Questions — Edexcel FP1 (269 questions)

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Edexcel FP1 2010 June Q1
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
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
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
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. 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
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. $$f ( n ) = 2 ^ { n } + 6 ^ { n }$$
  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
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 \(B\).
Edexcel FP1 2010 June Q9
9. (a) Prove by induction that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ Using the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\),
(b) show that $$\sum _ { r = 1 } ^ { n } ( r + 2 ) ( r + 3 ) = \frac { 1 } { 3 } n \left( n ^ { 2 } + a n + b \right) ,$$ where \(a\) and \(b\) are integers to be found.
(c) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 2 ) ( r + 3 ) = \frac { 1 } { 3 } n \left( 7 n ^ { 2 } + 27 n + 26 \right)$$
Edexcel FP1 2011 June Q1
1. $$f ( x ) = 3 ^ { x } + 3 x - 7$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between \(x = 1\) and \(x = 2\).
  2. Starting with the interval \([ 1,2 ]\), use interval bisection twice to find an interval of width 0.25 which contains \(\alpha\).
Edexcel FP1 2011 June Q2
2. $$z _ { 1 } = - 2 + \mathrm { i }$$
  1. Find the modulus of \(z _ { 1 }\).
  2. Find, in radians, the argument of \(z _ { 1 }\), giving your answer to 2 decimal places. The solutions to the quadratic equation $$z ^ { 2 } - 10 z + 28 = 0$$ are \(z _ { 2 }\) and \(z _ { 3 }\).
  3. Find \(z _ { 2 }\) and \(z _ { 3 }\), giving your answers in the form \(p \pm i \sqrt { } q\), where \(p\) and \(q\) are integers.
  4. Show, on an Argand diagram, the points representing your complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).
Edexcel FP1 2011 June Q3
3. (a) Given that $$\mathbf { A } = \left( \begin{array} { c c } 1 & \sqrt { } 2
\sqrt { } 2 & - 1 \end{array} \right)$$
  1. find \(\mathbf { A } ^ { 2 }\),
  2. describe fully the geometrical transformation represented by \(\mathbf { A } ^ { 2 }\).
    (b) Given that $$\mathbf { B } = \left( \begin{array} { r r } 0 & - 1
    - 1 & 0 \end{array} \right)$$ describe fully the geometrical transformation represented by \(\mathbf { B }\).
    (c) Given that $$\mathbf { C } = \left( \begin{array} { c c } k + 1 & 12
    k & 9 \end{array} \right)$$ where \(k\) is a constant, find the value of \(k\) for which the matrix \(\mathbf { C }\) is singular.
Edexcel FP1 2011 June Q4
4. $$f ( x ) = x ^ { 2 } + \frac { 5 } { 2 x } - 3 x - 1 , \quad x \neq 0$$
  1. Use differentiation to find \(\mathrm { f } ^ { \prime } ( x )\). The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [0.7, 0.9].
  2. Taking 0.8 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.
Edexcel FP1 2011 June Q5
5. \(\mathbf { A } = \left( \begin{array} { r r } - 4 & a
b & - 2 \end{array} \right)\), where \(a\) and \(b\) are constants. Given that the matrix \(\mathbf { A }\) maps the point with coordinates \(( 4,6 )\) onto the point with coordinates \(( 2 , - 8 )\),
  1. find the value of \(a\) and the value of \(b\). A quadrilateral \(R\) has area 30 square units.
    It is transformed into another quadrilateral \(S\) by the matrix \(\mathbf { A }\).
    Using your values of \(a\) and \(b\),
  2. find the area of quadrilateral \(S\).
Edexcel FP1 2011 June Q6
6. Given that \(z = x + \mathrm { i } y\), find the value of \(x\) and the value of \(y\) such that $$z + 3 \mathrm { i } z ^ { * } = - 1 + 13 \mathrm { i }$$ where \(z ^ { * }\) is the complex conjugate of \(z\).
Edexcel FP1 2011 June Q7
7. (a) Use the results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n ( 2 n + 1 ) ( 2 n - 1 )$$ for all positive integers \(n\).
(b) Hence show that $$\sum _ { r = n + 1 } ^ { 3 n } ( 2 r - 1 ) ^ { 2 } = \frac { 2 } { 3 } n \left( a n ^ { 2 } + b \right)$$ where \(a\) and \(b\) are integers to be found.
Edexcel FP1 2011 June Q8
8. The parabola \(C\) has equation \(y ^ { 2 } = 48 x\). The point \(P \left( 12 t ^ { 2 } , 24 t \right)\) is a general point on \(C\).
