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Edexcel FP1 2012 January Q3
8 marks Moderate -0.3
3. A parabola \(C\) has cartesian equation \(y ^ { 2 } = 16 x\). The point \(P \left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).
  1. Write down the coordinates of the focus \(F\) and the equation of the directrix of \(C\).
  2. Show that the equation of the normal to \(C\) at \(P\) is \(y + t x = 8 t + 4 t ^ { 3 }\).
Edexcel FP1 2012 January Q4
11 marks Moderate -0.8
4. A right angled triangle \(T\) has vertices \(A ( 1,1 ) , B ( 2,1 )\) and \(C ( 2,4 )\). When \(T\) is transformed by the matrix \(\mathbf { P } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\), the image is \(T ^ { \prime }\).
  1. Find the coordinates of the vertices of \(T ^ { \prime }\).
  2. Describe fully the transformation represented by \(\mathbf { P }\). The matrices \(\mathbf { Q } = \left( \begin{array} { c c } 4 & - 2 \\ 3 & - 1 \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)\) represent two transformations. When \(T\) is transformed by the matrix \(\mathbf { Q R }\), the image is \(T ^ { \prime \prime }\).
  3. Find \(\mathbf { Q R }\).
  4. Find the determinant of \(\mathbf { Q R }\).
  5. Using your answer to part (d), find the area of \(T ^ { \prime \prime }\).
Edexcel FP1 2012 January Q5
6 marks Moderate -0.3
5. The roots of the equation $$z ^ { 3 } - 8 z ^ { 2 } + 22 z - 20 = 0$$ are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).
  1. Given that \(z _ { 1 } = 3 + \mathrm { i }\), find \(z _ { 2 }\) and \(z _ { 3 }\).
  2. Show, on a single Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).
Edexcel FP1 2012 January Q6
11 marks Standard +0.3
6.
  1. Prove by induction $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$
  2. Using the result in part (a), show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } - 2 \right) = \frac { 1 } { 4 } n \left( n ^ { 3 } + 2 n ^ { 2 } + n - 8 \right)$$
  3. Calculate the exact value of \(\sum _ { r = 20 } ^ { 50 } \left( r ^ { 3 } - 2 \right)\).
Edexcel FP1 2012 January Q7
7 marks Standard +0.3
7. A sequence can be described by the recurrence formula $$u _ { n + 1 } = 2 u _ { n } + 1 , \quad n \geqslant 1 , \quad u _ { 1 } = 1$$
  1. Find \(u _ { 2 }\) and \(u _ { 3 }\).
  2. Prove by induction that \(u _ { n } = 2 ^ { n } - 1\)
Edexcel FP1 2012 January Q8
6 marks Moderate -0.3
8. $$\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 2 & 3 \end{array} \right)$$
  1. Show that \(\mathbf { A }\) is non-singular.
  2. Find \(\mathbf { B }\) such that \(\mathbf { B A } ^ { 2 } = \mathbf { A }\).
Edexcel FP1 2012 January Q9
9 marks Standard +0.8
9. The rectangular hyperbola \(H\) has cartesian equation \(x y = 9\) The points \(P \left( 3 p , \frac { 3 } { p } \right)\) and \(Q \left( 3 q , \frac { 3 } { q } \right)\) lie on \(H\), where \(p \neq \pm q\).
  1. Show that the equation of the tangent at \(P\) is \(x + p ^ { 2 } y = 6 p\).
  2. Write down the equation of the tangent at \(Q\). The tangent at the point \(P\) and the tangent at the point \(Q\) intersect at \(R\).
  3. Find, as single fractions in their simplest form, the coordinates of \(R\) in terms of \(p\) and \(q\).
Edexcel FP1 2013 January Q1
5 marks Moderate -0.3
  1. Show, using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), that
$$\sum _ { r = 1 } ^ { n } 3 ( 2 r - 1 ) ^ { 2 } = n ( 2 n + 1 ) ( 2 n - 1 ) , \text { for all positive integers } n .$$
Edexcel FP1 2013 January Q2
8 marks Moderate -0.3
2. $$z = \frac { 50 } { 3 + 4 \mathrm { i } }$$ Find, in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\),
  1. \(z\),
  2. \(z ^ { 2 }\). Find
  3. \(| z |\),
  4. \(\arg z ^ { 2 }\), giving your answer in degrees to 1 decimal place.
Edexcel FP1 2013 January Q3
6 marks Moderate -0.3
3. $$\mathrm { f } ( x ) = 2 x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } - 5 , \quad x > 0$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [4.5, 5.5].
