Questions - Edexcel FP1 - A-Level Maths

Show that an equation of the tangent to the parabola at $P$ is $$p y - x = a p ^ { 2 }$$
Write down the equation of the tangent at $Q$. The tangent at $P$ meets the tangent at $Q$ at the point $R$.
Find, in terms of $p$ and $q$, the coordinates of $R$, giving your answers in their simplest form. Given that $R$ lies on the directrix of $C$,
find the value of $p q$.

Edexcel FP1 2013 June Q7

7. $$z _ { 1 } = 2 + 3 \mathrm { i } , \quad z _ { 2 } = 3 + 2 \mathrm { i } , \quad z _ { 3 } = a + b \mathrm { i } , \quad a , b \in \mathbb { R }$$

Find the exact value of $\left| z _ { 1 } + z _ { 2 } \right|$. Given that $w = \frac { z _ { 1 } z _ { 3 } } { z _ { 2 } }$,
find $w$ in terms of $a$ and $b$, giving your answer in the form $x + \mathrm { i } y , \quad x , y \in \mathbb { R }$ Given also that $w = \frac { 17 } { 13 } - \frac { 7 } { 13 } \mathrm { i }$,
find the value of $a$ and the value of $b$,
find $\arg w$, giving your answer in radians to 3 decimal places.

Edexcel FP1 2013 June Q8

8. $$\mathbf { A } = \left( \begin{array} { c c } 6 & - 2
- 4 & 1 \end{array} \right)$$ and $\mathbf { I }$ is the $2 \times 2$ identity matrix.

Prove that $$\mathbf { A } ^ { 2 } = 7 \mathbf { A } + 2 \mathbf { I }$$
Hence show that $$\mathbf { A } ^ { - 1 } = \frac { 1 } { 2 } ( \mathbf { A } - 7 \mathbf { I } )$$ The transformation represented by $\mathbf { A }$ maps the point $P$ onto the point $Q$.
Given that $Q$ has coordinates $( 2 k + 8 , - 2 k - 5 )$, where $k$ is a constant,
find, in terms of $k$, the coordinates of $P$.

Edexcel FP1 2013 June Q9

9. (a) A sequence of numbers is defined by $$\begin{aligned} & u _ { 1 } = 8
& u _ { n + 1 } = 4 u _ { n } - 9 n , \quad n \geqslant 1 \end{aligned}$$ Prove by induction that, for $n \in \mathbb { Z } ^ { + }$, $$u _ { n } = 4 ^ { n } + 3 n + 1$$ (b) Prove by induction that, for $m \in \mathbb { Z } ^ { + }$, $$\left( \begin{array} { l l } 3 & - 4
1 & - 1 \end{array} \right) ^ { m } = \left( \begin{array} { c c } 2 m + 1 & - 4 m
m & 1 - 2 m \end{array} \right)$$

Edexcel FP1 2014 June Q1

The roots of the equation

$$2 z ^ { 3 } - 3 z ^ { 2 } + 8 z + 5 = 0$$ are $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$
Given that $z _ { 1 } = 1 + 2 i$, find $z _ { 2 }$ and $z _ { 3 }$

Edexcel FP1 2014 June Q2

2. $$\mathrm { f } ( x ) = 3 \cos 2 x + x - 2 , \quad - \pi \leqslant x < \pi$$

Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval [2,3].
[0pt]
Use linear interpolation once on the interval [2,3] to find an approximation to $\alpha$. Give your answer to 3 decimal places.
The equation $\mathrm { f } ( x ) = 0$ has another root $\beta$ in the interval $[ - 1,0 ]$. Starting with this interval, use interval bisection to find an interval of width 0.25 which contains $\beta$.

Edexcel FP1 2014 June Q3

3. (i) $$\mathbf { A } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { } 2 } & \frac { - 1 } { \sqrt { } 2 }
\frac { 1 } { \sqrt { } 2 } & \frac { 1 } { \sqrt { } 2 } \end{array} \right)$$

Describe fully the single transformation represented by the matrix $\mathbf { A }$. The matrix $\mathbf { B }$ represents an enlargement, scale factor - 2 , with centre the origin.
Write down the matrix $\mathbf { B }$.
(ii) $$\mathbf { M } = \left( \begin{array} { c c } 3 & k
- 2 & 3 \end{array} \right) , \quad \text { where } k \text { is a positive constant. }$$ Triangle $T$ has an area of 16 square units. Triangle $T$ is transformed onto the triangle $T ^ { \prime }$ by the transformation represented by the matrix M. Given that the area of the triangle $T ^ { \prime }$ is 224 square units, find the value of $k$.

