Questions — Edexcel (10514 questions)

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Edexcel F1 2018 January Q2
9 marks Standard +0.3
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
8 marks Standard +0.8
  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 \(n\), $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be determined.
  2. 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
8 marks Standard +0.8
  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
9 marks Standard +0.3
5.
  1. 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.
  2. 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
12 marks Standard +0.3
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 \(M\).
  4. Find the exact coordinates of the points \(L\) and \(M\), giving your answers in their simplest form.
Edexcel F1 2018 January Q7
11 marks Standard +0.3
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 \(( 0,0 )\).
    1. Find the value of \(p\).
    2. Find the value of \(\theta\).
Edexcel F1 2018 January Q8
11 marks Standard +0.3
8.
  1. 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 \(n\), $$u _ { n } = \frac { 3 } { 2 } n ^ { 2 } - \frac { 7 } { 2 } n + 5$$
  2. 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
5 marks Moderate -0.8
  1. 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]
  2. 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
5 marks Standard +0.3
  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
4 marks Moderate -0.8
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
8 marks Standard +0.8
  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}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel F1 2021 January Q5
7 marks Standard +0.3
5.
  1. 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 \(n\).
  2. 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.
    VI4V SIHI NI JIIIM ION OCVIAN SIHI NI IHMM I ON OOVAYV SIHI NI JIIIM ION OO
Edexcel F1 2021 January Q6
11 marks Moderate -0.3
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
9 marks Standard +0.3
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 \includegraphics[max width=\textwidth, alt={}, center]{f8660b02-384e-460f-a0e4-282ef5fef475-21_2255_50_314_34}
Edexcel F1 2021 January Q8
14 marks Standard +0.3
  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)
Edexcel F1 2021 January Q9
12 marks Standard +0.3
9.
  1. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n + 1 } = \frac { 1 } { 3 } \left( 2 u _ { n } - 1 \right) \quad u _ { 1 } = 1$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$u _ { n } = 3 \left( \frac { 2 } { 3 } \right) ^ { n } - 1$$
  2. \(\mathrm { f } ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }\) Prove by induction that, for \(n \in \mathbb { Z } ^ { + } , \mathrm { f } ( n )\) is a multiple of 7
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel F1 2023 January Q1
5 marks Moderate -0.8
  1. Given that
$$\mathbf { A } = \left( \begin{array} { r r r } 2 & - 1 & 3 \\ - 2 & 3 & 0 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r } 1 & k \\ 0 & - 3 \\ 2 k & 2 \end{array} \right)$$ where \(k\) is a non-zero constant,
  1. determine the matrix \(\mathbf { A B }\)
  2. determine the value of \(k\) for which \(\operatorname { det } ( \mathbf { A B } ) = 0\)
Edexcel F1 2023 January Q2
6 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.
Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that for all positive integers \(n\) $$\sum _ { r = 1 } ^ { n } ( 7 r - 5 ) ^ { 2 } = \frac { n } { 6 } ( 7 n + 1 ) ( A n + B )$$ where \(A\) and \(B\) are integers to be determined.
Edexcel F1 2023 January Q3
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.
$$\mathrm { f } ( z ) = 4 z ^ { 3 } + p z ^ { 2 } - 24 z + 108$$ where \(p\) is a constant.
Given that - 3 is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. determine the value of \(p\)
  2. using algebra, solve \(\mathrm { f } ( \mathrm { z } ) = 0\) completely, giving the roots in simplest form,
  3. determine the modulus of the complex roots of \(\mathrm { f } ( \mathrm { z } ) = 0\)
  4. show the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
Edexcel F1 2023 January Q4
8 marks Challenging +1.2
4. $$f ( x ) = 1 - \frac { 1 } { 8 x ^ { 4 } } + \frac { 2 } { 7 \sqrt { x ^ { 7 } } } \quad x > 0$$ The equation \(\mathrm { f } ( x ) = 0\) has a single root, \(\alpha\), that lies in the interval \([ 0.15,0.25 ]\)
    1. Determine \(\mathrm { f } ^ { \prime } ( x )\)
    2. Explain why 0.25 cannot be used as an initial approximation for \(\alpha\) in the Newton-Raphson process.
    3. Taking 0.15 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. Use linear interpolation once on the interval \([ 0.15,0.25 ]\) to find another approximation to \(\alpha\) Give your answer to 3 decimal places.