  2. Using \(x _ { 0 } = 5\) 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 significant figures.
Edexcel FP1 2013 January Q4
7 marks Moderate -0.8
4. The transformation \(U\), represented by the \(2 \times 2\) matrix \(\mathbf { P }\), is a rotation through \(90 ^ { \circ }\) anticlockwise about the origin.
  1. Write down the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line \(y = - x\).
  2. Write down the matrix \(\mathbf { Q }\). Given that \(U\) followed by \(V\) is transformation \(T\), which is represented by the matrix \(\mathbf { R }\), (c) express \(\mathbf { R }\) in terms of \(\mathbf { P }\) and \(\mathbf { Q }\),
    (d) find the matrix \(\mathbf { R }\),
    (e) give a full geometrical description of \(T\) as a single transformation.
Edexcel FP1 2013 January Q5
7 marks Moderate -0.8
5. $$f ( x ) = \left( 4 x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 6 x + 34 \right)$$
  1. Find the four roots of \(\mathrm { f } ( x ) = 0\) Give your answers in the form \(x = p + \mathrm { i } q\), where \(p\) and \(q\) are real.
  2. Show these four roots on a single Argand diagram.
Edexcel FP1 2013 January Q6
8 marks Moderate -0.8
6. \(\mathbf { X } = \left( \begin{array} { l l } 1 & a \\ 3 & 2 \end{array} \right)\), where \(a\) is a constant.
  1. Find the value of \(a\) for which the matrix \(\mathbf { X }\) is singular. $$\mathbf { Y } = \left( \begin{array} { r r } 1 & - 1 \\ 3 & 2 \end{array} \right)$$
  2. Find \(\mathbf { Y } ^ { - 1 }\). The transformation represented by \(\mathbf { Y }\) maps the point \(A\) onto the point \(B\).
    Given that \(B\) has coordinates ( \(1 - \lambda , 7 \lambda - 2\) ), where \(\lambda\) is a constant,
  3. find, in terms of \(\lambda\), the coordinates of point \(A\).
Edexcel FP1 2013 January Q7
14 marks Challenging +1.2
7. The rectangular hyperbola, \(H\), has cartesian equation \(x y = 25\) The point \(P \left( 5 p , \frac { 5 } { p } \right)\), and the point \(Q \left( 5 q , \frac { 5 } { q } \right)\), where \(p , q \neq 0 , p \neq q\), are points on the rectangular hyperbola \(H\).
  1. Show that the equation of the tangent at point \(P\) is $$p ^ { 2 } y + x = 10 p$$
  2. Write down the equation of the tangent at point \(Q\). The tangents at \(P\) and \(Q\) meet at the point \(N\).
    Given \(p + q \neq 0\),
  3. show that point \(N\) has coordinates \(\left( \frac { 10 p q } { p + q } , \frac { 10 } { p + q } \right)\). The line joining \(N\) to the origin is perpendicular to the line \(P Q\).
  4. Find the value of \(p ^ { 2 } q ^ { 2 }\).
Edexcel FP1 2013 January Q8
11 marks Standard +0.8
8.
  1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } r ( r + 3 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 5 )$$
  2. A sequence of positive integers is defined by $$\begin{aligned} u _ { 1 } & = 1 \\ u _ { n + 1 } & = u _ { n } + n ( 3 n + 1 ) , \quad n \in \mathbb { Z } ^ { + } \end{aligned}$$ Prove by induction that $$u _ { n } = n ^ { 2 } ( n - 1 ) + 1 , \quad n \in \mathbb { Z } ^ { + }$$
Edexcel FP1 2013 January Q9
9 marks Standard +0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7833e9c0-4a73-4ac6-8a77-51a5489e0614-10_624_716_210_614} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the parabola with equation \(y ^ { 2 } = 36 x\).
The point \(P ( 4,12 )\) lies on the parabola.
  1. Find an equation for the normal to the parabola at \(P\). This normal meets the \(x\)-axis at the point \(N\) and \(S\) is the focus of the parabola, as shown in Figure 1.
  2. Find the area of triangle \(P S N\).
Edexcel FP1 2014 January Q1
5 marks Moderate -0.8
\(\mathrm { f } ( x ) = 2 x - 5 \cos x , \quad\) where \(x\) is in radians.
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 1,1.4 ]\).
    [0pt]
  2. Starting with the interval [1,1.4], use interval bisection twice to find an interval of width 0.1 which contains \(\alpha\).
Edexcel FP1 2014 January Q2
7 marks Standard +0.3
2.
