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 2006 January Q1
1
  1. Express \(( 1 + 8 i ) ( 2 - i )\) in the form \(x + i y\), showing clearly how you obtain your answer.
  2. Hence express \(\frac { 1 + 8 i } { 2 + i }\) in the form \(x + i y\).
OCR FP1 2006 January Q2
2 Prove by induction that, for \(n \geqslant 1 , \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\).
OCR FP1 2006 January Q3
3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l l } 2 & 1 & 3
1 & 2 & 1
1 & 1 & 3 \end{array} \right)\).
  1. Find the value of the determinant of \(\mathbf { M }\).
  2. State, giving a brief reason, whether \(\mathbf { M }\) is singular or non-singular.
OCR FP1 2006 January Q4
4 Use the substitution \(x = u + 2\) to find the exact value of the real root of the equation $$x ^ { 3 } - 6 x ^ { 2 } + 12 x - 13 = 0$$
OCR FP1 2006 January Q5
5 Use the standard results for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } - 6 r ^ { 2 } + 2 r \right) = 2 n ^ { 3 } ( n + 1 )$$
OCR FP1 2006 January Q6
6 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 1 & 2
3 & 8 \end{array} \right)\).
  1. Find \(\mathbf { C } ^ { - 1 }\).
  2. Given that \(\mathbf { C } = \mathbf { A B }\), where \(\mathbf { A } = \left( \begin{array} { l l } 2 & 1
    1 & 3 \end{array} \right)\), find \(\mathbf { B } ^ { - 1 }\).
OCR FP1 2006 January Q7
7
  1. The complex number \(3 + 2 \mathrm { i }\) is denoted by \(w\) and the complex conjugate of \(w\) is denoted by \(w ^ { * }\). Find
    1. the modulus of \(w\),
    2. the argument of \(w ^ { * }\), giving your answer in radians, correct to 2 decimal places.
  2. Find the complex number \(u\) given that \(u + 2 u ^ { * } = 3 + 2 \mathrm { i }\).
  3. Sketch, on an Argand diagram, the locus given by \(| z + 1 | = | z |\).
OCR FP1 2006 January Q8
3 marks
8 The matrix \(\mathbf { T }\) is given by \(\mathbf { T } = \left( \begin{array} { r r } 2 & 0
0 & - 2 \end{array} \right)\).
  1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { T }\). [3]
  2. The transformation represented by matrix \(\mathbf { T }\) is equivalent to a transformation \(A\), followed by a transformation B. Give geometrical descriptions of possible transformations A and B, and state the matrices that represent them.
OCR FP1 2006 January Q9
9
  1. Show that \(\frac { 1 } { r } - \frac { 1 } { r + 2 } = \frac { 2 } { r ( r + 2 ) }\).
  2. Hence find an expression, in terms of \(n\), for $$\frac { 2 } { 1 \times 3 } + \frac { 2 } { 2 \times 4 } + \ldots + \frac { 2 } { n ( n + 2 ) }$$
  3. Hence find the value of
    (a) \(\sum _ { r = 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }\),
    (b) \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }\).
OCR FP1 2006 January Q10
10 The roots of the equation $$x ^ { 3 } - 9 x ^ { 2 } + 27 x - 29 = 0$$ are denoted by \(\alpha , \beta\) and \(\gamma\), where \(\alpha\) is real and \(\beta\) and \(\gamma\) are complex.
  1. Write down the value of \(\alpha + \beta + \gamma\).
  2. It is given that \(\beta = p + \mathrm { i } q\), where \(q > 0\). Find the value of \(p\), in terms of \(\alpha\).
  3. Write down the value of \(\alpha \beta \gamma\).
  4. Find the value of \(q\), in terms of \(\alpha\) only.
OCR FP1 2007 January Q1
\(\mathbf { 1 }\) The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 1
3 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } a & - 1
- 3 & - 2 \end{array} \right)\).
  1. Given that \(2 \mathbf { A } + \mathbf { B } = \left( \begin{array} { l l } 1 & 1
    3 & 2 \end{array} \right)\), write down the value of \(a\).
  2. Given instead that \(\mathbf { A B } = \left( \begin{array} { l l } 7 & - 4
    9 & - 7 \end{array} \right)\), find the value of \(a\). \end{itemize}
OCR FP1 2007 January Q2
2 Use an algebraic method to find the square roots of the complex number \(15 + 8 \mathrm { i }\).
OCR FP1 2007 January Q3
3 Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to find $$\sum _ { r = 1 } ^ { n } r ( r - 1 ) ( r + 1 ) ,$$ expressing your answer in a fully factorised form.
