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Edexcel FP1 2014 June Q4
9 marks Standard +0.3
4. The complex number \(z\) is given by $$z = \frac { p + 2 \mathrm { i } } { 3 + p \mathrm { i } }$$ where \(p\) is an integer.
  1. Express \(z\) in the form \(a + b \mathrm { i }\) where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\).
  2. Given that \(\arg ( z ) = \theta\), where \(\tan \theta = 1\) find the possible values of \(p\).
Edexcel FP1 2014 June Q5
8 marks Standard +0.3
  1. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that
$$\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right) = \frac { 1 } { 4 } n ( n + 1 ) ( n + 3 ) ( n - 2 )$$ (b) Calculate the value of \(\sum _ { r = 10 } ^ { 50 } r \left( r ^ { 2 } - 3 \right)\)
Edexcel FP1 2014 June Q6
10 marks Standard +0.3
6. $$\mathbf { A } = \left( \begin{array} { r r } 2 & 1 \\ - 1 & 0 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r } - 1 & 1 \\ 0 & 1 \end{array} \right)$$ Given that \(\mathbf { M } = ( \mathbf { A } + \mathbf { B } ) ( 2 \mathbf { A } - \mathbf { B } )\),
  1. calculate the matrix \(\mathbf { M }\),
  2. find the matrix \(\mathbf { C }\) such that \(\mathbf { M C } = \mathbf { A }\).
Edexcel FP1 2014 June Q7
11 marks Standard +0.8
7. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 4 a x , a > 0\) The points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(P ^ { \prime } \left( a p ^ { 2 } , - 2 a p \right)\) lie on \(C\).
  1. Show that an equation of the normal to \(C\) at the point \(P\) is $$y + p x = 2 a p + a p ^ { 3 }$$
  2. Write down an equation of the normal to \(C\) at the point \(P ^ { \prime }\). The normal to \(C\) at \(P\) meets the normal to \(C\) at \(P ^ { \prime }\) at the point \(Q\).
  3. Find, in terms of \(a\) and \(p\), the coordinates of \(Q\). Given that \(S\) is the focus of the parabola,
  4. find the area of the quadrilateral \(S P Q P ^ { \prime }\).
Edexcel FP1 2014 June Q8
5 marks Challenging +1.2
8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a positive constant. The point \(P \left( c t , \frac { c } { t } \right) , t \neq 0\), is a general point on \(H\). An equation for the tangent to \(H\) at \(P\) is given by $$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 2 c } { t }$$ The points \(A\) and \(B\) lie on \(H\).
The tangent to \(H\) at \(A\) and the tangent to \(H\) at \(B\) meet at the point \(\left( - \frac { 6 } { 7 } c , \frac { 12 } { 7 } c \right)\).
Find, in terms of \(c\), the coordinates of \(A\) and the coordinates of \(B\).
Edexcel FP1 2014 June Q9
12 marks Standard +0.8
9. (a) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } ( r + 1 ) 2 ^ { r - 1 } = n 2 ^ { n }$$ (b) A sequence of numbers is defined by $$\begin{gathered} u _ { 1 } = 0 , \quad u _ { 2 } = 32 , \\ u _ { n + 2 } = 6 u _ { n + 1 } - 8 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 4 ^ { n + 1 } - 2 ^ { n + 3 }$$
Edexcel FP1 2014 June Q1
8 marks Standard +0.3
  1. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by
$$z _ { 1 } = p + 2 i \text { and } z _ { 2 } = 1 - 2 i$$ where \(p\) is an integer.
  1. Find \(\frac { z _ { 1 } } { z _ { 2 } }\) in the form \(a + b\) i where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\). Given that \(\left| \frac { z _ { 1 } } { z _ { 2 } } \right| = 13\),
  2. find the possible values of \(p\).
Edexcel FP1 2014 June Q2
7 marks Standard +0.3
2. $$\mathrm { f } ( x ) = x ^ { 3 } - \frac { 5 } { 2 x ^ { \frac { 3 } { 2 } } } + 2 x - 3 , \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [1.1,1.5].
  2. Find f'(x).
  3. Using \(x _ { 0 } = 1.1\) 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.
Edexcel FP1 2014 June Q3
8 marks Moderate -0.3
3. Given that 2 and \(1 - 5 \mathrm { i }\) are roots of the equation $$x ^ { 3 } + p x ^ { 2 } + 30 x + q = 0 , \quad p , q \in \mathbb { R }$$
  1. write down the third root of the equation.
