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 2010 January Q7
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
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
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
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
\(\mathbf { 1 }\) 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
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
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
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
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
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
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 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
10
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    10
  2. ACHIEVEMENT
OCR FP1 2012 January Q1
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
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
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
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
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 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
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
  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
\(\mathbf { 9 }\) 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
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 2013 January Q1
1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } a & 1
1 & 4 \end{array} \right)\), where \(a \neq \frac { 1 } { 4 }\), and \(\mathbf { I }\) denotes the \(2 \times 2\) identity matrix. Find
  1. \(2 \mathbf { A } - 3 \mathbf { I }\),
  2. \(\mathrm { A } ^ { - 1 }\).
OCR FP1 2013 January Q2
2 Find \(\sum _ { r = 1 } ^ { n } ( r - 1 ) ( r + 1 )\), giving your answer in a fully factorised form.