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Edexcel F1 2016 January Q9
9. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21
Edexcel F1 2017 January Q1
\(\mathrm { f } ( x ) = 2 ^ { x } - 10 \sin x - 2\), where \(x\) is measured in radians
  1. Show that \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), between 2 and 3
    [0pt]
  2. Use linear interpolation once on the interval [2,3] to find an approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel F1 2017 January Q3
3. $$f ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 26 x ^ { 2 } + 32 x + 160$$ Given that \(x = - 1 + 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( x ) = 0\), use algebra to find the three other roots of \(\mathrm { f } ( x ) = 0\)
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel F1 2017 January Q4
4. (a) 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 \(n\), $$\sum _ { r = 1 } ^ { n } r ( 2 r + 1 ) ( 3 r + 1 ) = \frac { 1 } { 6 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
(b) Hence find the value of $$\sum _ { r = 10 } ^ { 20 } r ( 2 r + 1 ) ( 3 r + 1 )$$
Edexcel F1 2017 January Q5
  1. The complex number \(z\) is given by
$$z = - 7 + 3 i$$ Find
  1. \(| z |\)
  2. \(\arg z\), giving your answer in radians to 2 decimal places. Given that \(\frac { z } { 1 + \mathrm { i } } + w = 3 - 6 \mathrm { i }\)
  3. find the complex number \(w\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. You must show all your working.
  4. Show the points representing \(z\) and \(w\) on a single Argand diagram.
Edexcel F1 2017 January Q6
6. $$f ( x ) = x ^ { 3 } - \frac { 1 } { 2 x } + x ^ { \frac { 3 } { 2 } } , \quad x > 0$$ The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [0.6, 0.7].
  1. Taking 0.6 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
  2. Show that your answer to part (a) is correct to 3 decimal places.
Edexcel F1 2017 January Q7
7. (i) $$\mathbf { A } = \left( \begin{array} { r r } - 1 & 0
0 & 1 \end{array} \right)$$
  1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents a stretch, scale factor 3 , parallel to the \(x\)-axis.
  2. Find the matrix \(\mathbf { B }\).
    (ii) $$\mathbf { M } = \left( \begin{array} { r r } - 4 & 3
    - 3 & - 4 \end{array} \right)$$ The matrix \(\mathbf { M }\) represents an enlargement with scale factor \(k\) and centre ( 0,0 ), where \(k > 0\), followed by a rotation anticlockwise through an angle \(\theta\) about ( 0,0 ).
  3. Find the value of \(k\).
  4. Find the value of \(\theta\), giving your answer in radians to 2 decimal places.
  5. Find \(\mathbf { M } ^ { - 1 }\)
Edexcel F1 2017 January Q8
8. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(P \left( a t ^ { 2 } , 2 a t \right)\) lies on \(C\).
  1. Using calculus, show that the normal to \(C\) at \(P\) has equation $$y + t x = a t ^ { 3 } + 2 a t$$ The point \(S\) is the focus of the parabola \(C\).
    The point \(B\) lies on the positive \(x\)-axis and \(O B = 5 O S\), where \(O\) is the origin.
  2. Write down, in terms of \(a\), the coordinates of the point \(B\). A circle has centre \(B\) and touches the parabola \(C\) at two distinct points \(Q\) and \(R\). Given that \(t \neq 0\),
  3. find the coordinates of the points \(Q\) and \(R\).
  4. Hence find, in terms of \(a\), the area of triangle \(B Q R\).
Edexcel F1 2017 January Q9
  1. (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\)
$$\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } - 3 r ^ { 2 } + r \right) = n ^ { 3 } ( n + 1 )$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 5 ^ { 2 n } + 3 n - 1$$ is divisible by 9
Edexcel F1 2018 January Q1
1. $$f ( x ) = 3 x ^ { 2 } - \frac { 5 } { 3 \sqrt { x } } - 6 , \quad x > 0$$ The single root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [1.5, 1.6].
  1. Taking 1.5 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
    [0pt]
  2. Use linear interpolation once on the interval [1.5, 1.6] to find another approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel F1 2018 January Q2
2. $$f ( z ) = z ^ { 4 } - 6 z ^ { 3 } + 38 z ^ { 2 } - 94 z + 221$$
  1. Given that \(z = 2 + 3 i\) is a root of the equation \(f ( z ) = 0\), use algebra to find the three other roots of \(f ( z ) = 0\)
  2. Show the four roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
Edexcel F1 2018 January Q3
  1. (a) 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 \(n\),
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be determined.