OCR FP1 2007 January Q4
4
  1. Sketch, on an Argand diagram, the locus given by \(| z - 1 + \mathrm { i } | = \sqrt { 2 }\).
  2. Shade on your diagram the region given by \(1 \leqslant | z - 1 + \mathrm { i } | \leqslant \sqrt { 2 }\).
OCR FP1 2007 January Q5
5
  1. Verify that \(z ^ { 3 } - 8 = ( z - 2 ) \left( z ^ { 2 } + 2 z + 4 \right)\).
  2. Solve the quadratic equation \(z ^ { 2 } + 2 z + 4 = 0\), giving your answers exactly in the form \(x + \mathrm { i } y\). Show clearly how you obtain your answers.
  3. Show on an Argand diagram the roots of the cubic equation \(z ^ { 3 } - 8 = 0\).
OCR FP1 2007 January Q6
6 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = n ^ { 2 } + 3 n\), for all positive integers \(n\).
  1. Show that \(u _ { n + 1 } - u _ { n } = 2 n + 4\).
  2. Hence prove by induction that each term of the sequence is divisible by 2 .
OCR FP1 2007 January Q7
7 The quadratic equation \(x ^ { 2 } + 5 x + 10 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = 5\).
  3. Hence find a quadratic equation which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\).
OCR FP1 2007 January Q8
8
  1. Show that \(( r + 2 ) ! - ( r + 1 ) ! = ( r + 1 ) ^ { 2 } \times r !\).
  2. Hence find an expression, in terms of \(n\), for $$2 ^ { 2 } \times 1 ! + 3 ^ { 2 } \times 2 ! + 4 ^ { 2 } \times 3 ! + \ldots + ( n + 1 ) ^ { 2 } \times n ! .$$
  3. State, giving a brief reason, whether the series $$2 ^ { 2 } \times 1 ! + 3 ^ { 2 } \times 2 ! + 4 ^ { 2 } \times 3 ! + \ldots$$ converges.
OCR FP1 2007 January Q9
9 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r } 0 & 3
- 1 & 0 \end{array} \right)\).
  1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\). The transformation represented by \(\mathbf { C }\) is equivalent to a rotation, R , followed by another transformation, S.
  2. Describe fully the rotation R and write down the matrix that represents R .
  3. Describe fully the transformation S and write down the matrix that represents S .
OCR FP1 2007 January Q10
10 The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { r r r } a & 2 & 0
3 & 1 & 2
0 & - 1 & 1 \end{array} \right)\), where \(a \neq 2\).
  1. Find \(\mathbf { D } ^ { - 1 }\).
  2. Hence, or otherwise, solve the equations $$\begin{aligned} a x + 2 y & = 3
    3 x + y + 2 z & = 4
    - y + z & = 1 \end{aligned}$$
OCR FP1 2008 January Q1
1 The transformation S is a shear with the \(y\)-axis invariant (i.e. a shear parallel to the \(y\)-axis). It is given that the image of the point \(( 1,1 )\) is the point \(( 1,0 )\).
  1. Draw a diagram showing the image of the unit square under the transformation S .
  2. Write down the matrix that represents S .
OCR FP1 2008 January Q2
2 Given that \(\sum _ { r = 1 } ^ { n } \left( a r ^ { 2 } + b \right) \equiv n \left( 2 n ^ { 2 } + 3 n - 2 \right)\), find the values of the constants \(a\) and \(b\).
OCR FP1 2008 January Q3
3 The cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + 24 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Use the substitution \(x = \frac { 1 } { u }\) to find a cubic equation in \(u\) with integer coefficients.
  2. Hence, or otherwise, find the value of \(\frac { 1 } { \alpha \beta } + \frac { 1 } { \beta \gamma } + \frac { 1 } { \gamma \alpha }\).
OCR FP1 2008 January Q4
4 The complex number \(3 - 4 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find
  1. \(2 z + 5 z ^ { * }\),
  2. \(( z - \mathrm { i } ) ^ { 2 }\),
  3. \(\frac { 3 } { z }\).
OCR FP1 2008 January Q5
5 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l } 3
1
2 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l } 4
0
3 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { l l l } 2 & 4 & - 1 \end{array} \right)\). Find
  1. \(\mathbf { A } - 4 \mathbf { B }\),
  2. BC ,
  3. CA .