Questions — OCR FP1 (210 questions)

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OCR FP1 2011 January Q5
3 marks Moderate -0.8
5 Given that \(\mathbf { A }\) and \(\mathbf { B }\) are non-singular square matrices, simplify $$\mathbf { A B } \left( \mathbf { A } ^ { - 1 } \mathbf { B } \right) ^ { - 1 } .$$
OCR FP1 2011 January Q6
8 marks Standard +0.3
6
  1. Sketch on a single Argand diagram the loci given by
    1. \(\quad | z | = | z - 8 |\),
    2. \(\quad \arg ( z + 2 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
    3. Indicate by shading the region of the Argand diagram for which $$| z | \leqslant | z - 8 | \quad \text { and } \quad 0 \leqslant \arg ( z + 2 i ) \leqslant \frac { 1 } { 4 } \pi$$
OCR FP1 2011 January Q8
9 marks Standard +0.3
8 The quadratic equation \(2 x ^ { 2 } - x + 3 = 0\) has roots \(\alpha\) and \(\beta\), and the quadratic equation \(x ^ { 2 } - p x + q = 0\) has roots \(\alpha + \frac { 1 } { \alpha }\) and \(\beta + \frac { 1 } { \beta }\).
  1. Show that \(p = \frac { 5 } { 6 }\).
  2. Find the value of \(q\).
OCR FP1 2011 January Q9
9 marks Standard +0.3
9 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1 \\ 3 & a & 1 \\ 4 & 2 & 1 \end{array} \right)\).
  1. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
  2. Hence find the values of \(a\) for which \(\mathbf { M } ^ { - 1 }\) does not exist.
  3. Determine whether the simultaneous equations $$\begin{aligned} & 6 x - 6 y + z = 3 k \\ & 3 x + 6 y + z = 0 \\ & 4 x + 2 y + z = k \end{aligned}$$ where \(k\) is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.
  4. Show that \(\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }\).
  5. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
  6. Show that \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }\).
OCR FP1 2011 January Q10
11 marks Moderate -0.5
10
10
  • ACHIEVEMENT
    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 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 \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
    OCR FP1 2009 June Q6
    11 marks Standard +0.3
    6 The complex number \(3 - 3 \mathrm { i }\) is denoted by \(a\).
    1. Find \(| a |\) and \(\arg a\).
    2. Sketch on a single Argand diagram the loci given by
      1. \(| z - a | = 3 \sqrt { 2 }\),
      2. \(\quad \arg ( z - a ) = \frac { 1 } { 4 } \pi\).
      3. Indicate, by shading, the region of the Argand diagram for which $$| z - a | \geqslant 3 \sqrt { 2 } \quad \text { and } \quad 0 \leqslant \arg ( z - a ) \leqslant \frac { 1 } { 4 } \pi$$
    OCR FP1 2009 June Q7
    10 marks Moderate -0.3
    7
    1. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \left\{ ( r + 1 ) ^ { 4 } - r ^ { 4 } \right\} = ( n + 1 ) ^ { 4 } - 1$$
    2. Show that \(( r + 1 ) ^ { 4 } - r ^ { 4 } \equiv 4 r ^ { 3 } + 6 r ^ { 2 } + 4 r + 1\).
    3. Hence show that $$4 \sum _ { r = 1 } ^ { n } r ^ { 3 } = n ^ { 2 } ( n + 1 ) ^ { 2 }$$
    OCR FP1 2009 June Q8
    11 marks Standard +0.8
    8 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 3 & 2 \\ 1 & 1 \end{array} \right)\).
    1. Draw a diagram showing the image of the unit square under the transformation represented by \(\mathbf { C }\). The transformation represented by \(\mathbf { C }\) is equivalent to a transformation S followed by another transformation T.
    2. Given that S is a shear with the \(y\)-axis invariant in which the image of the point ( 1,1 ) is ( 1,2 ), write down the matrix that represents \(S\).
    3. Find the matrix that represents transformation T and describe fully the transformation T .
    OCR FP1 2009 June Q9
    10 marks Standard +0.3
    9 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & 2 \end{array} \right)\).
    1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\).
    2. Hence find the values of \(a\) for which \(\mathbf { A }\) is singular.
    3. State, giving a brief reason in each case, whether the simultaneous equations $$\begin{aligned} a x + y + z & = 2 a \\ x + a y + z & = - 1 \\ x + y + 2 z & = - 1 \end{aligned}$$ have any solutions when
      1. \(a = 0\),
      2. \(a = 1\).
    OCR FP1 2009 June Q10
    10 marks Standard +0.3
    10 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 3\) and \(u _ { n + 1 } = 3 u _ { n } - 2\).
    1. Find \(u _ { 2 }\) and \(u _ { 3 }\) and verify that \(\frac { 1 } { 2 } \left( u _ { 4 } - 1 \right) = 27\).
    2. Hence suggest an expression for \(u _ { n }\).
    3. Use induction to prove that your answer to part (ii) is correct.