Questions — Edexcel (10514 questions)

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Edexcel F1 2015 January Q2
7 marks Standard +0.8
2. $$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + \frac { 1 } { 2 \sqrt { x ^ { 5 } } } + 2 , \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 2,3 ]\).
  2. Taking 3 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel F1 2015 January Q3
6 marks Standard +0.8
3. Given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers, solve the equation $$( z - 2 i ) \left( z ^ { * } - 2 i \right) = 21 - 12 i$$ where \(z ^ { * }\) is the complex conjugate of \(z\).
Edexcel F1 2015 January Q4
14 marks Challenging +1.2
4. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 12 x\) The point \(P \left( 3 p ^ { 2 } , 6 p \right)\) lies on \(C\), where \(p \neq 0\)
  1. Show that the equation of the normal to the curve \(C\) at the point \(P\) is $$y + p x = 6 p + 3 p ^ { 3 }$$ This normal crosses the curve \(C\) again at the point \(Q\).
    Given that \(p = 2\) and that \(S\) is the focus of the parabola, find
  2. the coordinates of the point \(Q\),
  3. the area of the triangle \(P Q S\).
Edexcel F1 2015 January Q5
8 marks Standard +0.3
5. The quadratic equation $$4 x ^ { 2 } + 3 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\).
  2. Find the value of \(\left( \alpha ^ { 2 } + \beta ^ { 2 } \right)\).
  3. Find a quadratic equation which has roots $$( 4 \alpha - \beta ) \text { and } ( 4 \beta - \alpha )$$ 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 2015 January Q6
10 marks Moderate -0.8
6.
  1. $$\mathbf { A } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r } - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$
    1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\).
    2. Describe fully the single transformation represented by the matrix \(\mathbf { B }\). The transformation represented by \(\mathbf { A }\) followed by the transformation represented by \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\).
    3. Find \(\mathbf { C }\).
    4. \(\mathbf { M } = \left( \begin{array} { c c } 2 k + 5 & - 4 \\ 1 & k \end{array} \right)\), where \(k\) is a real number. Show that \(\operatorname { det } \mathbf { M } \neq 0\) for all values of \(k\).
Edexcel F1 2015 January Q7
11 marks Standard +0.8
7. Given that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } ( r + a ) ( r + b ) = \frac { 1 } { 6 } n ( 2 n + 11 ) ( n - 1 )$$ where \(a\) and \(b\) are constants and \(a > b\),
  1. find the value of \(a\) and the value of \(b\).
  2. Find the value of $$\sum _ { r = 9 } ^ { 20 } ( r + a ) ( r + b )$$
Edexcel F1 2015 January Q8
12 marks Standard +0.8
  1. A sequence of numbers is defined by $$\begin{gathered} u _ { 1 } = 5 \quad u _ { 2 } = 13 \\ u _ { n + 2 } = 5 u _ { n + 1 } - 6 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 2 ^ { n } + 3 ^ { n }$$
  2. Prove by induction that for \(n \geqslant 2\), where \(n \in \mathbb { Z }\), $$f ( n ) = 7 ^ { 2 n } - 48 n - 1$$ is divisible by 2304 \includegraphics[max width=\textwidth, alt={}, center]{864a8956-ead0-4abd-91f4-1caa6d17f5e8-14_106_58_2403_1884}
Edexcel F1 2016 January Q1
9 marks Moderate -0.3
1. $$z = 3 + 2 \mathrm { i } , \quad w = 1 - \mathrm { i }$$ Find in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,
  1. \(z w\)
  2. \(\frac { z } { w ^ { * } }\), showing clearly how you obtained your answer. Given that $$| z + k | = \sqrt { 53 } \text {, where } k \text { is a real constant }$$
  3. find the possible values of \(k\).
