Questions F1 (198 questions)

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Edexcel F1 2024 June Q7
8 marks Standard +0.3
  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
6 marks Standard +0.3
  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
13 marks Challenging +1.2
  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
5 marks Standard +0.3
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
9 marks Standard +0.3
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
9 marks Standard +0.8
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
7 marks Moderate -0.3
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
8 marks Standard +0.8
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
8 marks Standard +0.8
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
9 marks Standard +0.3
  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
10 marks Standard +0.8
  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
10 marks Challenging +1.2
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
\includegraphics[max width=\textwidth, alt={}]{41065a55-38c3-4b16-a02d-ece9f02ef32c-36_2820_1967_102_100}
Edexcel F1 2018 Specimen Q1
4 marks Moderate -0.3
  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
5 marks Standard +0.8
  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
7 marks Standard +0.3
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. \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-06_2250_51_317_1980}
Edexcel F1 2018 Specimen Q4
5 marks Moderate -0.5
  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 marks Moderate -0.3
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.
Edexcel F1 2018 Specimen Q6
10 marks Standard +0.3
  1. The rectangular hyperbola \(H\) has equation \(x y = 25\)
    1. Verify that, for \(t \neq 0\), the point \(P \left( 5 t , \frac { 5 } { t } \right)\) is a general point on \(H\).
    The point \(A\) on \(H\) has parameter \(t = \frac { 1 } { 2 }\)
  2. Show that the normal to \(H\) at the point \(A\) has equation $$8 y - 2 x - 75 = 0$$ This normal at \(A\) meets \(H\) again at the point \(B\).
  3. Find the coordinates of \(B\). \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-13_2261_50_312_36}
    VIAV SIHI NI BIIIM ION OCVGHV SIHI NI GHIYM ION OCVJ4V SIHI NI JIIYM ION OC
Edexcel F1 2018 Specimen Q7
10 marks Moderate -0.3
7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)$$
  1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\)
  2. Write down the matrix \(\mathbf { Q }\). Given that the transformation \(V\) followed by the transformation \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
  3. find the matrix \(\mathbf { R }\).
  4. Show that there is a value of \(k\) for which the transformation \(T\) maps each point on the straight line \(y = k x\) onto itself, and state the value of \(k\).
    \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-17_2261_54_312_34}
    VIAV SIHI NI BIIIM ION OCVGHV SIHI NI GHIYM ION OCVJ4V SIHI NI JIIYM ION OC
Edexcel F1 2018 Specimen Q8
9 marks Moderate -0.3
8. $$\mathrm { f } ( z ) = z ^ { 4 } + 6 z ^ { 3 } + 76 z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are real constants.
Given that \(- 3 + 8 \mathrm { i }\) is a complex root of the equation \(\mathrm { f } ( z ) = 0\)
  1. write down another complex root of this equation.
  2. Hence, or otherwise, find the other roots of the equation \(\mathrm { f } ( z ) = 0\)
  3. Show on a single Argand diagram all four roots of the equation \(\mathrm { f } ( z ) = 0\)
    VIAV SIHI NI BIIIM ION OCVGHV SIHI NI GHIYM ION OCVJ4V SIHI NI JIIYM ION OC
Edexcel F1 2018 Specimen Q9
9 marks Standard +0.8
  1. The quadratic equation
$$2 x ^ { 2 } + 4 x - 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the quadratic equation,
  1. find the exact value of
    1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
    2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  2. Find a quadratic equation which has roots ( \(\alpha ^ { 2 } + \beta\) ) and ( \(\beta ^ { 2 } + \alpha\) ), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers.
    VIAV SIHI NI BIIIM ION OCVGHV SIHI NI GHIYM ION OCVJ4V SIHI NI JIIYM ION OC
Edexcel F1 2018 Specimen Q10
11 marks Standard +0.3
  1. (i) A sequence of positive numbers is defined by
$$\begin{aligned} u _ { 1 } & = 5 \\ u _ { n + 1 } & = 3 u _ { n } + 2 , \quad n \geqslant 1 \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 2 \times ( 3 ) ^ { n } - 1$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } \frac { 4 r } { 3 ^ { r } } = 3 - \frac { ( 3 + 2 n ) } { 3 ^ { n } }$$
Edexcel F1 Specimen Q1
7 marks Moderate -0.8
  1. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by
$$z _ { 1 } = 2 + 8 \mathrm { i } \quad \text { and } \quad z _ { 2 } = 1 - \mathrm { i }$$ Find, showing your working,
  1. \(\frac { z _ { 1 } } { z _ { 2 } }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real,
  2. the value of \(\left| \frac { z _ { 1 } } { z _ { 2 } } \right|\),
  3. the value of \(\arg \frac { z _ { 1 } } { z _ { 2 } }\), giving your answer in radians to 2 decimal places.
Edexcel F1 Specimen Q2
10 marks Standard +0.3
2. $$f ( x ) = 5 x ^ { 2 } - 4 x ^ { \frac { 3 } { 2 } } - 6 , \quad x \geqslant 0$$ The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ 1.6,1.8 ]\)
  1. Use linear interpolation once on the interval \([ 1.6,1.8 ]\) to find an approximation to \(\alpha\). Give your answer to 3 decimal places.
  2. Taking 1.7 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.
Edexcel F1 Specimen Q3
7 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa5a23b5-d52c-4bae-97c7-2eb7220a3dc4-04_736_659_299_660} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 8 x\).
The point \(P\) lies on \(C\), where \(y > 0\), and the point \(Q\) lies on \(C\), where \(y < 0\) The line segment \(P Q\) is parallel to the \(y\)-axis. Given that the distance \(P Q\) is 12 ,
  1. write down the \(y\) coordinate of \(P\),
  2. find the \(x\) coordinate of \(P\). Figure 1 shows the point \(S\) which is the focus of \(C\). The line \(l\) passes through the point \(P\) and the point \(S\).
  3. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.