(b) Given that $$\sum _ { r = 5 } ^ { 25 } r ^ { 2 } ( r + 1 ) + \sum _ { r = 1 } ^ { k } 3 ^ { r } = 140543$$ find the value of the integer \(k\).
Edexcel F1 2018 January Q4
  1. The quadratic equation
$$3 x ^ { 2 } + 2 x + 5 = 0$$ has roots \(\alpha\) and \(\beta\). Without solving the equation,
  1. find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
  2. show that \(\alpha ^ { 3 } + \beta ^ { 3 } = \frac { 82 } { 27 }\)
  3. find a quadratic equation which has roots $$\left( \alpha + \frac { \alpha } { \beta ^ { 2 } } \right) \text { and } \left( \beta + \frac { \beta } { \alpha ^ { 2 } } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers.
Edexcel F1 2018 January Q5
5. (i) Given that $$\frac { 2 z + 3 } { z + 5 - 2 i } = 1 + i$$ find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants.
(ii) Given that $$w = ( 3 + \lambda \mathrm { i } ) ( 2 + \mathrm { i } )$$ where \(\lambda\) is a real constant, and that $$| w | = 15$$ find the possible values of \(\lambda\).
Edexcel F1 2018 January Q6
  1. The parabola \(C\) has equation \(y ^ { 2 } = 32 x\) and the point \(S\) is the focus of this parabola. The point \(P ( 2,8 )\) lies on \(C\) and the point \(T\) lies on the directrix of \(C\). The line segment \(P T\) is parallel to the \(x\)-axis.
    1. Write down the coordinates of \(S\).
    2. Find the length of \(P T\).
    3. Using calculus, show that the tangent to \(C\) at the point \(P\) has equation
    $$y = 2 x + 4$$ The hyperbola \(H\) has equation \(x y = 4\). The tangent to \(C\) at \(P\) meets \(H\) at the points \(L\) and \(M\).
  2. Find the exact coordinates of the points \(L\) and \(M\), giving your answers in their simplest form.
Edexcel F1 2018 January Q7
7. (i) $$\mathbf { A } = \left( \begin{array} { r r } 6 & k
- 3 & - 4 \end{array} \right) , \text { where } k \text { is a real constant, } k \neq 8$$ Find, in terms of \(k\),
  1. \(\mathbf { A } ^ { - 1 }\)
  2. \(\mathbf { A } ^ { 2 }\) Given that \(\mathbf { A } ^ { 2 } + 3 \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 5 & 9
    - 3 & - 5 \end{array} \right)\)
  3. find the value of \(k\).
    (ii) $$\mathbf { M } = \left( \begin{array} { c c } - \frac { 1 } { 2 } & - \sqrt { 3 }
    \frac { \sqrt { 3 } } { 2 } & - 1 \end{array} \right)$$ The matrix \(\mathbf { M }\) represents a one way stretch, parallel to the \(y\)-axis, scale factor \(p\), where \(p > 0\), followed by a rotation anticlockwise through an angle \(\theta\) about \(( 0,0 )\).
  4. Find the value of \(p\).
  5. Find the value of \(\theta\).
Edexcel F1 2018 January Q8
8. (i) A sequence of numbers is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = u _ { n } + 3 n - 2 \quad n \geqslant 1 \end{aligned}$$ Prove by induction that, for all positive integers \(n\), $$u _ { n } = \frac { 3 } { 2 } n ^ { 2 } - \frac { 7 } { 2 } n + 5$$ (ii) Prove by induction that, for all positive integers \(n\), $$f ( n ) = 3 ^ { 2 n + 3 } + 40 n - 27 \text { is divisible by } 64$$
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Edexcel F1 2021 January Q1
  1. (a) Show that the equation \(4 x - 2 \sin x - 1 = 0\), where \(x\) is in radians, has a root \(\alpha\) in the interval [0.2, 0.6]
    [0pt] (b) Starting with the interval [0.2, 0.6], use interval bisection twice to find an interval of width 0.1 in which \(\alpha\) lies.
