Questions — Edexcel F1 (197 questions)

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Edexcel F1 2018 Specimen Q6
  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
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
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
  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
  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 \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } \frac { 4 r } { 3 ^ { r } } = 3 - \frac { ( 3 + 2 n ) } { 3 ^ { n } }$$
\(\_\_\_\_\) VJYV SIHI NI JIIYM ION OO \(\quad\) VJYV SIHI NI JIIYM ION OC \(\quad\) VJYV SIHI NI JLIVM ION OC \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-31_2255_51_316_36}
Edexcel F1 Specimen Q1
  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
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
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.
Edexcel F1 Specimen Q4
  1. The quadratic equation
$$5 x ^ { 2 } - 4 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 6 } { 5 }\)
  3. Find a quadratic equation with integer coefficients, which has roots $$\alpha + \frac { 1 } { \alpha } \text { and } \beta + \frac { 1 } { \beta }$$
Edexcel F1 Specimen Q5
  1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
$$\mathrm { f } ( n ) = 5 ^ { n } + 8 n + 3 \text { is divisible by } 4$$
Edexcel F1 Specimen Q6
6. (a) Using the formulae for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), show that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 3 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + k )$$ where \(k\) is a constant to be found.
(b) Hence evaluate \(\sum _ { r = 21 } ^ { 40 } r ( r + 1 ) ( r + 3 )\)
Edexcel F1 Specimen Q7
  1. The point \(\mathrm { P } \left( 6 \mathrm { t } , \frac { 6 } { \mathrm { t } } \right) , t \neq 0\), lies on the rectangular hyperbola \(H\) with equation \(x y = 36\) (a) Show that an equation for the tangent to \(H\) at \(P\) is
$$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 12 } { t }$$ The tangent to \(H\) at the point \(A\) and the tangent to \(H\) at the point \(B\) meet at the point \(( - 9,12 )\).
(b) Find the coordinates of \(A\) and \(B\).
Edexcel F1 Specimen Q8
8. (i) The transformation \(U\) is represented by the matrix \(\mathbf { P }\) where, $$P = \left( \begin{array} { r r } - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 }
\frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$$
  1. Describe fully the transformation \(U\). The transformation \(V\), represented by the matrix \(\mathbf { Q }\), is a stretch scale factor 3 parallel to the \(x\)-axis.
  2. Write down the matrix \(\mathbf { Q }\). Transformation \(U\) followed by transformation \(V\) is a transformation which is represented by matrix \(\mathbf { R }\).
  3. Find the matrix \(\mathbf { R }\).
    (ii) $$S = \left( \begin{array} { r r } 1 & - 3
    3 & 1 \end{array} \right)$$ Given that the matrix \(\mathbf { S }\) represents an enlargement, with a positive scale factor and centre \(( 0,0 )\), followed by a rotation with centre \(( 0,0 )\),
  4. find the scale factor of the enlargement,
  5. find the angle and direction of rotation, giving your answer in degrees to 1 decimal place.
Edexcel F1 2017 January Q2
The quadratic equation $$2 x ^ { 2 } - x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the equation,
  1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
  2. find the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\)
  3. find a quadratic equation which has roots $$\left( 2 \alpha - \frac { 1 } { \beta } \right) \text { and } \left( 2 \beta - \frac { 1 } { \alpha } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.
Edexcel F1 2021 June Q1
1．（i） $$f ( x ) = x ^ { 3 } + 4 x - 6$$ （a）Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval［1，1．5］
（b）Taking 1.5 as a first approximation，apply the Newton Raphson process twice to \(\mathrm { f } ( x )\) to obtain an approximate value of \(\alpha\) ．Give your answer to 3 decimal places． Show your working clearly．
（ii） $$g ( x ) = 4 x ^ { 2 } + x - \tan x$$ where \(x\) is measured in radians． The equation \(\mathrm { g } ( x ) = 0\) has a single root \(\beta\) in the interval［1．4，1．5］
Use linear interpolation on the values at the end points of this interval to obtain an approximation to \(\beta\) ．Give your answer to 3 decimal places．
Edexcel F1 2021 June Q2
2. The complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) are given by $$\mathrm { z } _ { 1 } = 2 - \mathrm { i } \quad \mathrm { z } _ { 2 } = p - \mathrm { i } \quad \mathrm { z } _ { 3 } = p + \mathrm { i }$$ where \(p\) is a real number.
