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Edexcel FP1 AS 2022 June Q4

9 marks Challenging +1.2

The parabola $C$ has equation $y ^ { 2 } = 10 x$

The point $F$ is the focus of $C$.

Write down the coordinates of $F$. The point $P$ on $C$ has $y$ coordinate $q$, where $q > 0$
Show that an equation for the tangent to $C$ at $P$ is given by $$10 x - 2 q y + q ^ { 2 } = 0$$ The tangent to $C$ at $P$ intersects the directrix of $C$ at the point $A$.
The point $B$ lies on the directrix such that $P B$ is parallel to the $x$-axis.
Show that the point of intersection of the diagonals of quadrilateral $P B A F$ always lies on the $y$-axis.

Edexcel FP1 AS 2022 June Q5

11 marks Standard +0.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1be33445-e669-49af-a97e-a8ae84d63463-12_762_1129_246_468} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The points $A ( 3,2 , - 4 ) , B ( 9 , - 4,2 ) , C ( - 6 , - 10,8 )$ and $D ( - 4 , - 5,10 )$ are the vertices of a tetrahedron. The plane with equation $z = 0$ cuts the tetrahedron into two pieces, one on each side of the plane. The edges $A B , A C$ and $A D$ of the tetrahedron intersect the plane at the points $M , N$ and $P$ respectively, as shown in Figure 1. Determine

the coordinates of the points $M , N$ and $P$,
the area of triangle $M N P$,
the exact volume of the solid $B C D P N M$.

Edexcel FP1 AS 2023 June Q1

5 marks Moderate -0.3

(a) Use algebra to determine the values of $x$ for which

$$\frac { 5 x } { x - 2 } \geqslant 12$$ (b) Hence, given that $x$ is an integer, deduce the value of $x$.

Edexcel FP1 AS 2023 June Q2

7 marks Standard +0.8

(a) Use the substitution $t = \tan \left( \frac { x } { 2 } \right)$ to show that the equation

$$3 \cos x - 2 \sin x = 1$$ can be written in the form $$2 t ^ { 2 } + 2 t - 1 = 0$$ (b) Hence solve, for $- 180 ^ { \circ } < x < 180 ^ { \circ }$, the equation $$3 \cos x - 2 \sin x = 1$$ giving your answers to one decimal place.

Edexcel FP1 AS 2023 June Q3

5 marks Challenging +1.2

The rectangular hyperbola $H$ has equation $x y = c ^ { 2 }$ where $c$ is a positive constant.

The line $l$ has equation $x - 2 y = c$ The points $P$ and $Q$ are the points of intersection of $H$ and $l$

Determine, in terms of $c$, the coordinates of $P$ and the coordinates of $Q$ The point $R$ is the midpoint of $P Q$
Show that, as $C$ varies, the coordinates of $R$ satisfy the equation $$x y = - \frac { c ^ { 2 } } { a }$$ where $a$ is a constant to be determined.

Edexcel FP1 AS 2023 June Q4

6 marks Standard +0.3

A teacher made a cup of coffee. The temperature $\theta ^ { \circ } \mathrm { C }$ of the coffee, $t$ minutes after it was made, is modelled by the differential equation

$$\frac { \mathrm { d } \theta } { \mathrm {~d} t } + 0.05 ( \theta - 20 ) = 0$$ Given that

the initial temperature of the coffee was $95 ^ { \circ } \mathrm { C }$
the coffee can only be safely drunk when its temperature is below $70 ^ { \circ } \mathrm { C }$
the teacher made the cup of coffee at 1.15 pm
the teacher needs to be able to start drinking the coffee by 1.20 pm
use two iterations of the approximation formula

$$\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }$$ to estimate whether the teacher will be able to start drinking the coffee at 1.20 pm .

Edexcel FP1 AS 2023 June Q5

9 marks Standard +0.3

The points $A , B$ and $C$ are the vertices of a triangle.

