Questions — OCR FP1 (201 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR FP1 2007 June Q4
4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & 1
3 & 5 \end{array} \right)\).
  1. Find \(\mathbf { A } ^ { - 1 }\). The matrix \(\mathbf { B } ^ { - 1 }\) is given by \(\mathbf { B } ^ { - 1 } = \left( \begin{array} { r r } 1 & 1
    4 & - 1 \end{array} \right)\).
  2. Find \(( \mathbf { A B } ) ^ { - 1 }\).
OCR FP1 2007 June Q5
5
  1. Show that $$\frac { 1 } { r } - \frac { 1 } { r + 1 } = \frac { 1 } { r ( r + 1 ) }$$
  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 the value of \(\sum _ { r = n + 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) }\).
OCR FP1 2007 June Q6
6 The cubic equation \(3 x ^ { 3 } - 9 x ^ { 2 } + 6 x + 2 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. (a) Write down the values of \(\alpha + \beta + \gamma\) and \(\alpha \beta + \beta \gamma + \gamma \alpha\).
    (b) Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
  2. (a) Use the substitution \(x = \frac { 1 } { u }\) to find a cubic equation in \(u\) with integer coefficients.
    (b) Use your answer to part (ii) (a) to find the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\).
OCR FP1 2007 June Q7
7 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 0
0 & a & 4
2 & 3 & 1 \end{array} \right)\).
  1. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
  2. In the case when \(a = 2\), state whether \(\mathbf { M }\) is singular or non-singular, justifying your answer.
  3. In the case when \(a = 4\), determine whether the simultaneous equations $$\begin{aligned} a x + 4 y \quad = & 6
    a y + 4 z & = 8
    2 x + 3 y + z & = 1 \end{aligned}$$ have any solutions.
OCR FP1 2007 June Q8
8 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 3 | = 3\) and arg \(( z - 1 ) = \frac { 1 } { 4 } \pi\) respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Indicate, by shading, the region of the Argand diagram for which $$| z - 3 | \leqslant 3 \text { and } 0 \leqslant \arg ( z - 1 ) \leqslant \frac { 1 } { 4 } \pi$$
OCR FP1 2007 June Q9
9
  1. Write down the matrix, \(\mathbf { A }\), that represents an enlargement, centre ( 0,0 ), with scale factor \(\sqrt { 2 }\).
  2. The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 }
    - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { B }\).
  3. Given that \(\mathbf { C } = \mathbf { A B }\), show that \(\mathbf { C } = \left( \begin{array} { r r } 1 & 1
    - 1 & 1 \end{array} \right)\).
  4. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\).
  5. Write down the determinant of \(\mathbf { C }\) and explain briefly how this value relates to the transformation represented by \(\mathbf { C }\).
OCR FP1 2007 June Q10
10
  1. Use an algebraic method to find the square roots of the complex number \(16 + 30 \mathrm { i }\).
  2. Use your answers to part (i) to solve the equation \(z ^ { 2 } - 2 z - ( 15 + 30 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\).
OCR FP1 2008 June Q1
1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 4 & 1
5 & 2 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find
  1. \(\mathbf { A } - 3 \mathbf { I }\),
  2. \(\mathrm { A } ^ { - 1 }\).
OCR FP1 2008 June Q2
2 The complex number \(3 + 4 \mathrm { i }\) is denoted by \(a\).
  1. Find \(| a |\) and \(\arg a\).
  2. Sketch on a single Argand diagram the loci given by
    (a) \(| z - a | = | a |\),
    (b) \(\arg ( z - 3 ) = \arg a\).
OCR FP1 2008 June Q3
3
  1. Show that \(\frac { 1 } { r ! } - \frac { 1 } { ( r + 1 ) ! } = \frac { r } { ( r + 1 ) ! }\).
  2. Hence find an expression, in terms of \(n\), for $$\frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { n } { ( n + 1 ) ! }$$
OCR FP1 2008 June Q4
4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 1
0 & 1 \end{array} \right)\). Prove by induction that, for \(n \geqslant 1\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & \frac { 1 } { 2 } \left( 3 ^ { n } - 1 \right)
0 & 1 \end{array} \right)$$
OCR FP1 2008 June Q5
5 Find \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 )\), expressing your answer in a fully factorised form.
OCR FP1 2008 June Q6
6 The cubic equation \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are real, has roots ( \(3 + \mathrm { i }\) ) and 2 .