  1. Find the equation of the directrix of \(C\).
  2. Show that the equation of the tangent to \(C\) at \(P \left( 12 t ^ { 2 } , 24 t \right)\) is $$x - t y + 12 t ^ { 2 } = 0$$ The tangent to \(C\) at the point \(( 3,12 )\) meets the directrix of \(C\) at the point \(X\).
  3. Find the coordinates of \(X\).
Edexcel FP1 2011 June Q9
9. Prove by induction, that for \(n \in \mathbb { Z } ^ { + }\),
  1. \(\left( \begin{array} { l l } 3 & 0
    6 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & 0
    3 \left( 3 ^ { n } - 1 \right) & 1 \end{array} \right)\),
  2. \(\mathrm { f } ( n ) = 7 ^ { 2 n - 1 } + 5\) is divisible by 12 .
Edexcel FP1 2012 June Q1
1. $$f ( x ) = 2 x ^ { 3 } - 6 x ^ { 2 } - 7 x - 4$$
  1. Show that \(\mathrm { f } ( 4 ) = 0\)
  2. Use algebra to solve \(\mathrm { f } ( x ) = 0\) completely.
Edexcel FP1 2012 June Q2
2. (a) Given that $$\mathbf { A } = \left( \begin{array} { l l l } 3 & 1 & 3
4 & 5 & 5 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r } 1 & 1
1 & 2
0 & - 1 \end{array} \right)$$ find \(\mathbf { A B }\).
(b) Given that $$\mathbf { C } = \left( \begin{array} { l l } 3 & 2
8 & 6 \end{array} \right) , \quad \mathbf { D } = \left( \begin{array} { r r } 5 & 2 k
4 & k \end{array} \right) , \text { where } k \text { is a constant }$$ and $$\mathbf { E } = \mathbf { C } + \mathbf { D }$$ find the value of \(k\) for which \(\mathbf { E }\) has no inverse.
Edexcel FP1 2012 June Q3
3. $$f ( x ) = x ^ { 2 } + \frac { 3 } { 4 \sqrt { } x } - 3 x - 7 , \quad x > 0$$ A root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ 3,5 ]\).
Taking 4 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 2 decimal places.
Edexcel FP1 2012 June Q4
4. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } + 6 r - 3 \right) = \frac { 1 } { 4 } n ^ { 2 } \left( n ^ { 2 } + 2 n + 13 \right)$$ for all positive integers \(n\).
(b) Hence find the exact value of $$\sum _ { r = 16 } ^ { 30 } \left( r ^ { 3 } + 6 r - 3 \right)$$
Edexcel FP1 2012 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5e512646-962b-424b-af5f-a6c6b332e0c9-06_732_654_258_646} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 8 x\). The point \(P\) lies on \(C\), where \(y > 0\), and the point \(Q\) lies on \(C\), where \(y < 0\) The line segment \(P Q\) is parallel to the \(y\)-axis. Given that the distance \(P Q\) is 12 ,
  1. write down the \(y\)-coordinate of \(P\),
  2. find the \(x\)-coordinate of \(P\). Figure 1 shows the point \(S\) which is the focus of \(C\).
    The line \(l\) passes through the point \(P\) and the point \(S\).
  3. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
Edexcel FP1 2012 June Q6
6. $$f ( x ) = \tan \left( \frac { x } { 2 } \right) + 3 x - 6 , \quad - \pi < x < \pi$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 1,2 ]\).
  2. Use linear interpolation once on the interval \([ 1,2 ]\) to find an approximation to \(\alpha\). Give your answer to 2 decimal places.
Edexcel FP1 2012 June Q7
7. $$z = 2 - \mathrm { i } \sqrt { } 3$$
  1. Calculate \(\arg z\), giving your answer in radians to 2 decimal places. Use algebra to express
  2. \(z + z ^ { 2 }\) in the form \(a + b \mathrm { i } \sqrt { } 3\), where \(a\) and \(b\) are integers,
  3. \(\frac { z + 7 } { z - 1 }\) in the form \(c + d \mathrm { i } \sqrt { } 3\), where \(c\) and \(d\) are integers. Given that $$w = \lambda - 3 \mathrm { i }$$ where \(\lambda\) is a real constant, and \(\arg ( 4 - 5 \mathrm { i } + 3 w ) = - \frac { \pi } { 2 }\),
  4. find the value of \(\lambda\).