Given that

$\overrightarrow { A B } = \left( \begin{array} { l } p \\ 4 \\ 6 \end{array} \right)$ and $\overrightarrow { A C } = \left( \begin{array} { l } q \\ 4 \\ 5 \end{array} \right)$ where $p$ and $q$ are constants
$\overrightarrow { A B } \times \overrightarrow { A C }$ is parallel to $2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }$
1. determine the value of $p$ and the value of $q$
2. Hence, determine the exact area of triangle $A B C$

Edexcel FP1 AS 2023 June Q6

8 marks Standard +0.3

The parabola $C$ has equation $y ^ { 2 } = 4 a x$ where $a$ is a positive constant.

The point $P \left( a t ^ { 2 } , 2 a t \right) , t \neq 0$, lies on $C$ The normal to $C$ at $P$ is parallel to the line with equation $y = 2 x$

For the point $P$, show that $t = - 2$ The normal to $C$ at $P$ intersects $C$ again when $x = 9$
Determine the value of $a$, giving a reason for your answer.

Edexcel FP1 AS 2024 June Q1

7 marks Standard +0.3

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
1. Sketch the graph of the curve with equation
$$y = \frac { 1 } { x ^ { 2 } }$$
Solve, using algebra, the inequality $$3 - 2 x ^ { 2 } > \frac { 1 } { x ^ { 2 } }$$

Edexcel FP1 AS 2024 June Q2

6 marks Standard +0.3

An area of woodland contains a mixture of blue and yellow flowers.

A study found that the proportion, $x$, of blue flowers in the woodland area satisfies the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x t ( 0.8 - x ) } { x ^ { 2 } + 5 t } \quad t > 0$$ where $t$ is the number of years since the start of the study.
Given that exactly 3 years after the start of the study half of the flowers in the woodland area were blue,

use one application of the approximation formula $\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }$ to estimate the proportion of blue flowers in the woodland area half a year later.
Deduce from the differential equation the proportion of flowers that will be blue in the long term.

Edexcel FP1 AS 2024 June Q3

6 marks Standard +0.3

Vectors $\mathbf { u }$ and $\mathbf { v }$ are given by

$$\mathbf { u } = 5 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \mathbf { v } = a \mathbf { i } - 6 \mathbf { j } + 2 \mathbf { k }$$ where $a$ is a constant.

Determine, in terms of $a$, the vector product $\mathbf { u } \times \mathbf { v }$ Given that
- $\overrightarrow { A B } = 2 \mathbf { u }$
- $\overrightarrow { A C } = \mathbf { v }$
- the area of triangle $A B C$ is 15
- determine the possible values of $a$.

Edexcel FP1 AS 2024 June Q4

12 marks Standard +0.3

(a) Given that $t = \tan \frac { X } { 2 }$ prove that

$$\cos x \equiv \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } }$$ (b) Show that the equation $$3 \tan x - 10 \cos x = 10$$ can be written in the form $$( t + 2 ) \left( a t ^ { 2 } + b t + c \right) = 0$$ where $t = \tan \frac { X } { 2 }$ and $a , b$ and $c$ are integers to be determined.
(c) Hence solve, for $- 180 ^ { \circ } < x < 180 ^ { \circ }$, the equation $$3 \tan x - 10 \cos x = 10$$

Edexcel FP1 AS 2024 June Q5

9 marks Challenging +1.2

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

The point $P$ on $C$ has $y$ coordinate $p$, where $p$ is a positive constant.

Show that an equation of the tangent to $C$ at $P$ is given by $$2 p y = 16 x + p ^ { 2 }$$ $$\left[ Y \text { ou may quote without proof that for the general parabola } y ^ { 2 } = 4 a x , \frac { d y } { d x } = \frac { 2 a } { y } \right]$$
Write down the equation of the directrix of $C$. The line $l$ is the reflection of the tangent to $C$ at $P$ in the directrix of $C$.
Given that $l$ passes through the focus of $C$,
determine the exact value of $p$.

