Questions — Edexcel (9685 questions)

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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. (a) 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.
(b) 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 \(z\).
Edexcel FP1 2014 January Q10
11 marks Standard +0.3
10. (i) A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), is defined by $$u _ { n + 1 } = 5 u _ { n } + 3 , \quad u _ { 1 } = 3$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = \frac { 3 } { 4 } \left( 5 ^ { n } - 1 \right)$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$f ( n ) = 5 \left( 5 ^ { n } \right) - 4 n - 5 \text { is divisible by } 16 .$$ \includegraphics[max width=\textwidth, alt={}, center]{9093bb1d-4f32-44e7-b0e7-b8c4f8a844e1-32_109_127_2473_1818} \includegraphics[max width=\textwidth, alt={}, center]{9093bb1d-4f32-44e7-b0e7-b8c4f8a844e1-32_205_1828_2553_122}
Edexcel FP1 2009 June Q1
8 marks Moderate -0.8
  1. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by
$$z _ { 1 } = 2 - i \quad \text { and } \quad z _ { 2 } = - 8 + 9 i$$
  1. Show \(z _ { 1 }\) and \(z _ { 2 }\) on a single Argand diagram. Find, showing your working,
  2. the value of \(\left| z _ { 1 } \right|\),
  3. the value of \(\arg z _ { 1 }\), giving your answer in radians to 2 decimal places,
  4. \(\frac { Z _ { 2 } } { Z _ { 1 } }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real.
Edexcel FP1 2009 June Q2
9 marks Standard +0.8
2. (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 FP1 2009 June Q3
7 marks Moderate -0.8
3. $$f ( x ) = \left( x ^ { 2 } + 4 \right) \left( x ^ { 2 } + 8 x + 25 \right)$$
  1. Find the four roots of \(\mathrm { f } ( x ) = 0\).
  2. Find the sum of these four roots.
Edexcel FP1 2009 June Q4
10 marks Moderate -0.3
4. Given that \(\alpha\) is the only real root of the equation $$x ^ { 3 } - x ^ { 2 } - 6 = 0$$
  1. show that \(2.2 < \alpha < 2.3\)
  2. Taking 2.2 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 6\) to obtain a second approximation to \(\alpha\), giving your answer to 3 decimal places.
    [0pt]
  3. Use linear interpolation once on the interval [2.2, 2.3] to find another approximation to \(\alpha\), giving your answer to 3 decimal places.
Edexcel FP1 2009 June Q5
8 marks Standard +0.3
5. \(\mathbf { R } = \left( \begin{array} { l l } a & 2 \\ a & b \end{array} \right)\), where \(a\) and \(b\) are constants and \(a > 0\).
  1. Find \(\mathbf { R } ^ { 2 }\) in terms of \(a\) and \(b\). Given that \(\mathbf { R } ^ { 2 }\) represents an enlargement with centre ( 0,0 ) and scale factor 15 ,
  2. find the value of \(a\) and the value of \(b\).
Edexcel FP1 2009 June Q6
11 marks Standard +0.3
6. The parabola \(C\) has equation \(y ^ { 2 } = 16 x\).
  1. Verify that the point \(P \left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).
  2. Write down the coordinates of the focus \(S\) of \(C\).
  3. Show that the normal to \(C\) at \(P\) has equation $$y + t x = 8 t + 4 t ^ { 3 }$$ The normal to \(C\) at \(P\) meets the \(x\)-axis at the point \(N\).
  4. Find the area of triangle \(P S N\) in terms of \(t\), giving your answer in its simplest form.
Edexcel FP1 2009 June Q7
8 marks Moderate -0.8
7. \(\mathbf { A } = \left( \begin{array} { r r } a & - 2 \\ - 1 & 4 \end{array} \right)\), where \(a\) is a constant.
  1. Find the value of \(a\) for which the matrix \(\mathbf { A }\) is singular. $$\mathbf { B } = \left( \begin{array} { r r } 3 & - 2 \\ - 1 & 4 \end{array} \right)$$
  2. Find \(\mathbf { B } ^ { - 1 }\). The transformation represented by \(\mathbf { B }\) maps the point \(P\) onto the point \(Q\).
    Given that \(Q\) has coordinates \(( k - 6,3 k + 12 )\), where \(k\) is a constant,
  3. show that \(P\) lies on the line with equation \(y = x + 3\).