Edexcel F1 2023 January Q5
9 marks Standard +0.8
  1. The quadratic equation
$$4 x ^ { 2 } + 3 x + k = 0$$ where \(k\) is an integer, has roots \(\alpha\) and \(\beta\)
  1. Write down, in terms of \(k\) where appropriate, the value of \(\alpha + \beta\) and the value of \(\alpha \beta\)
  2. Determine, in simplest form in terms of \(k\), the value of \(\frac { \alpha } { \beta ^ { 2 } } + \frac { \beta } { \alpha ^ { 2 } }\)
  3. Determine a quadratic equation which has roots $$\frac { \alpha } { \beta ^ { 2 } } \text { and } \frac { \beta } { \alpha ^ { 2 } }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integer values in terms of \(k\)
Edexcel F1 2023 January Q6
9 marks Challenging +1.8
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
The rectangular hyperbola \(H\) has equation \(x y = 20\) The point \(P \left( 2 t \sqrt { a } , \frac { 2 \sqrt { a } } { t } \right) , t \neq 0\), where \(a\) is a constant, is a general point on \(H\)
  1. State the value of \(a\)
  2. Show that the normal to \(H\) at the point \(P\) has equation $$t y - t ^ { 3 } x - 2 \sqrt { 5 } \left( 1 - t ^ { 4 } \right) = 0$$ The points \(A\) and \(B\) lie on \(H\) The point \(A\) has parameter \(t = c\) and the point \(B\) has parameter \(t = - \frac { 1 } { 2 c }\), where \(c\) is a constant. The normal to \(H\) at \(A\) meets \(H\) again at \(B\)
  3. Determine the possible values of \(C\)
Edexcel F1 2023 January Q7
11 marks Standard +0.3
  1. (i)
$$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$ The matrix \(\mathbf { P }\) represents a geometrical transformation \(U\)
  1. Describe \(U\) fully as a single geometrical transformation. The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through \(240 ^ { \circ }\) anticlockwise about the origin followed by an enlargement about ( 0,0 ) with scale factor 6
  2. Determine the matrix \(\mathbf { Q }\), giving each entry in exact numerical form. Given that \(U\) followed by \(V\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\)
  3. determine the matrix \(\mathbf { R }\) (ii) The transformation \(W\) is represented by the matrix $$\left( \begin{array} { c c } - 2 & 2 \sqrt { 3 } \\ 2 \sqrt { 3 } & 2 \end{array} \right)$$ Show that there is a real number \(\lambda\) for which \(W\) maps the point \(( \lambda , 1 )\) onto the point ( \(4 \lambda , 4\) ), giving the exact value of \(\lambda\) \(\_\_\_\_\) VIAV SIHI NI JIIHM ION OC
    VILU SIHI NI JLIYM ION OC
    VEYV SIHI NI ELIYM ION OC
Edexcel F1 2023 January Q8
11 marks Challenging +1.8
  1. A parabola \(C\) has equation \(y ^ { 2 } = 4 a x\) where \(a\) is a positive constant.
The point \(S\) is the focus of \(C\) The line \(l _ { 1 }\) with equation \(y = k\) where \(k\) is a positive constant, intersects \(C\) at the point \(P\)
  1. Show that $$P S = \frac { k ^ { 2 } + 4 a ^ { 2 } } { 4 a }$$ The line \(l _ { 2 }\) passes through \(P\) and intersects the directrix of \(C\) on the \(x\)-axis.
    The line \(l _ { 2 }\) intersects the \(y\)-axis at the point \(A\)
  2. Show that the \(y\) coordinate of \(A\) is \(\frac { 4 a ^ { 2 } k } { k ^ { 2 } + 4 a ^ { 2 } }\) The line \(l _ { 1 }\) intersects the directrix of \(C\) at the point \(B\) Given that the areas of triangles \(B P A\) and \(O S P\), where \(O\) is the origin, satisfy the ratio $$\text { area } B P A \text { : area } O S P = 4 k ^ { 2 } : 1$$
  3. determine the exact value of \(a\)
Edexcel F1 2023 January Q9
6 marks Challenging +1.2
  1. Prove by induction that for all positive integers \(n\)
$$\sum _ { r = 1 } ^ { n } \log ( 2 r - 1 ) = \log \left( \frac { ( 2 n ) ! } { 2 ^ { n } n ! } \right)$$