Questions — OCR (4628 questions)

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OCR FP1 2009 January Q3
6 marks Moderate -0.8
3 Find \(\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } + 6 r ^ { 2 } + 2 r \right)\), expressing your answer in a fully factorised form.
OCR FP1 2009 January Q4
4 marks Standard +0.3
4 Given that \(\mathbf { A }\) and \(\mathbf { B }\) are \(2 \times 2\) non-singular matrices and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix, simplify $$\mathbf { B } ( \mathbf { A B } ) ^ { - 1 } \mathbf { A } - \mathbf { I } .$$
OCR FP1 2009 January Q5
5 marks Standard +0.3
5 By using the determinant of an appropriate matrix, or otherwise, find the value of \(k\) for which the simultaneous equations $$\begin{aligned} 2 x - y + z & = 7 \\ 3 y + z & = 4 \\ x + k y + k z & = 5 \end{aligned}$$ do not have a unique solution for \(x , y\) and \(z\).
OCR FP1 2009 January Q6
9 marks Moderate -0.8
6
  1. The transformation P is represented by the matrix \(\left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\). Give a geometrical description of transformation P .
  2. The transformation Q is represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\). Give a geometrical description of transformation Q.
  3. The transformation R is equivalent to transformation P followed by transformation Q . Find the matrix that represents R .
  4. Give a geometrical description of the single transformation that is represented by your answer to part (iii).
OCR FP1 2009 January Q7
7 marks
7 It is given that \(u _ { n } = 13 ^ { n } + 6 ^ { n - 1 }\), where \(n\) is a positive integer.
  1. Show that \(u _ { n } + u _ { n + 1 } = 14 \times 13 ^ { n } + 7 \times 6 ^ { n - 1 }\).
  2. Prove by induction that \(u _ { n }\) is a multiple of 7 .
OCR FP1 2009 January Q8
10 marks Standard +0.3
8
  1. Show that \(( \alpha - \beta ) ^ { 2 } \equiv ( \alpha + \beta ) ^ { 2 } - 4 \alpha \beta\). The quadratic equation \(x ^ { 2 } - 6 k x + k ^ { 2 } = 0\), where \(k\) is a positive constant, has roots \(\alpha\) and \(\beta\), with \(\alpha > \beta\).
  2. Show that \(\alpha - \beta = 4 \sqrt { 2 } k\).
  3. Hence find a quadratic equation with roots \(\alpha + 1\) and \(\beta - 1\).
OCR FP1 2009 January Q9
9 marks Standard +0.8
9
  1. Show that \(\frac { 1 } { 2 r - 3 } - \frac { 1 } { 2 r + 1 } = \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }\).
  2. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 2 } ^ { n } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }$$
  3. Show that \(\sum _ { r = 2 } ^ { \infty } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 } = \frac { 4 } { 3 }\).
  4. Use an algebraic method to find the square roots of the complex number \(2 + \mathrm { i } \sqrt { 5 }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.
  5. Hence find, in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are exact real numbers, the roots of the equation $$z ^ { 4 } - 4 z ^ { 2 } + 9 = 0$$
  6. Show, on an Argand diagram, the roots of the equation in part (ii).
  7. Given that \(\alpha\) is the root of the equation in part (ii) such that \(0 < \arg \alpha < \frac { 1 } { 2 } \pi\), sketch on the same Argand diagram the locus given by \(| z - \alpha | = | z |\).
OCR FP1 2010 January Q1
5 marks Moderate -0.8
1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } a & 2 \\ 3 & 4 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
  1. Find A-4I.
  2. Given that \(\mathbf { A }\) is singular, find the value of \(a\).
OCR FP1 2010 January Q2
5 marks Standard +0.3
2 The cubic equation \(2 x ^ { 3 } + 3 x - 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Use the substitution \(x = u - 1\) to find a cubic equation in \(u\) with integer coefficients.