Edexcel FP1 2009 June Q8
14 marks Standard +0.3
8. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
  1. \(\mathrm { f } ( n ) = 5 ^ { n } + 8 n + 3\) is divisible by 4 ,
  2. \(\left( \begin{array} { l l } 3 & - 2 \\ 2 & - 1 \end{array} \right) ^ { n } = \left( \begin{array} { l r } 2 n + 1 & - 2 n \\ 2 n & 1 - 2 n \end{array} \right)\)
Edexcel FP1 2010 June Q1
7 marks Moderate -0.5
1. $$z = 2 - 3 \mathrm { i }$$
  1. Show that \(z ^ { 2 } = - 5 - 12 \mathrm { i }\). Find, showing your working,
  2. the value of \(\left| z ^ { 2 } \right|\),
  3. the value of \(\arg \left( z ^ { 2 } \right)\), giving your answer in radians to 2 decimal places.
  4. Show \(z\) and \(z ^ { 2 }\) on a single Argand diagram.
Edexcel FP1 2010 June Q2
5 marks Moderate -0.8
2. \(\mathbf { M } = \left( \begin{array} { c c } 2 a & 3 \\ 6 & a \end{array} \right)\), where \(a\) is a real constant.
  1. Given that \(a = 2\), find \(\mathbf { M } ^ { - 1 }\).
  2. Find the values of \(a\) for which \(\mathbf { M }\) is singular.
Edexcel FP1 2010 June Q3
10 marks Standard +0.3
3. $$\mathrm { f } ( x ) = x ^ { 3 } - \frac { 7 } { x } + 2 , \quad x > 0$$
  1. Show that \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between 1.4 and 1.5
    [0pt]
  2. Starting with the interval [1.4,1.5], use interval bisection twice to find an interval of width 0.025 that contains \(\alpha\).
  3. Taking 1.45 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x ) = x ^ { 3 } - \frac { 7 } { x } + 2\) to obtain a second approximation to \(\alpha\), giving your answer to 3 decimal places.
Edexcel FP1 2010 June Q4
7 marks Moderate -0.8
4. $$f ( x ) = x ^ { 3 } + x ^ { 2 } + 44 x + 150$$ Given that \(\mathrm { f } ( x ) = ( x + 3 ) \left( x ^ { 2 } + a x + b \right)\), where \(a\) and \(b\) are real constants,
  1. find the value of \(a\) and the value of \(b\).
  2. Find the three roots of \(\mathrm { f } ( x ) = 0\).
  3. Find the sum of the three roots of \(\mathrm { f } ( x ) = 0\).
Edexcel FP1 2010 June Q5
5 marks Moderate -0.8
5. The parabola \(C\) has equation \(y ^ { 2 } = 20 x\).
  1. Verify that the point \(P \left( 5 t ^ { 2 } , 10 t \right)\) is a general point on \(C\). The point \(A\) on \(C\) has parameter \(t = 4\).
    The line \(l\) passes through \(A\) and also passes through the focus of \(C\).
  2. Find the gradient of \(l\).
Edexcel FP1 2010 June Q6
9 marks Moderate -0.8
6. Write down the \(2 \times 2\) matrix that represents
  1. an enlargement with centre \(( 0,0 )\) and scale factor 8 ,
  2. a reflection in the \(x\)-axis. Hence, or otherwise,
  3. find the matrix \(\mathbf { T }\) that represents an enlargement with centre ( 0,0 ) and scale factor 8, followed by a reflection in the \(x\)-axis. $$\mathbf { A } = \left( \begin{array} { l l } 6 & 1 \\ 4 & 2 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r } k & 1 \\ c & - 6 \end{array} \right) , \text { where } k \text { and } c \text { are constants. }$$
  4. Find \(\mathbf { A B }\). Given that \(\mathbf { A B }\) represents the same transformation as \(\mathbf { T }\),
  5. find the value of \(k\) and the value of \(c\).
Edexcel FP1 2010 June Q7
7 marks Standard +0.8
7. $$f ( n ) = 2 ^ { n } + 6 ^ { n }$$
  1. Show that \(\mathrm { f } ( k + 1 ) = 6 \mathrm { f } ( k ) - 4 \left( 2 ^ { k } \right)\).
  2. Hence, or otherwise, prove by induction that, for \(n \in \mathbb { Z } ^ { + } , \mathrm { f } ( n )\) is divisible by 8 .
Edexcel FP1 2010 June Q8
11 marks Challenging +1.2
8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where c is a positive constant. The point \(A\) on \(H\) has \(x\)-coordinate \(3 c\).
  1. Write down the \(y\)-coordinate of \(A\).
  2. Show that an equation of the normal to \(H\) at \(A\) is $$3 y = 27 x - 80 c$$ The normal to \(H\) at \(A\) meets \(H\) again at the point \(B\).
  3. Find, in terms of \(c\), the coordinates of \(B\).