  1. Find \(\frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }\) in the form \(a + b \mathrm { i }\) where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\). Given that \(\left| \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } } \right| = 2 \sqrt { 5 }\)
  2. find the possible values of \(p\).
Edexcel F1 2021 June Q3
  1. The triangle \(T\) has vertices \(A ( 2,1 ) , B ( 2,3 )\) and \(C ( 0,1 )\).
The triangle \(T ^ { \prime }\) is the image of \(T\) under the transformation represented by the matrix $$\mathbf { P } = \left( \begin{array} { r r } 0 & 1
- 1 & 0 \end{array} \right)$$
  1. Find the coordinates of the vertices of \(T ^ { \prime }\)
  2. Describe fully the transformation represented by \(\mathbf { P }\) The \(2 \times 2\) matrix \(\mathbf { Q }\) represents a reflection in the \(x\)-axis and the \(2 \times 2\) matrix \(\mathbf { R }\) represents a rotation through \(90 ^ { \circ }\) anticlockwise about the origin.
  3. Write down the matrix \(\mathbf { Q }\) and the matrix \(\mathbf { R }\)
  4. Find the matrix \(\mathbf { R Q }\)
  5. Give a full geometrical description of the single transformation represented by the answer to part (d).
Edexcel F1 2021 June Q4
  1. A rectangular hyperbola \(H\) has equation \(x y = 25\)
The point \(P \left( 5 t , \frac { 5 } { t } \right) , t \neq 0\), is a general point on \(H\).
  1. Show that the equation of the tangent to \(H\) at \(P\) is \(t ^ { 2 } y + x = 10 t\) The distinct points \(Q\) and \(R\) lie on \(H\). The tangent to \(H\) at the point \(Q\) and the tangent to \(H\) at the point \(R\) meet at the point \(( 15 , - 5 )\).
  2. Find the coordinates of the points \(Q\) and \(R\).
Edexcel F1 2021 June Q5
5. $$f ( x ) = \left( 9 x ^ { 2 } + d \right) \left( x ^ { 2 } - 8 x + ( 10 d + 1 ) \right)$$ where \(d\) is a positive constant.
  1. Find the four roots of \(\mathrm { f } ( x )\) giving your answers in terms of \(d\). Given \(d = 4\)
  2. Express these four roots in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
  3. Show these four roots on a single Argand diagram. \includegraphics[max width=\textwidth, alt={}, center]{d7689f4a-a41e-45be-911b-4a74e81997eb-21_2647_1840_118_111}
Edexcel F1 2021 June Q6
6. The parabola \(C\) has Cartesian equation \(y ^ { 2 } = 8 x\) The point \(P \left( 2 p ^ { 2 } , 4 p \right)\) and the point \(Q \left( 2 q ^ { 2 } , 4 q \right)\), where \(p , q \neq 0 , p \neq q\), are points on \(C\).
  1. Show that an equation of the normal to \(C\) at \(P\) is $$y + p x = 2 p ^ { 3 } + 4 p$$
  2. Write down an equation of the normal to \(C\) at \(Q\) The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(N\)
  3. Show that \(N\) has coordinates $$\left( 2 \left( p ^ { 2 } + p q + q ^ { 2 } + 2 \right) , - 2 p q ( p + q ) \right)$$ The line \(O N\), where \(O\) is the origin, is perpendicular to the line \(P Q\)
  4. Find the value of \(( p + q ) ^ { 2 } - 3 p q\)
Edexcel F1 2021 June Q7
7. (a) Prove by induction that for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { n } { 6 } ( n + 1 ) ( 2 n + 1 )$$ (b) Hence show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + 2 \right) = \frac { n } { 6 } \left( a n ^ { 2 } + b n + c \right)$$ where \(a , b\) and \(c\) are integers to be found.
(c) Using your answers to part (b), find the value of $$\sum _ { r = 10 } ^ { 25 } \left( r ^ { 2 } + 2 \right)$$
Edexcel F1 2021 June Q8
8. Prove by induction that \(4 ^ { n + 2 } + 5 ^ { 2 n + 1 }\) is divisible by 21 for all positive integers \(n\).
\includegraphics[max width=\textwidth, alt={}]{d7689f4a-a41e-45be-911b-4a74e81997eb-32_2644_1837_118_114}