Questions — Edexcel F1 (198 questions)

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Edexcel F1 2020 June Q8
12 marks Standard +0.8
  1. (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\)
$$\sum _ { r = 1 } ^ { n } \frac { 2 r ^ { 2 } - 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { n ^ { 2 } } { ( n + 1 ) ^ { 2 } }$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 12 ^ { n } + 2 \times 5 ^ { n - 1 }$$ is divisible by 7
VILU SIHI NI JIIIM ION OCVIUV SIHI NI III M M I ON OOVIAV SIHI NI JIIIM I ION OC
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VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
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END
Edexcel F1 2022 June Q1
6 marks Moderate -0.8
1. $$z _ { 1 } = 3 + 3 i \quad z _ { 2 } = p + q i \quad p , q \in \mathbb { R }$$ Given that \(\left| z _ { 1 } z _ { 2 } \right| = 15 \sqrt { 2 }\)
  1. determine \(\left| z _ { 2 } \right|\) Given also that \(p = - 4\)
  2. determine the possible values of \(q\)
  3. Show \(z _ { 1 }\) and the possible positions for \(z _ { 2 }\) on the same Argand diagram.
Edexcel F1 2022 June Q2
9 marks Standard +0.3
2. $$f ( x ) = 10 - 2 x - \frac { 1 } { 2 \sqrt { x } } - \frac { 1 } { x ^ { 3 } } \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [0.4, 0.5]
  2. Determine \(\mathrm { f } ^ { \prime } ( x )\).
  3. Using \(x _ { 0 } = 0.5\) as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 3 decimal places. The equation \(\mathrm { f } ( x ) = 0\) has another root \(\beta\) in the interval [4.8, 4.9]
    [0pt]
  4. Use linear interpolation once on the interval [4.8, 4.9] to find an approximation to \(\beta\), giving your answer to 3 decimal places.
Edexcel F1 2022 June Q3
4 marks Moderate -0.3
  1. \(\mathbf { M } = \left( \begin{array} { c c } k & k \\ 3 & 5 \end{array} \right) \quad\) where \(k\) is a non-zero constant
    1. Determine \(\mathbf { M } ^ { - 1 }\), giving your answer in simplest form in terms of \(k\).
    Hence, given that \(\mathbf { N } ^ { - 1 } = \left( \begin{array} { c c } k & k \\ 4 & - 1 \end{array} \right)\)
  2. determine \(( \mathbf { M N } ) ^ { - 1 }\), giving your answer in simplest form in terms of \(k\).
Edexcel F1 2022 June Q4
8 marks Standard +0.3
4. $$f ( z ) = 2 z ^ { 4 } - 19 z ^ { 3 } + A z ^ { 2 } + B z - 156$$ where \(A\) and \(B\) are constants.
The complex number \(5 - \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. Write down another complex root of this equation.
  2. Solve the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) completely.
  3. Determine the value of \(A\) and the value of \(B\).
Edexcel F1 2022 June Q5
10 marks Standard +0.8
  1. The quadratic equation
$$2 x ^ { 2 } - 3 x + 5 = 0$$ has roots \(\alpha\) and \(\beta\) Without solving the equation,
  1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
  2. determine the value of
    1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
    2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  3. find a quadratic equation which has roots $$\left( \alpha ^ { 3 } - \beta \right) \text { and } \left( \beta ^ { 3 } - \alpha \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers to be determined.