Edexcel FP1 2014 June Q4

4. The complex number $z$ is given by $$z = \frac { p + 2 \mathrm { i } } { 3 + p \mathrm { i } }$$ where $p$ is an integer.

Express $z$ in the form $a + b \mathrm { i }$ where $a$ and $b$ are real. Give your answer in its simplest form in terms of $p$.
Given that $\arg ( z ) = \theta$, where $\tan \theta = 1$ find the possible values of $p$.

Edexcel FP1 2014 June Q5

(a) Use the standard results for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 3 }$ to show that

$$\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right) = \frac { 1 } { 4 } n ( n + 1 ) ( n + 3 ) ( n - 2 )$$ (b) Calculate the value of $\sum _ { r = 10 } ^ { 50 } r \left( r ^ { 2 } - 3 \right)$

Edexcel FP1 2014 June Q6

6. $$\mathbf { A } = \left( \begin{array} { r r } 2 & 1
- 1 & 0 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r } - 1 & 1
0 & 1 \end{array} \right)$$ Given that $\mathbf { M } = ( \mathbf { A } + \mathbf { B } ) ( 2 \mathbf { A } - \mathbf { B } )$,

calculate the matrix $\mathbf { M }$,
find the matrix $\mathbf { C }$ such that $\mathbf { M C } = \mathbf { A }$.

Edexcel FP1 2014 June Q7

7. The parabola $C$ has cartesian equation $y ^ { 2 } = 4 a x , a > 0$ The points $P \left( a p ^ { 2 } , 2 a p \right)$ and $P ^ { \prime } \left( a p ^ { 2 } , - 2 a p \right)$ lie on $C$.

Show that an equation of the normal to $C$ at the point $P$ is $$y + p x = 2 a p + a p ^ { 3 }$$
Write down an equation of the normal to $C$ at the point $P ^ { \prime }$. The normal to $C$ at $P$ meets the normal to $C$ at $P ^ { \prime }$ at the point $Q$.
Find, in terms of $a$ and $p$, the coordinates of $Q$. Given that $S$ is the focus of the parabola,
find the area of the quadrilateral $S P Q P ^ { \prime }$.

Edexcel FP1 2014 June Q8

8. The rectangular hyperbola $H$ has equation $x y = c ^ { 2 }$, where $c$ is a positive constant. The point $P \left( c t , \frac { c } { t } \right) , t \neq 0$, is a general point on $H$. An equation for the tangent to $H$ at $P$ is given by $$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 2 c } { t }$$ The points $A$ and $B$ lie on $H$.
The tangent to $H$ at $A$ and the tangent to $H$ at $B$ meet at the point $\left( - \frac { 6 } { 7 } c , \frac { 12 } { 7 } c \right)$.
Find, in terms of $c$, the coordinates of $A$ and the coordinates of $B$.

Edexcel FP1 2014 June Q9

9. (a) Prove by induction that, for $n \in \mathbb { Z } ^ { + }$, $$\sum _ { r = 1 } ^ { n } ( r + 1 ) 2 ^ { r - 1 } = n 2 ^ { n }$$ (b) A sequence of numbers is defined by $$\begin{gathered} u _ { 1 } = 0 , \quad u _ { 2 } = 32 ,
u _ { n + 2 } = 6 u _ { n + 1 } - 8 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for $n \in \mathbb { Z } ^ { + }$, $$u _ { n } = 4 ^ { n + 1 } - 2 ^ { n + 3 }$$

Edexcel FP1 2014 June Q1

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

$$z _ { 1 } = p + 2 i \text { and } z _ { 2 } = 1 - 2 i$$ where $p$ is an integer.

Find $\frac { z _ { 1 } } { z _ { 2 } }$ in the form $a + b$ i where $a$ and $b$ are real. Give your answer in its simplest form in terms of $p$. Given that $\left| \frac { z _ { 1 } } { z _ { 2 } } \right| = 13$,
find the possible values of $p$.

Edexcel FP1 2014 June Q2

2. $$\mathrm { f } ( x ) = x ^ { 3 } - \frac { 5 } { 2 x ^ { \frac { 3 } { 2 } } } + 2 x - 3 , \quad x > 0$$

Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval [1.1,1.5].
Find f'(x).
Using $x _ { 0 } = 1.1$ 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 decimal places.

Edexcel FP1 2014 June Q3

3. Given that 2 and $1 - 5 \mathrm { i }$ are roots of the equation $$x ^ { 3 } + p x ^ { 2 } + 30 x + q = 0 , \quad p , q \in \mathbb { R }$$

write down the third root of the equation.
Find the value of $p$ and the value of $q$.
Show the three roots of this equation on a single Argand diagram.

