Questions — Edexcel (10514 questions)

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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
  1. (i)
$$f ( x ) = x - 4 - \cos ( 5 \sqrt { x } ) \quad x > 0$$
  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) $$\operatorname { g } ( x ) = \frac { 1 } { 10 } x ^ { 2 } - \frac { 1 } { 2 x ^ { 2 } } + x - 11 \quad x > 0$$
  1. Determine \(\mathrm { g } ^ { \prime } ( x )\). The equation \(\mathrm { g } ( x ) = 0\) has a root \(\beta\) in the interval [6,7]
  2. 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. Use the standard results for summations to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r \left( 2 r ^ { 2 } - 3 r - 1 \right) = \frac { 1 } { 2 } n ( n + 1 ) ^ { 2 } ( n - 2 )$$
  2. 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. Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$\left( \begin{array} { r r } 5 & - 1 \\ 4 & 1 \end{array} \right) ^ { n } = 3 ^ { n - 1 } \left( \begin{array} { c c } 2 n + 3 & - n \\ 4 n & 3 - 2 n \end{array} \right)$$
  2. 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
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.
Edexcel F1 2014 June Q6
8 marks Standard +0.8
6. It is given that \(\alpha\) and \(\beta\) are roots of the equation \(3 x ^ { 2 } + 5 x - 1 = 0\)
  1. Find the exact value of \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  2. Find a quadratic equation which has roots \(\frac { \alpha ^ { 2 } } { \beta }\) and \(\frac { \beta ^ { 2 } } { \alpha }\), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a\), \(b\) and \(c\) are integers.
Edexcel F1 2014 June Q7
11 marks Moderate -0.3
7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$
  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 \(x\)-axis.
  2. Write down the matrix \(\mathbf { Q }\). Given that \(V\) followed by \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
  3. find the matrix \(\mathbf { R }\).
  4. Show that there is a real number \(k\) for which the transformation \(T\) maps the point \(( 1 , k )\) onto itself. Give the exact value of \(k\) in its simplest form.
Edexcel F1 2014 June Q8
14 marks Standard +0.3
8. The hyperbola \(H\) has cartesian equation \(x y = 16\) The parabola \(P\) has parametric equations \(x = 8 t ^ { 2 } , y = 16 t\).
  1. Find, using algebra, the coordinates of the point \(A\) where \(H\) meets \(P\). Another point \(B ( 8,2 )\) lies on the hyperbola \(H\).
  2. Find the equation of the normal to \(H\) at the point (8, 2), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
  3. Find the coordinates of the points where this normal at \(B\) meets the parabola \(P\).
Edexcel F1 2014 June Q9
11 marks Standard +0.3
9.
  1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 2 ) = \frac { n ( n + 1 ) ( n + 2 ) ( n + 3 ) } { 4 }$$
  2. Prove by induction that, $$4 ^ { n } + 6 n + 8 \text { is divisible by } 18$$ for all positive integers \(n\). \includegraphics[max width=\textwidth, alt={}, center]{df5ab400-5cb1-4b51-8b0a-52dc3587f81a-16_62_44_2476_1889}
Edexcel F1 2015 June Q1
5 marks Moderate -0.8
  1. Given that
$$2 z ^ { 3 } - 5 z ^ { 2 } + 7 z - 6 \equiv ( 2 z - 3 ) \left( z ^ { 2 } + a z + b \right)$$ where \(a\) and \(b\) are real constants,
  1. find the value of \(a\) and the value of \(b\).
  2. Given that \(z\) is a complex number, find the three exact roots of the equation $$2 z ^ { 3 } - 5 z ^ { 2 } + 7 z - 6 = 0$$
Edexcel F1 2015 June Q2
5 marks Standard +0.3
  1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that
$$\sum _ { r = 1 } ^ { n } ( 3 r - 2 ) ^ { 2 } = \frac { n } { 2 } \left( a n ^ { 2 } + b n + c \right)$$ where \(a , b\) and \(c\) are integers to be found.
Edexcel F1 2015 June Q3
6 marks Standard +0.3
3. It is given that \(\alpha\) and \(\beta\) are roots of the equation $$2 x ^ { 2 } - 7 x + 4 = 0$$
  1. Find the exact value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
  2. Find a quadratic equation which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel F1 2015 June Q4
6 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{04f06398-ff29-4690-a6fe-825d089fba39-05_663_665_228_644} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(C\) and the point \(Q\) lies on the directrix of \(C\). The point \(P\) lies on \(C\) where \(y > 0\) and the line segment \(Q P\) is parallel to the \(x\)-axis. Given that the length of \(P S\) is 13
  1. write down the length of \(P Q\). Given that the point \(P\) has \(x\) coordinate 9
    find
  2. the value of \(a\),
  3. the area of triangle \(P S Q\).
Edexcel F1 2015 June Q5
7 marks Standard +0.3
  1. In the interval \(2 < x < 3\), the equation
$$6 - x ^ { 2 } \cos \left( \frac { x } { 5 } \right) = 0 , \text { where } x \text { is measured in radians }$$ has exactly one root \(\alpha\).
[0pt]
  1. Starting with the interval [2,3], use interval bisection twice to find an interval of width 0.25 which contains \(\alpha\).
    [0pt]
  2. Use linear interpolation once on the interval [2,3] to find an approximation to \(\alpha\). Give your answer to 2 decimal places.
Edexcel F1 2015 June Q6
10 marks Challenging +1.2
6. The rectangular hyperbola, \(H\), has cartesian equation $$x y = 36$$ The three points \(P \left( 6 p , \frac { 6 } { p } \right) , Q \left( 6 q , \frac { 6 } { q } \right)\) and \(R \left( 6 r , \frac { 6 } { r } \right)\), where \(p , q\) and \(r\) are distinct, non-zero values, lie on the hyperbola \(H\).
  1. Show that an equation of the line \(P Q\) is $$p q y + x = 6 ( p + q )$$ Given that \(P R\) is perpendicular to \(Q R\),
  2. show that the normal to the curve \(H\) at the point \(R\) is parallel to the line \(P Q\).