  1. Write down the other root of the equation.
  2. Find the values of \(a , b\) and \(c\).
OCR FP1 2008 June Q7
7 Describe fully the geometrical transformation represented by each of the following matrices:
  1. \(\left( \begin{array} { l l } 6 & 0
    0 & 6 \end{array} \right)\),
  2. \(\left( \begin{array} { l l } 0 & 1
    1 & 0 \end{array} \right)\),
  3. \(\left( \begin{array} { l l } 1 & 0
    0 & 6 \end{array} \right)\),
  4. \(\left( \begin{array} { r r } 0.8 & 0.6
    - 0.6 & 0.8 \end{array} \right)\).
OCR FP1 2008 June Q8
8 The quadratic equation \(x ^ { 2 } + k x + 2 k = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\). Find a quadratic equation with roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\).
OCR FP1 2008 June Q9
9
  1. Use an algebraic method to find the square roots of the complex number \(5 + 12 \mathrm { i }\).
  2. Find \(( 3 - 2 \mathrm { i } ) ^ { 2 }\).
  3. Hence solve the quartic equation \(x ^ { 4 } - 10 x ^ { 2 } + 169 = 0\).
OCR FP1 2008 June Q10
10 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } a & 8 & 10
2 & 1 & 2
4 & 3 & 6 \end{array} \right)\). The matrix \(\mathbf { B }\) is such that \(\mathbf { A B } = \left( \begin{array} { l l l } a & 6 & 1
1 & 1 & 0
1 & 3 & 0 \end{array} \right)\).
  1. Show that \(\mathbf { A B }\) is non-singular.
  2. Find \(( \mathbf { A B } ) ^ { - 1 }\).
  3. Find \(\mathbf { B } ^ { - 1 }\).
OCR FP1 2013 June Q1
1 The complex number \(3 + a \mathrm { i }\), where \(a\) is real, is denoted by \(z\). Given that \(\arg z = \frac { 1 } { 6 } \pi\), find the value of \(a\) and hence find \(| z |\) and \(z ^ { * } - 3\).
OCR FP1 2013 June Q2
2 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 5 & 1 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l } 2 & - 5 \end{array} \right)\) and \(\mathbf { C } = \binom { 3 } { 2 }\).
  1. Find \(3 \mathbf { A } - 4 \mathbf { B }\).
  2. Find CB. Determine whether \(\mathbf { C B }\) is singular or non-singular, giving a reason for your answer.
OCR FP1 2013 June Q3
3 Use an algebraic method to find the square roots of \(11 + ( 12 \sqrt { 5 } ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.
OCR FP1 2013 June Q4
4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 2 & 2
0 & 1 \end{array} \right)\). Prove by induction that, for \(n \geqslant 1\), $$\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 2 ^ { n + 1 } - 2
0 & 1 \end{array} \right) .$$
OCR FP1 2013 June Q5
5 Find \(\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } - 3 r ^ { 2 } + r \right)\), giving your answer in a fully factorised form.
OCR FP1 2013 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{2ba2e0bf-d20a-41ab-a77c-86a08e700b40-2_885_803_1425_630} The Argand diagram above shows a half-line \(l\) and a circle \(C\). The circle has centre 3 i and passes through the origin.
  1. Write down, in complex number form, the equations of \(l\) and \(C\).
    [0pt]
  2. Write down inequalities that define the region shaded in the diagram. [The shaded region includes the boundaries.]
OCR FP1 2013 June Q7
7
  1. Find the matrix that represents a rotation through \(90 ^ { \circ }\) clockwise about the origin.
  2. Find the matrix that represents a reflection in the \(x\)-axis.
  3. Hence find the matrix that represents a rotation through \(90 ^ { \circ }\) clockwise about the origin, followed by a reflection in the \(x\)-axis.
  4. Describe a single transformation that is represented by your answer to part (iii).
OCR FP1 2013 June Q8
8 The cubic equation \(k x ^ { 3 } + 6 x ^ { 2 } + x - 3 = 0\), where \(k\) is a non-zero constant, has roots \(\alpha , \beta\) and \(\gamma\).
Find the value of \(( \alpha + 1 ) ( \beta + 1 ) + ( \beta + 1 ) ( \gamma + 1 ) + ( \gamma + 1 ) ( \alpha + 1 )\) in terms of \(k\).