\) in terms of \(p\).
OCR FP1 2013 January Q10
10 marks Standard +0.8
10 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 2\) and \(u _ { n + 1 } = \frac { u _ { n } } { 1 + u _ { n } }\) for \(n \geqslant 1\).
  1. Find \(u _ { 2 }\) and \(u _ { 3 }\), and show that \(u _ { 4 } = \frac { 2 } { 7 }\).
  2. Hence suggest an expression for \(u _ { n }\).
  3. Use induction to prove that your answer to part (ii) is correct.
OCR FP1 2009 June Q1
3 marks Moderate -0.5
1 Evaluate \(\sum _ { r = 101 } ^ { 250 } r ^ { 3 }\).
OCR FP1 2009 June Q2
4 marks Easy -1.2
2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 5 & 0 \\ 0 & 2 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find the values of the constants \(a\) and \(b\) for which \(a \mathbf { A } + b \mathbf { B } = \mathbf { I }\).
OCR FP1 2009 June Q3
4 marks Easy -1.2
3 The complex numbers \(z\) and \(w\) are given by \(z = 5 - 2 \mathrm { i }\) and \(w = 3 + 7 \mathrm { i }\). Giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain them, find
  1. \(4 z - 3 w\),
  2. \(z ^ { * } w\).
OCR FP1 2009 June Q4
4 marks Moderate -0.5
4 The roots of the quadratic equation \(x ^ { 2 } + x - 8 = 0\) are \(p\) and \(q\). Find the value of \(p + q + \frac { 1 } { p } + \frac { 1 } { q }\).
OCR FP1 2009 June Q5
5 marks Standard +0.3
5 The cubic equation \(x ^ { 3 } + 5 x ^ { 2 } + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Use the substitution \(x = \sqrt { u }\) to find a cubic equation in \(u\) with integer coefficients.
  2. Hence find the value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).