Edexcel F1 2022 June Q6
11 marks Challenging +1.3
  1. The parabola \(C\) has equation \(y ^ { 2 } = 36 x\)
The point \(P \left( 9 t ^ { 2 } , 18 t \right)\), where \(t \neq 0\), lies on \(C\)
  1. Use calculus to show that the normal to \(C\) at \(P\) has equation $$y + t x = 9 t ^ { 3 } + 18 t$$
  2. Hence find the equations of the two normals to \(C\) which pass through the point (54, 0), giving your answers in the form \(y = p x + q\) where \(p\) and \(q\) are constants to be determined. Given that
    • the normals found in part (b) intersect the directrix of \(C\) at the points \(A\) and \(B\)
    • the point \(F\) is the focus of \(C\)
    • determine the area of triangle \(A F B\)
Edexcel F1 2022 June Q7
9 marks Standard +0.3
7. $$A = \left( \begin{array} { c c } - \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$
  1. Determine the matrix \(\mathbf { A } ^ { 2 }\)
  2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\)
  3. Hence determine the smallest positive integer value of \(n\) for which \(\mathbf { A } ^ { n } = \mathbf { I }\) The matrix \(\mathbf { B }\) represents a stretch scale factor 4 parallel to the \(x\)-axis.
  4. Write down the matrix \(\mathbf { B }\) The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { B }\) is represented by the matrix \(\mathbf { C }\)
  5. Determine the matrix \(\mathbf { C }\) The parallelogram \(P\) is transformed onto the parallelogram \(P ^ { \prime }\) by the matrix \(\mathbf { C }\)
  6. Given that the area of parallelogram \(P ^ { \prime }\) is 20 square units, determine the area of parallelogram \(P\)
Edexcel F1 2022 June Q8
8 marks Standard +0.3
  1. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that for all positive integers \(n\)
$$\sum _ { r = 0 } ^ { n } ( r + 1 ) ( r + 2 ) = \frac { 1 } { 3 } ( n + 1 ) ( n + 2 ) ( n + 3 )$$ (b) Hence determine the value of $$10 \times 11 + 11 \times 12 + 12 \times 13 + \ldots + 100 \times 101$$
Edexcel F1 2022 June Q9
10 marks Standard +0.8
  1. (i) A sequence of numbers is defined by
$$\begin{gathered} u _ { 1 } = 3 \\ u _ { n + 1 } = 2 u _ { n } - 2 ^ { n + 1 } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { N }\) $$u _ { n } = 5 \times 2 ^ { n - 1 } - n \times 2 ^ { n }$$ (ii) Prove by induction that, for \(n \in \mathbb { N }\) $$f ( n ) = 5 ^ { n + 2 } - 4 n - 9$$ is divisible by 16
Edexcel F1 2023 June Q1
4 marks Standard +0.3
  1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\)
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 2 ) = \frac { 1 } { 12 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
Edexcel F1 2023 June Q2
7 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
Given that \(x = 2 + 3 \mathrm { i }\) is a root of the equation $$2 x ^ { 4 } - 8 x ^ { 3 } + 29 x ^ { 2 } - 12 x + 39 = 0$$
  1. write down another complex root of this equation.
  2. Use algebra to determine the other 2 roots of the equation.
  3. Show all 4 roots on a single Argand diagram.
Edexcel F1 2023 June Q3
7 marks Challenging +1.2
  1. The rectangular hyperbola \(H\) has Cartesian equation \(x y = 9\)
The point \(P\) with coordinates \(\left( 3 t , \frac { 3 } { t } \right)\), where \(t \neq 0\), lies on \(H\)
  1. Use calculus to determine an equation for the normal to \(H\) at the point \(P\) Give your answer in the form \(t y - t ^ { 3 } x = \mathrm { f } ( t )\) Given that \(t = 2\)
  2. determine the coordinates of the point where the normal meets \(H\) again. Give your answer in simplest form.