  2. Find the value of \(p\) and the value of \(q\).
  3. Show the three roots of this equation on a single Argand diagram.
Edexcel FP1 2014 June Q4
7 marks Moderate -0.8
4. (i) Given that $$\mathbf { A } = \left( \begin{array} { r r } 1 & 2 \\ 3 & - 1 \\ 4 & 5 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r r } 2 & - 1 & 4 \\ 1 & 3 & 1 \end{array} \right)$$
  1. find \(\mathbf { A B }\).
  2. Explain why \(\mathbf { A B } \neq \mathbf { B A }\).
    (ii) Given that $$\mathbf { C } = \left( \begin{array} { c r } 2 k & - 2 \\ 3 & k \end{array} \right) \text {, where } k \text { is a real number }$$ find \(\mathbf { C } ^ { - 1 }\), giving your answer in terms of \(k\).
Edexcel FP1 2014 June Q5
9 marks Standard +0.8
5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$ (b) Hence show that $$\sum _ { r = 2 n + 1 } ^ { 4 n } ( 2 r - 1 ) ^ { 2 } = a n \left( b n ^ { 2 } - 1 \right)$$ where \(a\) and \(b\) are constants to be found.
Edexcel FP1 2014 June Q6
9 marks Standard +0.3
6. The rectangular hyperbola \(H\) has cartesian equation \(x y = c ^ { 2 }\). The point \(P \left( c t , \frac { c } { t } \right) , t > 0\), is a general point on \(H\).
  1. Show that an equation of the tangent to \(H\) at the point \(P\) is $$t ^ { 2 } y + x = 2 c t$$ An equation of the normal to \(H\) at the point \(P\) is \(t ^ { 3 } x - t y = c t ^ { 4 } - c\) Given that the normal to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and the tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(B\),
  2. find, in terms of \(c\) and \(t\), the coordinates of \(A\) and the coordinates of \(B\). Given that \(c = 4\),
  3. find, in terms of \(t\), the area of the triangle \(A P B\). Give your answer in its simplest form.
Edexcel FP1 2014 June Q7
10 marks Standard +0.3
7. (i) In each of the following cases, find a \(2 \times 2\) matrix that represents
  1. a reflection in the line \(y = - x\),
  2. a rotation of \(135 ^ { \circ }\) anticlockwise about \(( 0,0 )\),
  3. a reflection in the line \(y = - x\) followed by a rotation of \(135 ^ { \circ }\) anticlockwise about \(( 0,0 )\).
    (ii) The triangle \(T\) has vertices at the points \(( 1 , k ) , ( 3,0 )\) and \(( 11,0 )\), where \(k\) is a constant. Triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the matrix $$\left( \begin{array} { r r } 6 & - 2 \\ 1 & 2 \end{array} \right)$$ Given that the area of triangle \(T ^ { \prime }\) is 364 square units, find the value of \(k\).
Edexcel FP1 2014 June Q8
11 marks Standard +0.8
8. The points \(P \left( 4 k ^ { 2 } , 8 k \right)\) and \(Q \left( k ^ { 2 } , 4 k \right)\), where \(k\) is a constant, lie on the parabola \(C\) with equation \(y ^ { 2 } = 16 x\). The straight line \(l _ { 1 }\) passes through the points \(P\) and \(Q\).
  1. Show that an equation of the line \(l _ { 1 }\) is given by $$3 k y - 4 x = 8 k ^ { 2 }$$ The line \(l _ { 2 }\) is perpendicular to the line \(l _ { 1 }\) and passes through the focus of the parabola \(C\). The line \(l _ { 2 }\) meets the directrix of \(C\) at the point \(R\).
  2. Find, in terms of \(k\), the \(y\) coordinate of the point \(R\).
Edexcel FP1 2014 June Q9
6 marks
9. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$f ( n ) = 8 ^ { n } - 2 ^ { n }$$ is divisible by 6
Edexcel FP1 2015 June Q1
5 marks Moderate -0.5
1. $$f ( x ) = 9 x ^ { 3 } - 33 x ^ { 2 } - 55 x - 25$$ Given that \(x = 5\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), use an algebraic method to solve \(\mathrm { f } ( x ) = 0\) completely.
(5)
Edexcel FP1 2015 June Q2
7 marks Standard +0.3
2. In the interval \(13 < x < 14\), the equation $$3 + x \sin \left( \frac { x } { 4 } \right) = 0 , \text { where } x \text { is measured in radians, }$$ has exactly one root, \(\alpha\).
[0pt]
  1. Starting with the interval [13,14], use interval bisection twice to find an interval of width 0.25 which contains \(\alpha\).
