Questions - Edexcel FP1 - A-Level Maths

Find $\mathrm { f } ^ { \prime } ( x )$. The equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval [4.5, 5.5].
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

4. The transformation $U$, represented by the $2 \times 2$ matrix $\mathbf { P }$, is a rotation through $90 ^ { \circ }$ anticlockwise about the origin.

Write down the matrix $\mathbf { P }$. The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf { Q }$, is a reflection in the line $y = - x$.
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 $\mathbf { Q }$,
find the matrix $\mathbf { R }$,
give a full geometrical description of $T$ as a single transformation.

Edexcel FP1 2013 January Q5

5. $$f ( x ) = \left( 4 x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 6 x + 34 \right)$$

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.
Show these four roots on a single Argand diagram.

Edexcel FP1 2013 January Q6

6. $\mathbf { X } = \left( \begin{array} { l l } 1 & a
3 & 2 \end{array} \right)$, where $a$ is a constant.

Find the value of $a$ for which the matrix $\mathbf { X }$ is singular. $$\mathbf { Y } = \left( \begin{array} { r r } 1 & - 1
3 & 2 \end{array} \right)$$
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,
find, in terms of $\lambda$, the coordinates of point $A$.

Edexcel FP1 2013 January Q7

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 $H$.

Show that the equation of the tangent at point $P$ is $$p ^ { 2 } y + x = 10 p$$
Write down the equation of the tangent at point $Q$. The tangents at $P$ and $Q$ meet at the point $N$.
Given $p + q \neq 0$,
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 $P Q$.
Find the value of $p ^ { 2 } q ^ { 2 }$.

Edexcel FP1 2013 January Q8

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. \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.

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.
Find the area of triangle $P S N$.

Edexcel FP1 2014 January Q1

$\mathrm { f } ( x ) = 2 x - 5 \cos x , \quad$ 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 $\alpha$.

Edexcel FP1 2014 January Q2

$$\mathbf { A } = \left( \begin{array} { c c } - 4 & 10
- 3 & k \end{array} \right) , \quad \text { where } k \text { is a constant. }$$ 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 $k$.
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

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

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

5. $$z = 5 + \mathrm { i } \sqrt { 3 } , \quad w = \sqrt { 3 } - \mathrm { i }$$

Find the value of $| w |$. Find in the form $a + \mathrm { i } b$, where $a$ and $b$ are real constants,
$z w$, showing clearly how you obtained your answer,
$\frac { z } { w }$, showing clearly how you obtained your answer. Given that $$\arg ( z + \lambda ) = \frac { \pi } { 3 } , \quad \text { where } \lambda \text { is a real constant, }$$
find the value of $\lambda$.

Edexcel FP1 2014 January Q6

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

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.

Find $\mathbf { P } ^ { - 1 }$, leaving your answer in terms of $a$ and $b$. Given that $$\mathbf { M } = \mathbf { P Q }$$
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

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 $C$.

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 $p > 0$,
find, in terms of $a$, the coordinates of $D$. Given also that the directrix of $C$ cuts the $x$-axis at the point $X$,
find, in terms of $a$, the area of the triangle XPD, giving your answer in its simplest form.

Edexcel FP1 2014 January Q9

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

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

The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by

$$z _ { 1 } = 2 - i \quad \text { and } \quad z _ { 2 } = - 8 + 9 i$$

Show $z _ { 1 }$ and $z _ { 2 }$ on a single Argand diagram. Find, showing your working,
the value of $\left| z _ { 1 } \right|$,
the value of $\arg z _ { 1 }$, giving your answer in radians to 2 decimal places,
$\frac { Z _ { 2 } } { Z _ { 1 } }$ in the form $a + b \mathrm { i }$, where $a$ and $b$ are real.

Edexcel FP1 2009 June Q2

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

3. $$f ( x ) = \left( x ^ { 2 } + 4 \right) \left( x ^ { 2 } + 8 x + 25 \right)$$

Find the four roots of $\mathrm { f } ( x ) = 0$.
Find the sum of these four roots.

Edexcel FP1 2009 June Q4

4. Given that $\alpha$ is the only real root of the equation $$x ^ { 3 } - x ^ { 2 } - 6 = 0$$

show that $2.2 < \alpha < 2.3$
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]
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

5. $\mathbf { R } = \left( \begin{array} { l l } a & 2
a & b \end{array} \right)$, where $a$ and $b$ are constants and $a > 0$.

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 ,
find the value of $a$ and the value of $b$.

Edexcel FP1 2009 June Q6

6. The parabola $C$ has equation $y ^ { 2 } = 16 x$.

Verify that the point $P \left( 4 t ^ { 2 } , 8 t \right)$ is a general point on $C$.
Write down the coordinates of the focus $S$ of $C$.
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 $N$.
Find the area of triangle $P S N$ in terms of $t$, giving your answer in its simplest form.

Edexcel FP1 2009 June Q7

7. $\mathbf { A } = \left( \begin{array} { r r } a & - 2
- 1 & 4 \end{array} \right)$, where $a$ is a constant.

Find the value of $a$ for which the matrix $\mathbf { A }$ is singular. $$\mathbf { B } = \left( \begin{array} { r r } 3 & - 2
- 1 & 4 \end{array} \right)$$
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,
show that $P$ lies on the line with equation $y = x + 3$.

Edexcel FP1 2009 June Q8

8. Prove by induction that, for $n \in \mathbb { Z } ^ { + }$,

$\mathrm { f } ( n ) = 5 ^ { n } + 8 n + 3$ is divisible by 4 ,
$\left( \begin{array} { l l } 3 & - 2
2 & - 1 \end{array} \right) ^ { n } = \left( \begin{array} { l r } 2 n + 1 & - 2 n
2 n & 1 - 2 n \end{array} \right)$

Questions — Edexcel FP1 (269 questions)

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