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Edexcel F1 2021 January Q9
9. (i) 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 \(n \in \mathbb { Z } ^ { + }\) $$u _ { n } = 3 \left( \frac { 2 } { 3 } \right) ^ { n } - 1$$ (ii) \(\mathrm { f } ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }\) 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 2022 January Q1
1. $$\mathbf { M } = \left( \begin{array} { r r } 3 x & 7
4 x + 1 & 2 - x \end{array} \right)$$ Find the range of values of \(x\) for which the determinant of the matrix \(\mathbf { M }\) is positive.
VILV SIHI NI IIII M I ON OC
WIHW SIHI NI IIIIM I ON OC
WARV SIHI NI IIII M I ON OC
Edexcel F1 2022 January Q2
2. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by $$z _ { 1 } = 3 + 5 i \text { and } z _ { 2 } = - 2 + 6 i$$
  1. Show \(z _ { 1 }\) and \(z _ { 2 }\) on a single Argand diagram.
  2. Without using your calculator and showing all stages of your working,
    1. determine the value of \(\left| z _ { 1 } \right|\)
    2. express \(\frac { z _ { 1 } } { z _ { 2 } }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are fully simplified fractions.
  3. Hence determine the value of \(\arg \frac { Z _ { 1 } } { Z _ { 2 } }\) Give your answer in radians to 2 decimal places.
Edexcel F1 2022 January Q3
3. The parabola \(C\) has equation \(y ^ { 2 } = 18 x\) The point \(S\) is the focus of \(C\)
  1. Write down the coordinates of \(S\) The point \(P\), with \(y > 0\), lies on \(C\) The shortest distance from \(P\) to the directrix of \(C\) is 9 units.
  2. Determine the exact perimeter of the triangle \(O P S\), where \(O\) is the origin. Give your answer in simplest form.
Edexcel F1 2022 January Q4
4. The equation $$x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + 225 = 0$$ where \(A , B\) and \(C\) are real constants, has
  • a complex root \(4 + 3 \mathrm { i }\)
  • a repeated positive real root
    1. Write down the other complex root of this equation.
    2. Hence determine a quadratic factor of \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + 225\)
    3. Deduce the real root of the equation.
    4. Hence determine the value of each of the constants \(A , B\) and \(C\)
Edexcel F1 2022 January Q5
5. $$\mathbf { P } = \left( \begin{array} { r r } \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 }
\frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right)$$ The matrix \(\mathbf { P }\) represents the transformation \(U\)
  1. Give a full description of \(U\) as a single geometrical transformation. The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line \(y = - x\)
  2. Write down the matrix \(\mathbf { Q }\) The transformation \(U\) followed by the transformation \(V\) is represented by the matrix \(\mathbf { R }\)
  3. Determine the matrix \(\mathbf { R }\) The transformation \(W\) is represented by the matrix \(3 \mathbf { R }\) The transformation \(W\) maps a triangle \(T\) to a triangle \(T ^ { \prime }\) The transformation \(W ^ { \prime }\) maps the triangle \(T ^ { \prime }\) back to the original triangle \(T\)
  4. Determine the matrix that represents \(W ^ { \prime }\)
Edexcel F1 2022 January Q6
6. The quadratic equation $$A x ^ { 2 } + 5 x - 12 = 0$$ where \(A\) is a constant, has roots \(\alpha\) and \(\beta\)
  1. Write down an expression in terms of \(A\) for
    1. \(\alpha + \beta\)
    2. \(\alpha \beta\) The equation $$4 x ^ { 2 } - 5 x + B = 0$$ where \(B\) is a constant, has roots \(\alpha - \frac { 3 } { \beta }\) and \(\beta - \frac { 3 } { \alpha }\)
  2. Determine the value of \(A\)
  3. Determine the value of \(B\) The rectangular hyperbola \(H\) has equation \(x y = 36\) The point \(P ( 4,9 )\) lies on \(H\)
  4. Show, using calculus, that the normal to \(H\) at \(P\) has equation $$4 x - 9 y + 65 = 0$$ The normal to \(H\) at \(P\) crosses \(H\) again at the point \(Q\)
  5. Determine an equation for the tangent to \(H\) at \(Q\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are rational constants. \section*{7. In this question you must show all stages of your working.
    Solutions relying entirely on calculator technology are not acceptable.
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Edexcel F1 2022 January Q8
8. $$\mathrm { f } ( x ) = 2 x ^ { - \frac { 2 } { 3 } } + \frac { 1 } { 2 } x - \frac { 1 } { 3 x - 5 } - \frac { 5 } { 2 } \quad x \neq \frac { 5 } { 3 }$$ The table below shows values of \(\mathrm { f } ( x )\) for some values of \(x\), with values of \(\mathrm { f } ( x )\) given to 4 decimal places where appropriate.
\(x\)12345
\(\mathrm { f } ( x )\)0.5- 0.28850.5834
  1. Complete the table giving the values to 4 decimal places. The equation \(\mathrm { f } ( x ) = 0\) has exactly one positive root, \(\alpha\). Using the values in the completed table and explaining your reasoning,
  2. determine an interval of width one that contains \(\alpha\).