Edexcel F1 2023 June Q4
8 marks Standard +0.3
  1. (i) \(\mathbf { A } = \left( \begin{array} { c c } - 3 & 8 \\ - 3 & k \end{array} \right) \quad\) where \(k\) is a constant The transformation represented by \(\mathbf { A }\) transforms triangle \(T\) to triangle \(T ^ { \prime }\) The area of triangle \(T ^ { \prime }\) is three times the area of triangle \(T\)
Determine the possible values of \(k\) (ii) \(\mathbf { B } = \left( \begin{array} { r r } a & - 4 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { B C } = \left( \begin{array} { l l l } 2 & 5 & 1 \\ 1 & 4 & 2 \end{array} \right)\) where \(a\) is a constant Determine, in terms of \(a\), the matrix \(\mathbf { C }\)
Edexcel F1 2023 June Q5
10 marks Standard +0.8
5. $$f ( x ) = x ^ { 2 } - 6 x + 3$$ The equation \(\mathrm { f } ( x ) = 0\) has roots \(\alpha\) and \(\beta\) Without solving the equation,
  1. determine the value of $$\left( \alpha ^ { 2 } + 1 \right) \left( \beta ^ { 2 } + 1 \right)$$
  2. find a quadratic equation which has roots $$\frac { \alpha } { \left( \alpha ^ { 2 } + 1 \right) } \text { and } \frac { \beta } { \left( \beta ^ { 2 } + 1 \right) }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers to be determined.
Edexcel F1 2023 June Q6
10 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
$$z _ { 1 } = 3 + 2 i \quad z _ { 2 } = 2 + 3 i \quad z _ { 3 } = a + b i \quad a , b \in \mathbb { R }$$
  1. Determine the exact value of \(\left| z _ { 1 } + z _ { 2 } \right|\) Given that \(w = \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }\)
  2. determine \(w\) in terms of \(a\) and \(b\), giving your answer in the form \(x + \mathrm { i } y\), where \(x , y \in \mathbb { R }\) Given also that \(w = \frac { 4 } { 13 } + \frac { 58 } { 13 } \mathrm { i }\)
  3. determine the value of \(a\) and the value of \(b\)
  4. determine arg \(w\), giving your answer in radians to 4 significant figures.
Edexcel F1 2023 June Q7
11 marks Standard +0.3
7. $$f ( x ) = x ^ { \frac { 3 } { 2 } } + x - 3$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), in the interval \([ 1,2 ]\) [0pt]
  2. Starting with the interval [1, 2], use interval bisection twice to show that \(\alpha\) lies in the interval [1.25, 1.5]
    1. Determine \(\mathrm { f } ^ { \prime } ( x )\)
    2. Using 1.375 as a first approximation for \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to determine a second approximation for \(\alpha\), giving your answer to 3 decimal places.
      [0pt]
  4. Use linear interpolation once on the interval [1.25,1.5] to obtain a different approximation for \(\alpha\), giving your answer to 3 decimal places.
Edexcel F1 2023 June Q8
13 marks Challenging +1.2
  1. The point \(P \left( 2 p ^ { 2 } , 4 p \right)\) lies on the parabola with equation \(y ^ { 2 } = 8 x\)
    1. Show that the point \(Q \left( \frac { 2 } { p ^ { 2 } } , \frac { - 4 } { p } \right)\), where \(p \neq 0\), lies on the parabola.
    2. Show that the chord \(P Q\) passes through the focus of the parabola.
    The tangent to the parabola at \(P\) and the tangent to the parabola at \(Q\) meet at the point \(R\)
  2. Determine, in simplest form, the coordinates of \(R\)
Edexcel F1 2023 June Q9
5 marks Standard +0.3
  1. Prove, by induction, that for \(n \in \mathbb { Z } , n \geqslant 2\)
$$4 ^ { n } + 6 n - 10$$ is divisible by 18
Edexcel F1 2024 June Q1
6 marks Moderate -0.8
    1. The matrix \(\mathbf { A }\) is defined by
$$\mathbf { A } = \left( \begin{array} { c c } 3 k & 4 k - 1 \\ 2 & 6 \end{array} \right)$$ where \(k\) is a constant.
  1. Determine the value of \(k\) for which \(\mathbf { A }\) is singular. Given that \(\mathbf { A }\) is non-singular,
  2. determine \(\mathbf { A } ^ { - 1 }\) in terms of \(k\), giving your answer in simplest form.
    (ii) The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \left( \begin{array} { l l } p & 0 \\ 0 & q \end{array} \right)$$ where \(p\) and \(q\) are integers.
    State the value of \(p\) and the value of \(q\) when \(\mathbf { B }\) represents
  3. an enlargement about the origin with scale factor - 2
  4. a reflection in the \(y\)-axis.
