Questions F1 (198 questions)

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Edexcel F1 2021 January Q9
12 marks Standard +0.3
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 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
$$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$ 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 \(n\)
$$\sum _ { r = 1 } ^ { n } \log ( 2 r - 1 ) = \log \left( \frac { ( 2 n ) ! } { 2 ^ { n } n ! } \right)$$
Edexcel F1 2024 January Q1
5 marks Moderate -0.3
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
9 marks Standard +0.8
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
6 marks Standard +0.8
  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
7 marks Moderate -0.3
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
9 marks Standard +0.8
  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
8 marks Moderate -0.8
$$f ( x ) = x - 4 - \cos ( 5 \sqrt { x } ) \quad x > 0$$
  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
7 marks Standard +0.8
  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 \(Q\).
Edexcel F1 2024 January Q8
8 marks Challenging +1.2
  1. (a) Use the standard results for summations to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } r \left( 2 r ^ { 2 } - 3 r - 1 \right) = \frac { 1 } { 2 } n ( n + 1 ) ^ { 2 } ( n - 2 )$$ (b) Hence show that, for all positive integers \(n\), $$\sum _ { r = n } ^ { 2 n } r \left( 2 r ^ { 2 } - 3 r - 1 \right) = \frac { 1 } { 2 } n ( n - 1 ) ( a n + b ) ( c n + d )$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be determined.
Edexcel F1 2024 January Q9
6 marks Standard +0.8
  1. Given that
$$\frac { 3 z - 1 } { 2 } = \frac { \lambda + 5 i } { \lambda - 4 i }$$ where \(\lambda\) is a real constant,
  1. determine \(z\), giving your answer in the form \(x + y i\), where \(x\) and \(y\) are real and in terms of \(\lambda\). Given also that \(\arg z = \frac { \pi } { 4 }\)
  2. find the possible values of \(\lambda\).
Edexcel F1 2024 January Q10
10 marks Standard +0.8
  1. (i) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)
$$\left( \begin{array} { r r } 5 & - 1 \\ 4 & 1 \end{array} \right) ^ { n } = 3 ^ { n - 1 } \left( \begin{array} { c c } 2 n + 3 & - n \\ 4 n & 3 - 2 n \end{array} \right)$$ (ii) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 8 ^ { 2 n + 1 } + 6 ^ { 2 n - 1 }$$ is divisible by 7
Edexcel F1 2014 June Q1
4 marks Moderate -0.8
  1. Find the value of
$$\sum _ { r = 1 } ^ { 200 } ( r + 1 ) ( r - 1 )$$
Edexcel F1 2014 June Q2
4 marks Moderate -0.8
2. Given that \(- 2 + 3 \mathrm { i }\) is a root of the equation $$z ^ { 2 } + p z + q = 0$$ where \(p\) and \(q\) are real constants,
  1. write down the other root of the equation.
  2. Find the value of \(p\) and the value of \(q\).
Edexcel F1 2014 June Q3
6 marks Standard +0.8
3. $$\mathbf { A } = \left( \begin{array} { l l } 4 & - 2 \\ a & - 3 \end{array} \right)$$ where \(a\) is a real constant and \(a \neq 6\)
  1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\). Given that \(\mathbf { A } + 2 \mathbf { A } ^ { - 1 } = \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
  2. find the value of \(a\).
Edexcel F1 2014 June Q4
10 marks Standard +0.3
4. $$\mathrm { f } ( x ) = x ^ { \frac { 3 } { 2 } } - 3 x ^ { \frac { 1 } { 2 } } - 3 , \quad x > 0$$ Given that \(\alpha\) is the only real root of the equation \(\mathrm { f } ( x ) = 0\),
  1. show that \(4 < \alpha < 5\)
  2. Taking 4.5 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.
    [0pt]
  3. Use linear interpolation once on the interval [4,5] to find another approximation to \(\alpha\), giving your answer to 3 decimal places.
Edexcel F1 2014 June Q5
7 marks Standard +0.8
  1. Given that \(z _ { 1 } = - 3 - 4 \mathrm { i }\) and \(z _ { 2 } = 4 - 3 \mathrm { i }\)
    1. show, on an Argand diagram, the point \(P\) representing \(z _ { 1 }\) and the point \(Q\) representing \(z _ { 2 }\)
    2. Given that \(O\) is the origin, show that \(O P\) is perpendicular to \(O Q\).
    3. Show the point \(R\) on your diagram, where \(R\) represents \(z _ { 1 } + z _ { 2 }\)
    4. Prove that \(O P R Q\) is a square.