  3. Hence use interval bisection twice to obtain an interval of width 0.25 that contains \(\alpha\). Given also that the equation \(\mathrm { f } ( x ) = 0\) has a negative root, \(\beta\), in the interval \([ - 1 , - 0.5 ]\)
  4. use linear interpolation once on this interval to find an approximation for \(\beta\). Give your answer to 3 significant figures.
Edexcel F1 2022 January Q9
9. (a) Prove by induction that, for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ (b) Using the standard summation formulae, show that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = \frac { 1 } { 4 } n ( n + A ) ( n + B ) ( n + C )$$ where \(A , B\) and \(C\) are constants to be determined.
(c) Determine the value of \(n\) for which $$3 \sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = 17 \sum _ { r = n } ^ { 2 n } r ^ { 2 }$$
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Edexcel F1 2023 January Q1
  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
  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
  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
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
  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
  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
$$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1
- 1 & 0 \end{array} \right)$$ 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
  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
  1. Prove by induction that for all positive integers \(n\)
$$\sum _ { r = 1 } ^ { n } \log ( 2 r - 1 ) = \log \left( \frac { ( 2 n ) ! } { 2 ^ { n } n ! } \right)$$
Edexcel F1 2024 January Q1
1. $$\mathbf { M } = \left( \begin{array} { c c } 2 k + 1 & k
k + 7 & k + 4 \end{array} \right) \quad \text { where } k \text { is a constant }$$
  1. Show that \(\mathbf { M }\) is non-singular for all real values of \(k\).
  2. Determine \(\mathbf { M } ^ { - 1 }\) in terms of \(k\).
Edexcel F1 2024 January Q2
2. $$f ( z ) = 2 z ^ { 3 } + p z ^ { 2 } + q z - 41$$ where \(p\) and \(q\) are integers.
The complex number \(5 - 4 \mathrm { i }\) 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 \(p\) and the value of \(q\). When plotted on an Argand diagram, the points representing the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) form the vertices of a triangle.
  4. Determine the area of this triangle.
Edexcel F1 2024 January Q3
  1. The 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 \(H\).
The tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).
  1. Determine, in terms of \(c\) and \(t\),
    1. the coordinates of \(A\),
    2. the coordinates of \(B\). Given that the area of triangle \(A O B\), where \(O\) is the origin, is 90 square units,
  2. determine the value of \(c\), giving your answer as a simplified surd.
Edexcel F1 2024 January Q4
4. $$\mathbf { A } = \left( \begin{array} { l l } 1 & 0
0 & 3 \end{array} \right)$$
  1. Describe the single geometrical transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents a rotation of \(210 ^ { \circ }\) anticlockwise about centre \(( 0,0 )\).
  2. Write down the matrix \(\mathbf { B }\), giving each element in exact form. The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { B }\) is represented by the matrix \(\mathbf { C }\).
  3. Find \(\mathbf { C }\). The hexagon \(H\) is transformed onto the hexagon \(H ^ { \prime }\) by the matrix \(\mathbf { C }\).
  4. Given that the area of hexagon \(H\) is 5 square units, determine the area of hexagon \(H ^ { \prime }\)
Edexcel F1 2024 January Q5
  1. The quadratic equation
$$2 x ^ { 2 } - 3 x + 7 = 0$$ 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 the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
  3. find a quadratic equation which has roots $$\left( \alpha - \frac { 1 } { \beta ^ { 2 } } \right) \text { and } \left( \beta - \frac { 1 } { \alpha ^ { 2 } } \right)$$ 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 2024 January Q6
$$f ( x ) = x - 4 - \cos ( 5 \sqrt { x } ) \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [2.5, 3.5]
    [0pt]
  2. Use linear interpolation once on the interval [2.5, 3.5] to find an approximation to \(\alpha\), giving your answer to 2 decimal places.
    (ii) $$\operatorname { g } ( x ) = \frac { 1 } { 10 } x ^ { 2 } - \frac { 1 } { 2 x ^ { 2 } } + x - 11 \quad x > 0$$
  3. Determine \(\mathrm { g } ^ { \prime } ( x )\). The equation \(\mathrm { g } ( x ) = 0\) has a root \(\beta\) in the interval [6,7]
  4. Using \(x _ { 0 } = 6\) as a first approximation to \(\beta\), apply the Newton-Raphson procedure once to \(\mathrm { g } ( x )\) to find a second approximation to \(\beta\), giving your answer to 3 decimal places.
Edexcel F1 2024 January Q7
  1. The parabola \(C\) has equation \(y ^ { 2 } = \frac { 4 } { 3 } x\)
The point \(P \left( \frac { 1 } { 3 } t ^ { 2 } , \frac { 2 } { 3 } t \right)\), where \(t \neq 0\), lies on \(C\).
  1. Use calculus to show that the normal to \(C\) at \(P\) has equation $$3 t x + 3 y = t ^ { 3 } + 2 t$$ The normal to \(C\) at the point where \(t = 9\) meets \(C\) again at the point \(Q\).
  2. Determine the exact coordinates of \(Q\).