Edexcel F1 2024 June Q2
9 marks Standard +0.3
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable. $$\mathrm { f } ( z ) = z ^ { 3 } - 13 z ^ { 2 } + 59 z + p \quad p \in \mathbb { Z }$$ Given that \(z = 3\) is a root of the equation \(f ( z ) = 0\)
  1. show that \(p = - 87\)
  2. Use algebra to determine the other roots of \(\mathrm { f } ( \mathrm { z } ) = 0\), giving your answers in simplest form. On an Argand diagram
    • the root \(z = 3\) is represented by the point \(P\)
    • the other roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) are represented by the points \(Q\) and \(R\)
    • the number \(z = - 9\) is represented by the point \(S\)
    • Show on a single Argand diagram the positions of \(P , Q , R\) and \(S\)
    • Determine the perimeter of the quadrilateral \(P Q S R\), giving your answer as a simplified surd.
Edexcel F1 2024 June Q3
7 marks Standard +0.3
3. $$\mathrm { f } ( x ) = x ^ { 3 } - 5 \sqrt { x } - 4 x + 7 \quad x \geqslant 0$$ The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 0.25,1 ]\)
  1. Use linear interpolation once on the interval [ \(0.25,1\) ] to determine an approximation to \(\alpha\), giving your answer to 3 decimal places. The equation \(\mathrm { f } ( x ) = 0\) has another root \(\beta\) in the interval [1.5, 2.5]
  2. Determine \(\mathrm { f } ^ { \prime } ( x )\)
  3. Hence, using \(x _ { 0 } = 1.75\) as a first approximation to \(\beta\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to determine a second approximation to \(\beta\), giving your answer to 3 decimal places.
Edexcel F1 2024 June Q4
8 marks Standard +0.3
  1. In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.} The complex number \(z\) is defined by $$2 = - 3 + 4 i$$
  1. Determine \(\left| z ^ { 2 } - 3 \right|\)
  2. Express \(\frac { 50 } { z ^ { * } }\) in the form \(k z\), where \(k\) is a positive integer.
  3. Hence find the value of \(\arg \left( \frac { 50 } { z ^ { * } } \right)\) Give your answer in radians to 3 significant figures.
Edexcel F1 2024 June Q5
9 marks Challenging +1.2
  1. The equation \(5 x ^ { 2 } - 4 x + 2 = 0\) has roots \(\frac { 1 } { p }\) and \(\frac { 1 } { q }\)
    1. Without solving the equation,
      1. show that \(p q = \frac { 5 } { 2 }\)
      2. determine the value of \(p + q\)
    2. Hence, without finding the values of \(p\) and \(q\), determine a quadratic equation with roots
    $$\frac { p } { p ^ { 2 } + 1 } \text { and } \frac { q } { q ^ { 2 } + 1 }$$ giving your answer in the form \(a x ^ { 2 } + b x + c = 0\) where \(a , b\) and \(c\) are integers.
Edexcel F1 2024 June Q6
9 marks Standard +0.3
  1. (a) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)
$$\left( \begin{array} { l l } 1 & r \\ 0 & 2 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & \left( 2 ^ { n } - 1 \right) r \\ 0 & 2 ^ { n } \end{array} \right)$$ where \(r\) is a constant. $$\mathbf { M } = \left( \begin{array} { l l } 4 & 0 \\ 0 & 5 \end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r } 1 & - 2 \\ 0 & 2 \end{array} \right) ^ { 4 }$$ The transformation represented by matrix \(\mathbf { M }\) followed by the transformation represented by matrix \(\mathbf { N }\) is represented by the matrix \(\mathbf { B }\) (b) (i) Determine \(\mathbf { N }\) in the form \(\left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\) where \(a , b , c\) and \(d\) are integers.
(ii) Determine B Hexagon \(S\) is transformed onto hexagon \(S ^ { \prime }\) by matrix \(\mathbf { B }\) (c) Given that the area of \(S ^ { \prime }\) is 720 square units, determine the area of \(S\)