    [0pt]
  2. Use linear interpolation once on the interval [13,14] to find an approximate value for \(\alpha\). Give your answer to 3 decimal places.
Edexcel FP1 2015 June Q3
8 marks Standard +0.3
3. (a) Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that $$\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 4 ) = \frac { n } { 3 } ( n + 4 ) ( n + 5 )$$ for all positive integers \(n\).
(b) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 1 ) ( r + 4 ) = \frac { n } { 3 } ( n + 1 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be found.
Edexcel FP1 2015 June Q4
8 marks Moderate -0.8
4. $$z _ { 1 } = 3 \mathrm { i } \text { and } z _ { 2 } = \frac { 6 } { 1 + \mathrm { i } \sqrt { 3 } }$$
  1. Express \(z _ { 2 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.
  2. Find the modulus and the argument of \(z _ { 2 }\), giving the argument in radians in terms of \(\pi\).
  3. Show the three points representing \(z _ { 1 } , z _ { 2 }\) and \(\left( z _ { 1 } + z _ { 2 } \right)\) respectively, on a single Argand diagram.
Edexcel FP1 2015 June Q5
9 marks Challenging +1.2
5. The rectangular hyperbola \(H\) has equation \(x y = 9\) The point \(A\) on \(H\) has coordinates \(\left( 6 , \frac { 3 } { 2 } \right)\).
  1. Show that the normal to \(H\) at the point \(A\) has equation $$2 y - 8 x + 45 = 0$$ The normal at \(A\) meets \(H\) again at the point \(B\).
  2. Find the coordinates of \(B\).
Edexcel FP1 2015 June Q6
12 marks Standard +0.3
  1. (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
$$\left( \begin{array} { r r } 1 & 0 \\ - 1 & 5 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & 0 \\ - \frac { 1 } { 4 } \left( 5 ^ { n } - 1 \right) & 5 ^ { n } \end{array} \right)$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
Edexcel FP1 2015 June Q7
12 marks Standard +0.3
$$\mathbf { A } = \left( \begin{array} { r r } 5 k & 3 k - 1 \\ - 3 & k + 1 \end{array} \right) , \text { where } k \text { is a real constant. }$$ Given that \(\mathbf { A }\) is a singular matrix, find the possible values of \(k\).
(ii) $$\mathbf { B } = \left( \begin{array} { l l } 10 & 5 \\ - 3 & 3 \end{array} \right)$$ A triangle \(T\) is transformed onto a triangle \(T ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { B }\). The vertices of triangle \(T ^ { \prime }\) have coordinates \(( 0,0 ) , ( - 20,6 )\) and \(( 10 c , 6 c )\), where \(c\) is a positive constant. The area of triangle \(T ^ { \prime }\) is 135 square units.
  1. Find the matrix \(\mathbf { B } ^ { - 1 }\)
  2. Find the coordinates of the vertices of the triangle \(T\), in terms of \(c\) where necessary.
  3. Find the value of \(c\).
Edexcel FP1 2015 June Q8
14 marks Challenging +1.2
  1. The point \(P \left( 3 p ^ { 2 } , 6 p \right)\) lies on the parabola with equation \(y ^ { 2 } = 12 x\) and the point \(S\) is the focus of this parabola.
    1. Prove that \(S P = 3 \left( 1 + p ^ { 2 } \right)\)
    The point \(Q \left( 3 q ^ { 2 } , 6 q \right) , p \neq q\), also lies on this parabola.
    The tangent to the parabola at the point \(P\) and the tangent to the parabola at the point \(Q\) meet at the point \(R\).
  2. Find the equations of these two tangents and hence find the coordinates of the point \(R\), giving the coordinates in their simplest form.
  3. Prove that \(S R ^ { 2 } = S P \cdot S Q\)
Edexcel FP1 2016 June Q1
3 marks Moderate -0.8
  1. Given that \(k\) is a real number and that
$$\mathbf { A } = \left( \begin{array} { c c } 1 + k & k \\ k & 1 - k \end{array} \right)$$ find the exact values of \(k\) for which \(\mathbf { A }\) is a singular matrix. Give your answers in their simplest form.
(3)
Edexcel FP1 2016 June Q2
6 marks Standard +0.3
2. $$f ( x ) = 3 x ^ { \frac { 3 } { 2 } } - 25 x ^ { - \frac { 1 } { 2 } } - 125 , \quad x > 0$$
  1. Find \(f ^ { \prime } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [12, 13].
  2. Using \(x _ { 0 } = 12.5\) 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.
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