Questions — OCR FP1 (201 questions)

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OCR FP1 2013 January Q3
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 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
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
  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
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
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 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
1 Evaluate \(\sum _ { r = 101 } ^ { 250 } r ^ { 3 }\).
OCR FP1 2009 June Q2
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
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 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 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
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
    (a) \(| z - a | = 3 \sqrt { 2 }\),
    (b) \(\quad \arg ( z - a ) = \frac { 1 } { 4 } \pi\).
  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
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
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
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
    (a) \(a = 0\),
    (b) \(a = 1\).
OCR FP1 2009 June Q10
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.
OCR FP1 2010 June Q1
1 Prove by induction that, for \(n \geqslant 1 , \sum _ { r = 1 } ^ { n } r ( r + 1 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 2 )\).
OCR FP1 2010 June Q2
2 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & - 4 \end{array} \right) , \mathbf { B } = \binom { 5 } { 3 }\) and \(\mathbf { C } = \left( \begin{array} { r r } 3 & 0
- 2 & 2 \end{array} \right)\). Find
  1. \(\mathbf { A B }\),
  2. \(\mathbf { B A } - 4 \mathbf { C }\).
OCR FP1 2010 June Q3
3 Find \(\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 }\), expressing your answer in a fully factorised form.
OCR FP1 2010 June Q4
4 The complex numbers \(a\) and \(b\) are given by \(a = 7 + 6 \mathrm { i }\) and \(b = 1 - 3 \mathrm { i }\). Showing clearly how you obtain your answers, find
  1. \(| a - 2 b |\) and \(\arg ( a - 2 b )\),
  2. \(\frac { b } { a }\), giving your answer in the form \(x + \mathrm { i } y\).
OCR FP1 2010 June Q5
5
  1. Write down the matrix that represents a reflection in the line \(y = x\).
  2. Describe fully the geometrical transformation represented by each of the following matrices: $$\begin{aligned} & \text { (i) } \left( \begin{array} { c c } 5 & 0
    0 & 1 \end{array} \right) \text {, }
    & \text { (ii) } \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { 1 } { 2 } \sqrt { 3 }
    - \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right) \text {. } \end{aligned}$$
OCR FP1 2010 June Q6
6
  1. Sketch on a single Argand diagram the loci given by
    (a) \(| z - 3 + 4 \mathrm { i } | = 5\),
    (b) \(| z | = | z - 6 |\).
  2. Indicate, by shading, the region of the Argand diagram for which $$| z - 3 + 4 i | \leqslant 5 \quad \text { and } \quad | z | \geqslant | z - 6 | .$$
OCR FP1 2010 June Q7
7 The quadratic equation \(x ^ { 2 } + 2 k x + k = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\frac { \alpha + \beta } { \alpha }\) and \(\frac { \alpha + \beta } { \beta }\).
  1. Show that \(\frac { 1 } { \sqrt { r + 2 } + \sqrt { r } } \equiv \frac { \sqrt { r + 2 } - \sqrt { r } } { 2 }\).
  2. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }$$
  3. State, giving a brief reason, whether the series \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }\) converges.
OCR FP1 2010 June Q9
9 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } a & a & - 1
0 & a & 2
1 & 2 & 1 \end{array} \right)\).
  1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\).
  2. Three simultaneous equations are shown below. $$\begin{aligned} a x + a y - z & = - 1
    a y + 2 z & = 2 a
    x + 2 y + z & = 1 \end{aligned}$$ For each of the following values of \(a\), determine whether the equations are consistent or inconsistent. If the equations are consistent, determine whether or not there is a unique solution.
    (a) \(a = 0\)
    (b) \(a = 1\)
    (c) \(a = 2\)