Edexcel F1 2016 January Q2
7 marks Standard +0.3
2. $$\mathrm { f } ( x ) = x ^ { 2 } - \frac { 3 } { \sqrt { x } } - \frac { 4 } { 3 x ^ { 2 } } , \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [1.6,1.7]
  2. Taking 1.6 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel F1 2016 January Q3
11 marks Standard +0.8
3. The quadratic equation $$x ^ { 2 } - 2 x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the equation,
    1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
    2. show that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 2\)
    3. find the value of \(\alpha ^ { 3 } + \beta ^ { 3 }\)
    1. show that \(\alpha ^ { 4 } + \beta ^ { 4 } = \left( \alpha ^ { 2 } + \beta ^ { 2 } \right) ^ { 2 } - 2 ( \alpha \beta ) ^ { 2 }\)
    2. find a quadratic equation which has roots $$\text { ( } \alpha ^ { 3 } - \beta \text { ) and ( } \beta ^ { 3 } - \alpha \text { ) }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.
Edexcel F1 2016 January Q4
8 marks Standard +0.3
4. $$\mathbf { A } = \left( \begin{array} { c c } - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ - \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \end{array} \right)$$
  1. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A }\).
  2. Hence find the smallest positive integer value of \(n\) for which $$\mathbf { A } ^ { n } = \mathbf { I }$$ where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. The transformation represented by the matrix \(\mathbf { A }\) followed by the transformation represented by the matrix \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\). Given that \(\mathbf { C } = \left( \begin{array} { r r } 2 & 4 \\ - 3 & - 5 \end{array} \right)\),
  3. find the matrix \(\mathbf { B }\).
Edexcel F1 2016 January Q5
8 marks Standard +0.8
5.
  1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } - 3 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 3 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be found. Given that $$\sum _ { r = 5 } ^ { 10 } \left( 8 r ^ { 3 } - 3 r + k r ^ { 2 } \right) = 22768$$
  2. find the exact value of the constant \(k\).
Edexcel F1 2016 January Q6
9 marks Challenging +1.8
6. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a non-zero constant. The point \(P \left( c p , \frac { c } { p } \right)\), where \(p \neq 0\), lies on \(H\).
  1. Show that the normal to \(H\) at \(P\) has equation $$y p - p ^ { 3 } x = c \left( 1 - p ^ { 4 } \right)$$ The normal to \(H\) at \(P\) meets \(H\) again at the point \(Q\).
  2. Find, in terms of \(c\) and \(p\), the coordinates of \(Q\).
Edexcel F1 2016 January Q7
9 marks Standard +0.3
7. $$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } - 15 x ^ { 2 } + 99 x - 130$$
  1. Given that \(x = 3 + 2 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( x ) = 0\), use algebra to find the three other roots of the equation \(\mathrm { f } ( x ) = 0\)
  2. Show the four roots of \(\mathrm { f } ( x ) = 0\) on a single Argand diagram.
Edexcel F1 2016 January Q8
8 marks Challenging +1.2
8. The parabola \(P\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(P\). The point \(B\), which does not lie on the parabola, has coordinates ( \(q , r\) ) where \(q\) and \(r\) are positive constants and \(q > a\). The line \(l\) passes through \(B\) and \(S\).
  1. Show that an equation of the line \(l\) is $$( q - a ) y = r ( x - a )$$ The line \(l\) intersects the directrix of \(P\) at the point \(C\). Given that the area of triangle \(O C S\) is three times the area of triangle \(O B S\), where \(O\) is the origin,
  2. show that the area of triangle \(O B C\) is \(\frac { 6 } { 5 } \mathrm { qr }\)
Edexcel F1 2016 January Q9
6 marks Standard +0.3
9. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21
Edexcel F1 2017 January Q1
5 marks Standard +0.8
\(\mathrm { f } ( x ) = 2 ^ { x } - 10 \sin x - 2\), where \(x\) is measured in radians
  1. Show that \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), between 2 and 3
    [0pt]
  2. Use linear interpolation once on the interval [2,3] to find an approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel F1 2017 January Q3
7 marks Standard +0.8
3. $$f ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 26 x ^ { 2 } + 32 x + 160$$ Given that \(x = - 1 + 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( x ) = 0\), use algebra to find the three other roots of \(\mathrm { f } ( x ) = 0\) (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel F1 2017 January Q4
7 marks Standard +0.8
4.
