Questions FP1 (1385 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 MEI FP1 2008 June Q4
4 Find the values of \(A , B , C\) and \(D\) in the identity \(3 x ^ { 3 } - x ^ { 2 } + 2 \equiv A ( x - 1 ) ^ { 3 } + \left( x ^ { 3 } + B x ^ { 2 } + C x + D \right)\).
OCR MEI FP1 2008 June Q5
5 You are given that \(\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 4
3 & 2 & 5
4 & 1 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r r } - 1 & 0 & 2
14 & - 14 & 7
- 5 & 7 & - 4 \end{array} \right)\).
  1. Calculate AB.
  2. Write down \(\mathbf { A } ^ { - 1 }\).
OCR MEI FP1 2008 June Q6
6 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } - 3 x + 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(2 \alpha , 2 \beta\) and \(2 \gamma\), expressing your answer in a form with integer coefficients.
OCR MEI FP1 2008 June Q7
7
  1. Show that \(\frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 } \equiv \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\) for all integers \(r\).
  2. Hence use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) }\). Section B (36 marks)
OCR MEI FP1 2008 June Q8
8 A curve has equation \(y = \frac { 2 x ^ { 2 } } { ( x - 3 ) ( x + 2 ) }\).
  1. Write down the equations of the three asymptotes.
  2. Determine whether the curve approaches the horizontal asymptote from above or below for
    (A) large positive values of \(x\),
    (B) large negative values of \(x\).
  3. Sketch the curve.
  4. Solve the inequality \(\frac { 2 x ^ { 2 } } { ( x - 3 ) ( x + 2 ) } < 0\).
OCR MEI FP1 2008 June Q9
9 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 2 - 2 \mathrm { j }\) and \(\beta = - 1 + \mathrm { j }\).
\(\alpha\) and \(\beta\) are both roots of a quartic equation \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0\), where \(A , B , C\) and \(D\) are real numbers.
  1. Write down the other two roots.
  2. Represent these four roots on an Argand diagram.
  3. Find the values of \(A , B , C\) and \(D\).
OCR MEI FP1 2008 June Q10
10
  1. Using the standard formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), prove that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + 1 )$$
  2. Prove the same result by mathematical induction.
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 \(q\).
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 $$\begin{array} { r } 2 x - y + z = 1
    3 y + z = 2
    x + y + a z = 2 \end{array}$$