Questions — OCR FP1 (201 questions)

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OCR FP1 2013 June Q9
9
  1. Show that \(\frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 } \equiv \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\).
  2. Hence show that \(\sum _ { r = 1 } ^ { 2 n } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) } = \frac { n } { 2 ( 3 n + 1 ) }\).
OCR FP1 2013 June Q10
10 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 1
1 & 3 & 2
4 & 1 & 1 \end{array} \right)\).
  1. Find the value of \(a\) for which \(\mathbf { A }\) is singular.
  2. Given that \(\mathbf { A }\) is non-singular, find \(\mathbf { A } ^ { - 1 }\) and hence solve the equations $$\begin{aligned} a x + 2 y + z & = 1
    x + 3 y + 2 z & = 2
    4 x + y + z & = 3 \end{aligned}$$
OCR FP1 Specimen Q1
1 Use formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 2 )$$
OCR FP1 Specimen Q2
2 The cubic equation \(x ^ { 3 } - 6 x ^ { 2 } + k x + 10 = 0\) has roots \(p - q , p\) and \(p + q\), where \(q\) is positive.
  1. By considering the sum of the roots, find \(p\).
  2. Hence, by considering the product of the roots, find \(q\).
  3. Find the value of \(k\).
OCR FP1 Specimen Q3
3 The complex number \(2 + \mathrm { i }\) is denoted by \(z\), and the complex conjugate of \(z\) is denoted by \(z ^ { * }\).
  1. Express \(z ^ { 2 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, showing clearly how you obtain your answer.
  2. Show that \(4 z - z ^ { 2 }\) simplifies to a real number, and verify that this real number is equal to \(z z ^ { * }\).
  3. Express \(\frac { z + 1 } { z - 1 }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, showing clearly how you obtain your answer.
OCR FP1 Specimen Q4
4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 3 ^ { 2 n } - 1$$
  1. Write down the value of \(u _ { 1 }\).
  2. Show that \(u _ { n + 1 } - u _ { n } = 8 \times 3 ^ { 2 n }\).
  3. Hence prove by induction that each term of the sequence is a multiple of 8 .
OCR FP1 Specimen Q5
5
  1. Show that $$\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 } = \frac { 2 } { 4 r ^ { 2 } - 1 }$$
  2. Hence find an expression in terms of \(n\) for $$\frac { 2 } { 3 } + \frac { 2 } { 15 } + \frac { 2 } { 35 } + \ldots + \frac { 2 } { 4 n ^ { 2 } - 1 }$$
  3. State the value of
    (a) \(\quad \sum _ { r = 1 } ^ { \infty } \frac { 2 } { 4 r ^ { 2 } - 1 }\),
    (b) \(\quad \sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { 4 r ^ { 2 } - 1 }\).
OCR FP1 Specimen Q6
6 In an Argand diagram, the variable point \(P\) represents the complex number \(z = x + \mathrm { i } y\), and the fixed point \(A\) represents \(a = 4 - 3 \mathrm { i }\).
  1. Sketch an Argand diagram showing the position of \(A\), and find \(| a |\) and \(\arg a\).
  2. Given that \(| z - a | = | a |\), sketch the locus of \(P\) on your Argand diagram.
  3. Hence write down the non-zero value of \(z\) corresponding to a point on the locus for which
    (a) the real part of \(z\) is zero,
    (b) \(\quad \arg z = \arg a\).
OCR FP1 Specimen Q7
7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 1 & - 2
2 & 1 \end{array} \right)\).
  1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { A }\).
  2. The value of \(\operatorname { det } \mathbf { A }\) is 5 . Show clearly how this value relates to your diagram in part (i). A represents a sequence of two elementary geometrical transformations, one of which is a rotation \(R\).
  3. Determine the angle of \(R\), and describe the other transformation fully.
  4. State the matrix that represents \(R\), giving the elements in an exact form.
OCR FP1 Specimen Q8
8 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & 2 & - 1
2 & 3 & - 1
2 & - 1 & 1 \end{array} \right)\), where \(a\) is a constant.
  1. Show that the determinant of \(\mathbf { M }\) is \(2 a\).
  2. Given that \(a \neq 0\), find the inverse matrix \(\mathbf { M } ^ { - 1 }\).
  3. Hence or otherwise solve the simultaneous equations $$\begin{array} { r } x + 2 y - z = 1
    2 x + 3 y - z = 2
    2 x - y + z = 0 \end{array}$$
  4. Find the value of \(k\) for which the simultaneous equations $$\begin{array} { r } 2 y - z = k
    2 x + 3 y - z = 2
    2 x - y + z = 0 \end{array}$$ have solutions.
  5. Do the equations in part (iv), with the value of \(k\) found, have a solution for which \(x = z\) ? Justify your answer.
OCR FP1 2009 January Q1
1 Express \(\frac { 2 + 3 \mathrm { i } } { 5 - \mathrm { i } }\) in the form \(x + \mathrm { i } y\), showing clearly how you obtain your answer.
OCR FP1 2009 January Q2
2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0
a & 5 \end{array} \right)\). Find
  1. \(\mathbf { A } ^ { - 1 }\),
  2. \(2 \mathbf { A } - \left( \begin{array} { l l } 1 & 2
    0 & 4 \end{array} \right)\).
OCR FP1 2009 January Q3
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 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 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
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 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
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
  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
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
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
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
4 Find \(\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 2 )\), expressing your answer in a fully factorised form.
OCR FP1 2010 January Q5
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
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 \(q\).