  1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( 2 r + 1 ) ( 3 r + 1 ) = \frac { 1 } { 6 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
  2. Hence find the value of $$\sum _ { r = 10 } ^ { 20 } r ( 2 r + 1 ) ( 3 r + 1 )$$
Edexcel F1 2017 January Q5
8 marks Moderate -0.3
  1. The complex number \(z\) is given by
$$z = - 7 + 3 i$$ Find
  1. \(| z |\)
  2. \(\arg z\), giving your answer in radians to 2 decimal places. Given that \(\frac { z } { 1 + \mathrm { i } } + w = 3 - 6 \mathrm { i }\)
  3. find the complex number \(w\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. You must show all your working.
  4. Show the points representing \(z\) and \(w\) on a single Argand diagram.
Edexcel F1 2017 January Q6
7 marks Standard +0.3
6. $$f ( x ) = x ^ { 3 } - \frac { 1 } { 2 x } + x ^ { \frac { 3 } { 2 } } , \quad x > 0$$ The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [0.6, 0.7].
  1. Taking 0.6 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. Show that your answer to part (a) is correct to 3 decimal places.
Edexcel F1 2017 January Q7
10 marks Standard +0.3
7. (i) $$\mathbf { A } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)$$
  1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents a stretch, scale factor 3 , parallel to the \(x\)-axis.
  2. Find the matrix \(\mathbf { B }\).
    (ii) $$\mathbf { M } = \left( \begin{array} { r r } - 4 & 3 \\ - 3 & - 4 \end{array} \right)$$ The matrix \(\mathbf { M }\) represents an enlargement with scale factor \(k\) and centre ( 0,0 ), where \(k > 0\), followed by a rotation anticlockwise through an angle \(\theta\) about ( 0,0 ).
    1. Find the value of \(k\).
    2. Find the value of \(\theta\), giving your answer in radians to 2 decimal places.
    3. Find \(\mathbf { M } ^ { - 1 }\)
Edexcel F1 2017 January Q8
12 marks Hard +2.3
8. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(P \left( a t ^ { 2 } , 2 a t \right)\) lies on \(C\).
  1. Using calculus, show that the normal to \(C\) at \(P\) has equation $$y + t x = a t ^ { 3 } + 2 a t$$ The point \(S\) is the focus of the parabola \(C\).
    The point \(B\) lies on the positive \(x\)-axis and \(O B = 5 O S\), where \(O\) is the origin.
  2. Write down, in terms of \(a\), the coordinates of the point \(B\). A circle has centre \(B\) and touches the parabola \(C\) at two distinct points \(Q\) and \(R\). Given that \(t \neq 0\),
  3. find the coordinates of the points \(Q\) and \(R\).
  4. Hence find, in terms of \(a\), the area of triangle \(B Q R\).
Edexcel F1 2017 January Q9
12 marks Standard +0.3
  1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } - 3 r ^ { 2 } + r \right) = n ^ { 3 } ( n + 1 )$$
  2. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 5 ^ { 2 n } + 3 n - 1$$ is divisible by 9
Edexcel F1 2018 January Q1
7 marks Standard +0.3
1. $$f ( x ) = 3 x ^ { 2 } - \frac { 5 } { 3 \sqrt { x } } - 6 , \quad x > 0$$ The single root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [1.5, 1.6].
  1. Taking 1.5 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.
    [0pt]
  2. Use linear interpolation once on the interval [1.5, 1.6] to find another approximation to \(\alpha\). Give your answer to 3 decimal places.