Questions FP1 (1491 questions)

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Edexcel FP1 Q9
14 marks Standard +0.3
9. (a) A sequence of numbers is defined by $$u _ { 1 } = 3 \text { and } u _ { n + 1 } = 3 u _ { n } + 4 \text { for } n \geqslant 1 .$$ Prove by induction that $$u _ { n } = 3 ^ { n } + 2 \left( 3 ^ { n - 1 } - 1 \right) \text { for } n \in \mathbb { Z } ^ { + } \text {. }$$ (b) $$\mathbf { A } = \left( \begin{array} { l l } 4 & 0 \\ 9 & 1 \end{array} \right)$$
  1. Prove by induction that $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 4 ^ { n } & 0 \\ 3 \left( 4 ^ { n } - 1 \right) & 1 \end{array} \right) \text { for } n \in \mathbb { Z } ^ { + } .$$
  2. Determine whether the result \(\mathbf { A } ^ { n } = \left( \begin{array} { c c } 4 ^ { n } & 0 \\ 3 \left( 4 ^ { n } - 1 \right) & 1 \end{array} \right)\) is also valid for \(n = - 1\).
Edexcel FP1 Specimen Q1
6 marks Moderate -0.8
1. $$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x - 4$$
  1. Use differentiation to find \(\mathrm { f } ^ { \prime } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \(1.4 < x < 1.5\)
  2. Taking 1.4 as a first approximation to \(\alpha\), use the Newton-Raphson procedure once to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel FP1 Specimen Q2
7 marks Moderate -0.5
2. The rectangle \(R\) has vertices at the points \(( 0,0 ) , ( 1,0 ) , ( 1,2 )\) and \(( 0,2 )\).
  1. Find the coordinates of the vertices of the image of \(R\) under the transformation given by the matrix \(\mathbf { A } = \left( \begin{array} { c c } a & 4 \\ - 1 & 1 \end{array} \right)\), where \(a\) is a constant.
  2. Find det \(\mathbf { A }\), giving your answer in terms of \(a\). Given that the area of the image of \(R\) is 18 ,
  3. find the value of \(a\).
Edexcel FP1 Specimen Q3
5 marks Standard +0.3
3. The matrix \(\mathbf { R }\) is given by \(\mathbf { R } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right)\)
  1. Find \(\mathbf { R } ^ { 2 }\).
  2. Describe the geometrical transformation represented by \(\mathbf { R } ^ { 2 }\).
  3. Describe the geometrical transformation represented by \(\mathbf { R }\).
Edexcel FP1 Specimen Q4
3 marks Moderate -0.8
4. $$f ( x ) = 2 ^ { x } - 6 x$$ The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [4,5]. Using the end points of this interval find, by linear interpolation, an approximation to \(\alpha\).
Edexcel FP1 Specimen Q5
9 marks Standard +0.3
5. (a) Show that \(\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 1 \right) = \frac { 1 } { 3 } ( n - 2 ) n ( n + 2 )\).
(b) Hence calculate the value of \(\sum _ { r = 10 } ^ { 40 } \left( r ^ { 2 } - r - 1 \right)\).
Edexcel FP1 Specimen Q6
10 marks Moderate -0.8
6. Given that \(z = - 3 + 4 \mathrm { i }\),
  1. find the modulus of \(z\),
  2. the argument of \(z\) in radians to 2 decimal places. Given also that \(w = \frac { - 14 + 2 \mathrm { i } } { z }\),
  3. use algebra to find \(w\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real. The complex numbers \(z\) and \(w\) are represented by points \(A\) and \(B\) on an Argand diagram.
  4. Show the points \(A\) and \(B\) on an Argand diagram.
Edexcel FP1 Specimen Q7
12 marks Standard +0.8
7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant. The point \(\left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).
  1. Find the value of \(a\).
  2. Show that the equation for the tangent to \(C\) at the point \(\left( 4 t ^ { 2 } , 8 t \right)\) is $$y t = x + 4 t ^ { 2 } .$$ The tangent to \(C\) at the point \(A\) meets the tangent to \(C\) at the point \(B\) on the directrix of \(C\) when \(y = 15\).
  3. Find the coordinates of \(A\) and the coordinates of \(B\).
Edexcel FP1 Specimen Q8
9 marks Moderate -0.3
8. $$\mathrm { f } ( x ) \equiv 2 x ^ { 3 } - 5 x ^ { 2 } + p x - 5 , p \in \mathbb { R }$$ Given that \(1 - 2 \mathrm { i }\) is a complex solution of \(\mathrm { f } ( x ) = 0\),
  1. write down the other complex solution of \(\mathrm { f } ( x ) = 0\),
  2. solve the equation \(\mathrm { f } ( x ) = 0\),
  3. find the value of \(p\).
Edexcel FP1 Specimen Q9
14 marks Standard +0.3
9. Use the method of mathematical induction to prove that, for \(n \in \mathbb { Z } ^ { + }\),
  1. \(\left( \begin{array} { c c } 2 & 1 \\ - 1 & 0 \end{array} \right) ^ { n } = \left( \begin{array} { c c } n + 1 & n \\ - n & 1 - n \end{array} \right)\)
  2. \(\mathrm { f } ( n ) = 4 ^ { n } + 6 n - 1\) is divisible by 3 .
OCR FP1 2006 January Q1
5 marks Moderate -0.8
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
5 marks Standard +0.3
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
4 marks Moderate -0.8
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
5 marks Standard +0.3
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
6 marks Moderate -0.5
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
7 marks Standard +0.3
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
10 marks Moderate -0.8
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
9 marks Standard +0.3
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
10 marks Standard +0.3
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
    1. \(\sum _ { r = 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }\),
    2. \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }\).
OCR FP1 2006 January Q10
11 marks Standard +0.3
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
3 marks Moderate -0.8
\(\mathbf { 1 }\) 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 \(a\).
OCR FP1 2007 January Q2
6 marks Moderate -0.3
2 Use an algebraic method to find the square roots of the complex number \(15 + 8 \mathrm { i }\).
OCR FP1 2007 January Q3
6 marks Moderate -0.3
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
6 marks Moderate -0.3
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
7 marks Moderate -0.8
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 \(z ^ { 3 } - 8 = 0\).