    (3)
Edexcel F1 2021 January Q2
  1. Given that \(x = \frac { 3 } { 8 } + \frac { \sqrt { 71 } } { 8 } \mathrm { i }\) is a root of the equation
$$4 x ^ { 3 } - 19 x ^ { 2 } + p x + q = 0$$
  1. write down the other complex root of the equation. Given that \(x = 4\) is also a root of the equation,
  2. find the value of \(p\) and the value of \(q\).
Edexcel F1 2021 January Q3
3. The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left( \begin{array} { c c } k + 5 & - 2
- 3 & k \end{array} \right)$$
  1. Determine the values of \(k\) for which \(\mathbf { M }\) is singular. Given that \(\mathbf { M }\) is non-singular,
  2. find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\).
Edexcel F1 2021 January Q4
  1. The equation \(2 x ^ { 2 } + 5 x + 7 = 0\) has roots \(\alpha\) and \(\beta\)
Without solving the equation
  1. determine the exact value of \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  2. form a quadratic equation, with integer coefficients, which has roots $$\frac { \alpha ^ { 2 } } { \beta } \text { and } \frac { \beta ^ { 2 } } { \alpha }$$ \includegraphics[max width=\textwidth, alt={}, center]{f8660b02-384e-460f-a0e4-282ef5fef475-09_2255_50_314_34}
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Edexcel F1 2021 January Q5
5. (a) Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that $$\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 5 ) = \frac { n } { 6 } ( n + 7 ) ( 2 n + 7 )$$ for all positive integers \(n\).
(b) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 1 ) ( r + 5 ) = \frac { 7 n } { 6 } ( n + 1 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be determined.
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Edexcel F1 2021 January Q6
6. The complex number \(z\) is defined by $$z = - \lambda + 3 i \quad \text { where } \lambda \text { is a positive real constant }$$ Given that the modulus of \(z\) is 5
  1. write down the value of \(\lambda\)
  2. determine the argument of \(z\), giving your answer in radians to one decimal place. In part (c) you must show detailed reasoning.
    Solutions relying on calculator technology are not acceptable.
  3. Express in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are real,
    1. \(\frac { z + 3 i } { 2 - 4 i }\)
    2. \(\mathrm { Z } ^ { 2 }\)
  4. Show on a single Argand diagram the points \(A\), \(B\), \(C\) and \(D\) that represent the complex numbers $$z , z ^ { * } , \frac { z + 3 i } { 2 - 4 i } \text { and } z ^ { 2 }$$
Edexcel F1 2021 January Q7
7. The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r } 4 & - 5
- 3 & 2 \end{array} \right)$$ The transformation represented by \(\mathbf { A }\) maps triangle \(T\) onto triangle \(T ^ { \prime }\) Given that the area of triangle \(T\) is \(23 \mathrm {~cm} ^ { 2 }\)
  1. determine the area of triangle \(T ^ { \prime }\)
    (2) The point \(P\) has coordinates ( \(3 p + 2,2 p - 1\) ) where \(p\) is a constant. The transformation represented by \(\mathbf { A }\) maps \(P\) onto the point \(P ^ { \prime }\) with coordinates \(( 17 , - 18 )\)
  2. Determine the value of \(p\). Given that $$\mathbf { B } = \left( \begin{array} { r r } 0 & 1
    - 1 & 0 \end{array} \right)$$
  3. describe fully the single geometrical transformation represented by matrix \(\mathbf { B }\) The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { C }\) is equivalent to the transformation represented by matrix \(\mathbf { B }\)
  4. Determine C
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Edexcel F1 2021 January Q8
  1. The hyperbola \(H\) has Cartesian equation \(x y = 25\)
The parabola \(P\) has parametric equations \(x = 10 t ^ { 2 } , y = 20 t\)
The hyperbola \(H\) intersects the parabola \(P\) at the point \(A\)
  1. Use algebra to determine the coordinates of \(A\) The point \(B\) with coordinates \(( 10,20 )\) lies on \(P\)
  2. Find an equation for the normal to \(P\) at \(B\) Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be determined.
  3. Use algebra to determine, in simplest form, the exact coordinates of the points where this normal intersects the hyperbola \(H\)
    (6)