Edexcel FP1 2014 June Q4

4. (i) Given that $$\mathbf { A } = \left( \begin{array} { r r } 1 & 2
3 & - 1
4 & 5 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r r } 2 & - 1 & 4
1 & 3 & 1 \end{array} \right)$$

find $\mathbf { A B }$.
Explain why $\mathbf { A B } \neq \mathbf { B A }$.
(ii) Given that $$\mathbf { C } = \left( \begin{array} { c r } 2 k & - 2
3 & k \end{array} \right) \text {, where } k \text { is a real number }$$ find $\mathbf { C } ^ { - 1 }$, giving your answer in terms of $k$.

Edexcel FP1 2014 June Q5

5. (a) Use the standard results for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ to show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$ (b) Hence show that $$\sum _ { r = 2 n + 1 } ^ { 4 n } ( 2 r - 1 ) ^ { 2 } = a n \left( b n ^ { 2 } - 1 \right)$$ where $a$ and $b$ are constants to be found.

Edexcel FP1 2014 June Q6

6. The rectangular hyperbola $H$ has cartesian equation $x y = c ^ { 2 }$. The point $P \left( c t , \frac { c } { t } \right) , t > 0$, is a general point on $H$.

Show that an equation of the tangent to $H$ at the point $P$ is $$t ^ { 2 } y + x = 2 c t$$ An equation of the normal to $H$ at the point $P$ is $t ^ { 3 } x - t y = c t ^ { 4 } - c$ Given that the normal to $H$ at $P$ meets the $x$-axis at the point $A$ and the tangent to $H$ at $P$ meets the $x$-axis at the point $B$,
find, in terms of $c$ and $t$, the coordinates of $A$ and the coordinates of $B$. Given that $c = 4$,
find, in terms of $t$, the area of the triangle $A P B$. Give your answer in its simplest form.

Edexcel FP1 2014 June Q7

7. (i) In each of the following cases, find a $2 \times 2$ matrix that represents

a reflection in the line $y = - x$,
a rotation of $135 ^ { \circ }$ anticlockwise about $( 0,0 )$,
a reflection in the line $y = - x$ followed by a rotation of $135 ^ { \circ }$ anticlockwise about $( 0,0 )$.
(ii) The triangle $T$ has vertices at the points $( 1 , k ) , ( 3,0 )$ and $( 11,0 )$, where $k$ is a constant. Triangle $T$ is transformed onto the triangle $T ^ { \prime }$ by the matrix $$\left( \begin{array} { r r } 6 & - 2
1 & 2 \end{array} \right)$$ Given that the area of triangle $T ^ { \prime }$ is 364 square units, find the value of $k$.

Edexcel FP1 2014 June Q8

8. The points $P \left( 4 k ^ { 2 } , 8 k \right)$ and $Q \left( k ^ { 2 } , 4 k \right)$, where $k$ is a constant, lie on the parabola $C$ with equation $y ^ { 2 } = 16 x$. The straight line $l _ { 1 }$ passes through the points $P$ and $Q$.

Show that an equation of the line $l _ { 1 }$ is given by $$3 k y - 4 x = 8 k ^ { 2 }$$ The line $l _ { 2 }$ is perpendicular to the line $l _ { 1 }$ and passes through the focus of the parabola $C$. The line $l _ { 2 }$ meets the directrix of $C$ at the point $R$.
Find, in terms of $k$, the $y$ coordinate of the point $R$.

Edexcel FP1 2014 June Q9

9. Prove by induction that, for $n \in \mathbb { Z } ^ { + }$, $$f ( n ) = 8 ^ { n } - 2 ^ { n }$$ is divisible by 6

Edexcel FP1 2015 June Q1

1. $$f ( x ) = 9 x ^ { 3 } - 33 x ^ { 2 } - 55 x - 25$$ Given that $x = 5$ is a solution of the equation $\mathrm { f } ( x ) = 0$, use an algebraic method to solve $\mathrm { f } ( x ) = 0$ completely.
(5)

Edexcel FP1 2015 June Q2

2. In the interval $13 < x < 14$, the equation $$3 + x \sin \left( \frac { x } { 4 } \right) = 0 , \text { where } x \text { is measured in radians, }$$ has exactly one root, $\alpha$.
[0pt]

Starting with the interval [13,14], use interval bisection twice to find an interval of width 0.25 which contains $\alpha$.
[0pt]
Use linear interpolation once on the interval [13,14] to find an approximate value for $\alpha$. Give your answer to 3 decimal places.

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