  2. Hence find the value of \(( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )\).
OCR FP1 2010 January Q3
5 marks Moderate -0.3
3 The complex number \(z\) satisfies the equation \(z + 2 \mathrm { i } z ^ { * } = 12 + 9 \mathrm { i }\). Find \(z\), giving your answer in the form \(x + \mathrm { i } y\).
OCR FP1 2010 January Q4
6 marks Standard +0.3
4 Find \(\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 2 )\), expressing your answer in a fully factorised form.
OCR FP1 2010 January Q5
6 marks Moderate -0.3
5
  1. The transformation T is represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\). Give a geometrical description of T .
  2. The transformation T is equivalent to a reflection in the line \(y = - x\) followed by another transformation S . Give a geometrical description of S and find the matrix that represents S .
OCR FP1 2010 January Q6
7 marks Standard +0.3
6 One root of the cubic equation \(x ^ { 3 } + p x ^ { 2 } + 6 x + q = 0\), where \(p\) and \(q\) are real, is the complex number 5-i.
  1. Find the real root of the cubic equation.
  2. Find the values of \(p\) and \(q\).
OCR FP1 2010 January Q7
7 marks Standard +0.8
7
  1. Show that \(\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } } \equiv \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }\).
  2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }\).
  3. Find \(\sum _ { r = 2 } ^ { \infty } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }\).
OCR FP1 2010 January Q8
9 marks Standard +0.8
8 The complex number \(a\) is such that \(a ^ { 2 } = 5 - 12 \mathrm { i }\).
  1. Use an algebraic method to find the two possible values of \(a\).
  2. Sketch on a single Argand diagram the two possible loci given by \(| z - a | = | a |\).
OCR FP1 2010 January Q9
11 marks Standard +0.3
9 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } 2 & - 1 & 1 \\ 0 & 3 & 1 \\ 1 & 1 & a \end{array} \right)\), where \(a \neq 1\).
  1. Find \(\mathbf { A } ^ { - 1 }\).
  2. Hence, or otherwise, solve the equations $$\begin{array} { r } 2 x - y + z = 1 \\ 3 y + z = 2 \\ x + y + a z = 2 \end{array}$$
OCR FP1 2010 January Q10
11 marks Standard +0.8
10 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).
  1. Find \(\mathbf { M } ^ { 2 }\) and \(\mathbf { M } ^ { 3 }\).
  2. Hence suggest a suitable form for the matrix \(\mathbf { M } ^ { n }\).
  3. Use induction to prove that your answer to part (ii) is correct.
  4. Describe fully the single geometrical transformation represented by \(\mathbf { M } ^ { 10 }\).
OCR FP1 2011 January Q1
7 marks Easy -1.8
\(\mathbf { 1 }\) The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l } 3 & - 1 \end{array} \right)\) and \(\mathbf { C } = \binom { 4 } { 2 }\). Find
  1. \(2 \mathbf { A } + \mathbf { B }\),
  2. \(\mathbf { A C }\),
  3. CB. \end{itemize}
OCR FP1 2011 January Q2
6 marks Moderate -0.8
2 The complex numbers \(z\) and \(w\) are given by \(z = 4 + 3 \mathrm { i }\) and \(w = 6 - \mathrm { i }\). Giving your answers in the form \(x + \mathrm { i } y\) and showing clearly how you obtain them, find
  1. \(3 z - 4 w\),
  2. \(\frac { z ^ { * } } { w }\).
OCR FP1 2011 January Q3
4 marks Standard +0.3
3 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 2\), and \(u _ { n + 1 } = 2 u _ { n } - 1\) for \(n \geqslant 1\). Prove by induction that \(u _ { n } = 2 ^ { n - 1 } + 1\).
OCR FP1 2011 January Q4
6 marks Standard +0.8
4 Given that \(\sum _ { r = 1 } ^ { n } \left( a r ^ { 3 } + b r \right) \equiv n ( n - 1 ) ( n + 1 ) ( n + 2 )\), find the values of the constants \(a\) and \(b\).
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
    (a) \(\quad | z | = | z - 8 |\),
    (b) \(\quad \arg ( z + 2 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
  2. 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 \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }\).