Questions — Edexcel (9685 questions)

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Edexcel FP1 2011 January Q5
7 marks Standard +0.8
5. (a) Use the results for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), to prove that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 5 ) = \frac { 1 } { 4 } n ( n + 1 ) ( n + 2 ) ( n + 7 )$$ for all positive integers \(n\).
(b) Hence, or otherwise, find the value of $$\sum _ { r = 20 } ^ { 50 } r ( r + 1 ) ( r + 5 )$$
Edexcel FP1 2011 January Q6
8 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d20fa710-2d91-4ac2-adbc-46ccdcb93380-07_789_791_228_566} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 36 x\). The point \(S\) is the focus of \(C\).
  1. Find the coordinates of \(S\).
  2. Write down the equation of the directrix of \(C\). Figure 1 shows the point \(P\) which lies on \(C\), where \(y > 0\), and the point \(Q\) which lies on the directrix of \(C\). The line segment \(Q P\) is parallel to the \(x\)-axis. Given that the distance \(P S\) is 25 ,
  3. write down the distance \(Q P\),
  4. find the coordinates of \(P\),
  5. find the area of the trapezium \(O S P Q\).
Edexcel FP1 2011 January Q7
9 marks Moderate -0.8
7. $$z = - 24 - 7 i$$
  1. Show \(z\) on an Argand diagram.
  2. Calculate \(\arg z\), giving your answer in radians to 2 decimal places. It is given that $$w = a + b \mathrm { i } , \quad a \in \mathbb { R } , b \in \mathbb { R }$$ Given also that \(| w | = 4\) and \(\arg w = \frac { 5 \pi } { 6 }\),
  3. find the values of \(a\) and \(b\),
  4. find the value of \(| z w |\).
Edexcel FP1 2011 January Q8
9 marks Standard +0.3
8. $$\mathbf { A } = \left( \begin{array} { r r } 2 & - 2 \\ - 1 & 3 \end{array} \right)$$
  1. Find \(\operatorname { det } \mathbf { A }\).
  2. Find \(\mathbf { A } ^ { - 1 }\). The triangle \(R\) is transformed to the triangle \(S\) by the matrix \(\mathbf { A }\). Given that the area of triangle \(S\) is 72 square units,
  3. find the area of triangle \(R\). The triangle \(S\) has vertices at the points \(( 0,4 ) , ( 8,16 )\) and \(( 12,4 )\).
  4. Find the coordinates of the vertices of \(R\).
Edexcel FP1 2011 January Q9
5 marks Standard +0.3
9. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , u _ { 4 } , \ldots\) is defined by $$u _ { n + 1 } = 4 u _ { n } + 2 , \quad u _ { 1 } = 2$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = \frac { 2 } { 3 } \left( 4 ^ { n } - 1 \right)$$
Edexcel FP1 2011 January Q10
12 marks Standard +0.8
10. The point \(P \left( 6 t , \frac { 6 } { t } \right) , t \neq 0\), lies on the rectangular hyperbola \(H\) with equation \(x y = 36\).
  1. Show that an equation for the tangent to \(H\) at \(P\) is $$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 12 } { t }$$ The tangent to \(H\) at the point \(A\) and the tangent to \(H\) at the point \(B\) meet at the point \(( - 9,12 )\).
  2. Find the coordinates of \(A\) and \(B\).
Edexcel FP1 2012 January Q1
7 marks Moderate -0.5
  1. Given that \(z _ { 1 } = 1 - \mathrm { i }\),
    1. find \(\arg \left( z _ { 1 } \right)\).
    Given also that \(z _ { 2 } = 3 + 4 \mathrm { i }\), find, in the form \(a + \mathrm { i } b , a , b \in \mathbb { R }\),
  2. \(z _ { 1 } z _ { 2 }\),
  3. \(\frac { z _ { 2 } } { z _ { 1 } }\). In part (b) and part (c) you must show all your working clearly.
Edexcel FP1 2012 January Q2
10 marks Moderate -0.3
2. (a) Show that \(\mathrm { f } ( x ) = x ^ { 4 } + x - 1\) has a real root \(\alpha\) in the interval [0.5, 1.0].
[0pt] (b) Starting with the interval [0.5, 1.0], use interval bisection twice to find an interval of width 0.125 which contains \(\alpha\).
(c) Taking 0.75 as a first approximation, apply the Newton Raphson process twice to \(\mathrm { f } ( x )\) to obtain an approximate value of \(\alpha\). Give your answer to 3 decimal places.
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. (a) Prove by induction $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ (b) Using the result in part (a), show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } - 2 \right) = \frac { 1 } { 4 } n \left( n ^ { 3 } + 2 n ^ { 2 } + n - 8 \right)$$ (c) 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 \(\mathbf { Q }\),
  3. find the matrix \(\mathbf { R }\),
  4. 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. (a) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } r ( r + 3 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 5 )$$ (b) 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
  1. \(\mathrm { f } ( x ) = 2 x - 5 \cos x , \quad\) 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 \(\alpha\).