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Edexcel F1 2023 June Q8
  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
  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
    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
  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
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
  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
  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
  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 \(S\)
Edexcel F1 2024 June Q7
  1. In this question use the standard results for summations.
    1. Show that for all positive integers \(n\)
    $$\sum _ { r = 1 } ^ { n } \left( 12 r ^ { 2 } + 2 r - 3 \right) = A n ^ { 3 } + B n ^ { 2 }$$ where \(A\) and \(B\) are integers to be determined.
  2. Hence determine the value of \(n\) for which $$\sum _ { r = 1 } ^ { 2 n } r ^ { 3 } - \sum _ { r = 1 } ^ { n } \left( 12 r ^ { 2 } + 2 r - 3 \right) = 270$$
Edexcel F1 2024 June Q8
  1. Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)
$$f ( n ) = 7 ^ { n - 1 } + 8 ^ { 2 n + 1 }$$ is divisible by 57
(6)
Edexcel F1 2024 June Q9
  1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.
The point \(P \left( c t , \frac { c } { t } \right)\), where \(t > 0\), lies on \(H\)
  1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$t ^ { 3 } x - t y = c \left( t ^ { 4 } - 1 \right)$$ The parabola \(C\) has equation \(y ^ { 2 } = 6 x\)
    The normal to \(H\) at the point with coordinates \(( 8,2 )\) meets \(C\) at the point \(Q\) where \(y > 0\)
  2. Determine the exact coordinates of \(Q\) Given that
    • the point \(R\) is the focus of \(C\)
    • the line \(l\) is the directrix of \(C\)
    • the line through \(Q\) and \(R\) meets \(l\) at the point \(S\)
    • determine the exact length of \(Q S\)
Edexcel F1 2021 October Q1
1. $$\mathbf { A } = \left( \begin{array} { r r } 3 & a
- 2 & - 2 \end{array} \right)$$ where \(a\) is a non-zero constant and \(a \neq 3\)
  1. Determine \(\mathbf { A } ^ { - 1 }\) giving your answer in terms of \(a\). Given that \(\mathbf { A } + \mathbf { A } ^ { - 1 } = \mathbf { I }\) where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
  2. determine the value of \(a\).
Edexcel F1 2021 October Q2
2. $$f ( x ) = 7 \sqrt { x } - \frac { 1 } { 2 } x ^ { 3 } - \frac { 5 } { 3 x } \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), in the interval [2.8, 2.9]
    1. Find \(\mathrm { f } ^ { \prime } ( x )\).
    2. Hence, using \(x _ { 0 } = 2.8\) as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to calculate a second approximation to \(\alpha\), giving your answer to 3 decimal places.
      [0pt]
  3. Use linear interpolation once on the interval [2.8, 2.9] to find another approximation to \(\alpha\). Give your answer to 3 decimal places.
    VIIN SIHILNI III IM ION OCVIAV SIHI NI III HM ION OOVIAV SIHI NI III IM I ON OC
Edexcel F1 2021 October Q3
3. The quadratic equation $$2 x ^ { 2 } - 5 x + 7 = 0$$ 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, giving each answer as a simplified fraction, the value of
    1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
    2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  3. find a quadratic equation that has roots $$\frac { 1 } { \alpha ^ { 2 } + \beta } \text { and } \frac { 1 } { \beta ^ { 2 } + \alpha }$$ 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 2021 October Q4
4. $$f ( z ) = 2 z ^ { 3 } - z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are integers. The complex number \(- 1 - 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. Write down another complex root of this equation.
  2. Determine the value of \(a\) and the value of \(b\).
  3. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
    VIIN SIHILNI III IM ION OCVIAV SIHI NI III HM ION OOVIAV SIHI NI III IM I ON OC
Edexcel F1 2021 October Q5
5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 } , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( r - 1 ) ( r - 3 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n - 1 ) ( 3 n - 10 )$$ (b) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n + 1 } r ( r - 1 ) ( r - 3 ) = \frac { 1 } { 12 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
Edexcel F1 2021 October Q6
6. The curve \(H\) has equation $$x y = a ^ { 2 } \quad x > 0$$ where \(a\) is a positive constant. The line with equation \(y = k x\), where \(k\) is a positive constant, intersects \(H\) at the point \(P\)
  1. Use calculus to determine, in terms of \(a\) and \(k\), an equation for the tangent to \(H\) at \(P\) The tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\)
  2. Determine the coordinates of \(A\) and the coordinates of \(B\), giving your answers in terms of \(a\) and \(k\)
  3. Hence show that the area of triangle \(A O B\), where \(O\) is the origin, is independent of \(k\)
Edexcel F1 2021 October Q7
  1. In part (i), the elements of each matrix should be expressed in exact numerical form.
    1. (a) Write down the \(2 \times 2\) matrix that represents a rotation of \(210 ^ { \circ }\) anticlockwise about the origin.