  1. $$\mathbf { A } = \left( \begin{array} { c c } - 4 & 10 \\ - 3 & k \end{array} \right) , \quad \text { where } k \text { is a constant. }$$ The triangle \(T\) is transformed to the triangle \(T ^ { \prime }\) by the transformation represented by \(\mathbf { A }\). Given that the area of triangle \(T ^ { \prime }\) is twice the area of triangle \(T\), find the possible values of \(k\).
  2. Given that $$\mathbf { B } = \left( \begin{array} { r r r } 1 & - 2 & 3 \\ - 2 & 5 & 1 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { r r } 2 & 8 \\ 0 & 2 \\ 1 & - 2 \end{array} \right)$$ find \(\mathbf { B C }\). \includegraphics[max width=\textwidth, alt={}, center]{9093bb1d-4f32-44e7-b0e7-b8c4f8a844e1-05_124_42_2608_1902}
Edexcel FP1 2014 January Q3
4 marks Standard +0.3
3. A rectangular hyperbola has parametric equations $$x = 2 t , \quad y = \frac { 2 } { t } , \quad t \neq 0$$ Points \(P\) and \(Q\) on this hyperbola have parameters \(t = \frac { 1 } { 2 }\) and \(t = 4\) respectively.
The line \(L\), which passes through the origin \(O\), is perpendicular to the chord \(P Q\).
Find an equation for \(L\).
Edexcel FP1 2014 January Q4
5 marks Standard +0.3
4. $$f ( x ) = 2 x ^ { \frac { 1 } { 2 } } - \frac { 6 } { x ^ { 2 } } - 3 , \quad x > 0$$ A root \(\beta\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ 3,4 ]\).
Taking 3.5 as a first approximation to \(\beta\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\beta\). Give your answer to 3 decimal places.
Edexcel FP1 2014 January Q5
8 marks Standard +0.3
5. $$z = 5 + \mathrm { i } \sqrt { 3 } , \quad w = \sqrt { 3 } - \mathrm { i }$$
  1. Find the value of \(| w |\). Find in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real constants,
  2. \(z w\), showing clearly how you obtained your answer,
  3. \(\frac { z } { w }\), showing clearly how you obtained your answer. Given that $$\arg ( z + \lambda ) = \frac { \pi } { 3 } , \quad \text { where } \lambda \text { is a real constant, }$$
  4. find the value of \(\lambda\).
Edexcel FP1 2014 January Q6
9 marks Standard +0.3
6.
  1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = \frac { 1 } { 4 } n ( n + 1 ) ( n - 1 ) ( n + a )$$ where \(a\) is an integer to be determined.
  2. Hence find the value of \(n\), where \(n > 1\), that satisfies $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = 10 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$
Edexcel FP1 2014 January Q7
6 marks Standard +0.3
7. $$\mathbf { P } = \left( \begin{array} { c c } 3 a & - 2 a \\ - b & 2 b \end{array} \right) , \quad \mathbf { M } = \left( \begin{array} { c c } - 6 a & 7 a \\ 2 b & - b \end{array} \right)$$ where \(a\) and \(b\) are non-zero constants.
  1. Find \(\mathbf { P } ^ { - 1 }\), leaving your answer in terms of \(a\) and \(b\). Given that $$\mathbf { M } = \mathbf { P Q }$$
  2. find the matrix \(\mathbf { Q }\), giving your answer in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{9093bb1d-4f32-44e7-b0e7-b8c4f8a844e1-19_95_77_2617_1804}
Edexcel FP1 2014 January Q8
12 marks Standard +0.8
8. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(P \left( a p ^ { 2 } , 2 a p \right)\) lies on the parabola \(C\).
  1. Show that an equation of the normal to \(C\) at \(P\) is $$y + p x = a p ^ { 3 } + 2 a p$$ The normal to \(C\) at the point \(P\) meets the \(x\)-axis at the point \(( 6 a , 0 )\) and meets the directrix of \(C\) at the point \(D\). Given that \(p > 0\),
  2. find, in terms of \(a\), the coordinates of \(D\). Given also that the directrix of \(C\) cuts the \(x\)-axis at the point \(X\),
  3. find, in terms of \(a\), the area of the triangle XPD, giving your answer in its simplest form.
Edexcel FP1 2014 January Q9
8 marks Standard +0.3
9. Given that \(z = x + \mathrm { i } y\), where \(x \in \mathbb { R } , y \in \mathbb { R }\), find the value of \(x\) and the value of \(y\) such that $$( 3 - i ) z ^ { * } + 2 i z = 9 - i$$ where \(z ^ { * }\) is the complex conjugate of \(z\).