Questions — Edexcel FP1 (269 questions)

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Edexcel FP1 2010 January Q6
6. Given that 2 and \(5 + 2 \mathrm { i }\) are roots of the equation $$x ^ { 3 } - 12 x ^ { 2 } + c x + d = 0 , \quad c , d \in \mathbb { R }$$
  1. write down the other complex root of the equation.
  2. Find the value of \(c\) and the value of \(d\).
  3. Show the three roots of this equation on a single Argand diagram.
Edexcel FP1 2010 January Q7
7. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a constant. The point \(P \left( c t , \frac { c } { t } \right)\) is a general point on \(H\).
  1. Show that the tangent to \(H\) at \(P\) has equation $$t ^ { 2 } y + x = 2 c t$$ The tangents to \(H\) at the points \(A\) and \(B\) meet at the point \(( 15 c , - c )\).
  2. Find, in terms of \(c\), the coordinates of \(A\) and \(B\).
Edexcel FP1 2010 January Q8
8. (a) Prove by induction that, for any positive integer \(n\), $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ (b) Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } + 3 r + 2 \right) = \frac { 1 } { 4 } n ( n + 2 ) \left( n ^ { 2 } + 7 \right)$$ (c) Hence evaluate \(\sum _ { r = 15 } ^ { 25 } \left( r ^ { 3 } + 3 r + 2 \right)\)
Edexcel FP1 2010 January Q9
9. $$\mathbf { M } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } }
\frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right)$$
  1. Describe fully the geometrical transformation represented by the matrix \(\mathbf { M }\). The transformation represented by \(\mathbf { M }\) maps the point \(A\) with coordinates \(( p , q )\) onto the point \(B\) with coordinates \(( 3 \sqrt { } 2,4 \sqrt { } 2 )\).
  2. Find the value of \(p\) and the value of \(q\).
  3. Find, in its simplest surd form, the length \(O A\), where \(O\) is the origin.
  4. Find \(\mathbf { M } ^ { 2 }\). The point \(B\) is mapped onto the point \(C\) by the transformation represented by \(\mathbf { M } ^ { 2 }\).
  5. Find the coordinates of \(C\).
Edexcel FP1 2011 January Q1
1. $$z = 5 - 3 \mathrm { i } , \quad w = 2 + 2 \mathrm { i }$$ Express in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants,
  1. \(z ^ { 2 }\),
  2. \(\frac { z } { w }\).
Edexcel FP1 2011 January Q2
2. $$\mathbf { A } = \left( \begin{array} { l l } 2 & 0
5 & 3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } - 3 & - 1
5 & 2 \end{array} \right)$$
  1. Find \(\mathbf { A B }\). Given that $$\mathbf { C } = \left( \begin{array} { r r } - 1 & 0
    0 & 1 \end{array} \right)$$
  2. describe fully the geometrical transformation represented by \(\mathbf { C }\),
  3. write down \(\mathbf { C } ^ { 100 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{d20fa710-2d91-4ac2-adbc-46ccdcb93380-03_99_97_2631_1784}
Edexcel FP1 2011 January Q3
3. $$f ( x ) = 5 x ^ { 2 } - 4 x ^ { \frac { 3 } { 2 } } - 6 , \quad x \geqslant 0$$ The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ 1.6,1.8 ]\).
  1. Use linear interpolation once on the interval \([ 1.6,1.8 ]\) to find an approximation to \(\alpha\). Give your answer to 3 decimal places.
  2. Differentiate \(\mathrm { f } ( x )\) to find \(\mathrm { f } ^ { \prime } ( x )\).
  3. Taking 1.7 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 January Q4
4. Given that \(2 - 4 \mathrm { i }\) is a root of the equation $$z ^ { 2 } + p z + q = 0 ,$$ where \(p\) and \(q\) are real constants,
  1. write down the other root of the equation,
  2. find the value of \(p\) and the value of \(q\).
Edexcel FP1 2011 January Q5
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
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
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
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
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
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
  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
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
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
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
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
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. 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
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. 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
  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
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.