      (b) Write down the \(2 \times 2\) matrix that represents a stretch parallel to the \(y\)-axis with scale factor 5
    The transformation \(T\) is a rotation of \(210 ^ { \circ }\) anticlockwise about the origin followed by a stretch parallel to the \(y\)-axis with scale factor 5
    (c) Determine the \(2 \times 2\) matrix that represents \(T\)
  2. $$\mathbf { M } = \left( \begin{array} { r r } k & k + 3
    - 5 & 1 - k \end{array} \right) \quad \text { where } k \text { is a constant }$$ (a) Find det \(\mathbf { M }\), giving your answer in simplest form in terms of \(k\). A closed shape \(R\) is transformed to a closed shape \(R ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { M }\). Given that the area of \(R\) is 2 square units and that the area of \(R ^ { \prime }\) is \(16 k\) square units,
    (b) determine the possible values of \(k\).
Edexcel F1 2021 October Q8
  1. The parabola \(C\) has equation \(y ^ { 2 } = 20 x\)
The point \(P\) on \(C\) has coordinates ( \(5 p ^ { 2 } , 10 p\) ) where \(p\) is a non-zero constant.
  1. Use calculus to show that the tangent to \(C\) at \(P\) has equation $$p y - x = 5 p ^ { 2 }$$ The tangent to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\).
  2. Write down the coordinates of \(A\). The point \(S\) is the focus of \(C\).
  3. Write down the coordinates of \(S\). The straight line \(l _ { 1 }\) passes through \(A\) and \(S\).
    The straight line \(l _ { 2 }\) passes through \(O\) and \(P\), where \(O\) is the origin. Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(B\),
  4. show that the coordinates of \(B\) satisfy the equation $$2 x ^ { 2 } + y ^ { 2 } = 10 x$$
Edexcel F1 2021 October Q9
9. (i) A sequence of numbers is defined by $$\begin{gathered} u _ { 1 } = 0 \quad u _ { 2 } = - 6
u _ { n + 2 } = 5 u _ { n + 1 } - 6 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$u _ { n } = 3 \times 2 ^ { n } - 2 \times 3 ^ { n }$$ (ii) Prove by induction that, for all positive integers \(n\), $$f ( n ) = 3 ^ { 3 n - 2 } + 2 ^ { 4 n - 1 }$$ is divisible by 11
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Edexcel F1 2018 Specimen Q1
  1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right) = \frac { n } { 4 } ( n + a ) ( n + b ) ( n + c )$$ where \(a\), \(b\) and \(c\) are integers to be found.
Edexcel F1 2018 Specimen Q2
  1. A parabola \(P\) has cartesian equation \(y ^ { 2 } = 28 x\). The point \(S\) is the focus of the parabola \(P\).
    1. Write down the coordinates of the point \(S\).
    Points \(A\) and \(B\) lie on the parabola \(P\). The line \(A B\) is parallel to the directrix of \(P\) and cuts the \(x\)-axis at the midpoint of \(O S\), where \(O\) is the origin.
  2. Find the exact area of triangle \(A B S\).
Edexcel F1 2018 Specimen Q3
3. $$\mathrm { f } ( x ) = x ^ { 2 } + \frac { 3 } { x } - 1 , \quad x < 0$$ The only real root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ - 2 , - 1 ]\).
  1. Taking - 1.5 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 2 decimal places.
  2. Show that your answer to part (a) gives \(\alpha\) correct to 2 decimal places.
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Edexcel F1 2018 Specimen Q4
  1. Given that
$$\mathbf { A } = \left( \begin{array} { c c } k & 3
- 1 & k + 2 \end{array} \right) \text {, where } k \text { is a constant }$$
  1. show that \(\operatorname { det } ( \mathbf { A } ) > 0\) for all real values of \(k\),
  2. find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
Edexcel F1 2018 Specimen Q5
5. $$2 z + z ^ { * } = \frac { 3 + 4 \mathrm { i } } { 7 + \mathrm { i } }$$ Find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. You must show all your working.