Edexcel FP1 AS Specimen Q1

6 marks Challenging +1.2

(a) Use the substitution $\mathrm { t } = \tan \left( \frac { \mathrm { x } } { 2 } \right)$ to show that

$$\sec x - \tan x \equiv \frac { 1 - t } { 1 + t } \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 } , n \in \mathbb { Z }$$ (b) Use the substitution $\mathrm { t } = \tan \left( \frac { \mathrm { x } } { 2 } \right)$ and the answer to part (a) to prove that $$\frac { 1 - \sin x } { 1 + \sin x } \equiv ( \sec x - \tan x ) ^ { 2 } \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 } , n \in \mathbb { Z }$$ \section*{Q uestion 1 continued}

Edexcel FP1 AS Specimen Q2

6 marks Standard +0.3

The value, V hundred pounds, of a particular stock thours after the opening of trading on a given day is modelled by the differential equation

$$\frac { d V } { d t } = \frac { V ^ { 2 } - t } { t ^ { 2 } + t V } \quad 0 < t < 8.5$$ A trader purchases $\pounds 300$ of the stock one hour after the opening of trading.
Use two iterations of the approximation formula $\left( \frac { \mathrm { dy } } { \mathrm { dx } } \right) _ { 0 } \approx \frac { \mathrm { y } _ { 1 } - \mathrm { y } _ { 0 } } { \mathrm {~h} }$ to estimate, to the nearest $\pounds$, the value of the trader's stock half an hour after it was purchased.

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Edexcel FP1 AS Specimen Q3

6 marks Standard +0.3

Use algebra to find the set of values of x for which

$$\frac { 1 } { x } < \frac { x } { x + 2 }$$

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Edexcel FP1 AS Specimen Q4

10 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ff1fc9b0-6514-44e0-a2a3-46aa6411ce10-08_538_807_251_630} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of a solid sculpture made of glass and concrete. The sculpture is modelled as a parallelepiped. The sculpture is made up of a concrete solid in the shape of a tetrahedron, shown shaded in Figure 1, whose vertices are $\mathrm { O } ( 0,0,0 ) , \mathrm { A } ( 2,0,0 ) , \mathrm { B } ( 0,3,1 )$ and $\mathrm { C } ( 1,1,2 )$, where the units are in metres. The rest of the solid parallelepiped is made of glass which is glued to the concrete tetrahedron.

Find the surface area of the glued face of the tetrahedron.
Find the volume of glass contained in this parallelepiped.
Give a reason why the volume of concrete predicted by this model may not be an accurate value for the volume of concrete that was used to make the sculpture. \section*{Q uestion 4 continued}

Edexcel FP1 AS Specimen Q5

12 marks Standard +0.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ff1fc9b0-6514-44e0-a2a3-46aa6411ce10-10_965_853_212_621} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Diagram not drawn to scale $$\left[ Y \text { ou may quote without proof that for the general parabola } y ^ { 2 } = 4 a x , \frac { d y } { d x } = \frac { 2 a } { y } \right]$$ The parabola C has equation $\mathrm { y } ^ { 2 } = 16 \mathrm { x }$

Deduce that the point $\mathrm { P } \left( 4 \mathrm { p } ^ { 2 } , 8 \mathrm { p } \right)$ is a general point on C . The line I is the tangent to C at the point P .
Show that an equation for I is $$p y = x + 4 p ^ { 2 }$$ The finite region R , shown shaded in Figure 2, is bounded by the line I , the x -axis and the parabola C.
The line $I$ intersects the directrix of $C$ at the point $B$, where the $y$ coordinate of $B$ is $\frac { 10 } { 3 }$ Given that $\mathrm { p } > 0$
show that the area of R is 36 \section*{Q uestion 5 continued}

Edexcel FP2 AS 2018 June Q1

5 marks Moderate -0.8

(i) Using a suitable algorithm and without performing any division, determine whether 23738 is divisible by 11
(ii) Use the Euclidean algorithm to find the highest common factor of 2322 and 654

Edexcel FP2 AS 2018 June Q2

10 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{285b6ae9-ca8f-46b7-b4ed-a3310fe4ebe6-04_568_634_248_717} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows an equilateral triangle $A B C$. The lines $x , y$ and $z$ and their point of intersection, $O$, are fixed in the plane. The triangle $A B C$ is transformed about these fixed lines and the fixed point $O$. The lines $x , y$ and $z$ each pass through a vertex of the triangle and the midpoint of the opposite side. The transformations $I , X , Y , Z , R _ { 1 }$ and $R _ { 2 }$ of the plane containing triangle $A B C$ are defined as follows:

I: Do nothing
$X$ : Reflect in the line $x$
$Y$ : Reflect in the line $y$
$Z$ : Reflect in the line $z$
$R _ { 1 }$ : Rotate $120 ^ { \circ }$ anticlockwise about $O$
$R _ { 2 }$ : Rotate $240 ^ { \circ }$ anticlockwise about $O$

The operation * is defined as 'followed by' on the set $T = \left\{ I , X , Y , Z , R _ { 1 } , R _ { 2 } \right\}$.
For example, $X { } ^ { * } Y$ means a reflection in the line $x$ followed by a reflection in the line $y$.

1. Complete the Cayley table on page 5 Given that the associative law is satisfied,
2. show that $T$ is a group under the operation *
Show that the element $R _ { 2 }$ has order 3
Explain why $T$ is not a cyclic group.
Write down the elements of a subgroup of $T$ that has order 3
\multirow{2}{*}{} Second transformation
* I $X$ $Y$ $Z$ $R _ { 1 }$ $R _ { 2 }$
\multirow{6}{*}{First Transformation} I
$X$ I $Z$
$Y$
$Z$
$R _ { 1 }$ $Y$
$R _ { 2 }$
\footnotetext{Turn over for a spare table if you need to re-write your Cayley table } \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Only use this grid if you need to re-write your Cayley table}
\multirow{2}{*}{} Second transformation
* I $X$ $Y$ Z $R _ { 1 }$ $R _ { 2 }$
\multirow{6}{*}{First Transformation} I
X I Z
Y
Z
$R _ { 1 }$ $Y$
$R _ { 2 }$
\end{table}

Edexcel FP2 AS 2018 June Q3

10 marks Standard +0.3

3 A tree at the bottom of a garden needs to be reduced in height. The tree is known to increase in height by 15 centimetres each year. On the first day of every year, the height is measured and the tree is immediately trimmed by $3 \%$ of this height. When the tree is measured, before trimming on the first day of year 1 , the height is 6 metres.
Let $H _ { n }$ be the height of the tree immediately before trimming on the first day of year $n$.

Explain, in the context of the problem, why the height of the tree may be modelled by the recurrence relation $$H _ { n + 1 } = 0.97 H _ { n } + 0.15 , \quad H _ { 1 } = 6 , \quad n \in \mathbb { Z } ^ { + }$$
Prove by induction that $H _ { n } = 0.97 ^ { n - 1 } + 5 , \quad n \geqslant 1$
Explain what will happen to the height of the tree immediately before trimming in the long term.
By what fixed percentage should the tree be trimmed each year if the height of the tree immediately before trimming is to be 4 metres in the long term?

Edexcel FP2 AS 2018 June Q4

7 marks Standard +0.3

4. $$\mathbf { A } = \left( \begin{array} { r r } 1 & 1 \\ - 2 & 4 \end{array} \right)$$ Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { D } = \mathbf { P } ^ { - 1 } \mathbf { A P }$

Edexcel FP2 AS 2018 June Q5

8 marks Challenging +1.2

A complex number $z$ is represented by the point $P$ on an Argand diagram.

Given that $\arg \left( \frac { z - 6 i } { z - 3 i } \right) = \frac { \pi } { 3 }$

sketch the locus of $P$ as $z$ varies,
find the exact maximum possible value of $| z |$

Edexcel FP2 AS 2019 June Q1

5 marks Moderate -0.3

Given that

$$\mathbf { A } = \left( \begin{array} { l l } 3 & 2 \\ 2 & 2 \end{array} \right)$$

find the characteristic equation for the matrix $\mathbf { A }$, simplifying your answer.
Hence find an expression for the matrix $\mathbf { A } ^ { - 1 }$ in the form $\lambda \mathbf { A } + \mu \mathbf { I }$, where $\lambda$ and $\mu$ are constants to be found.

Edexcel FP2 AS 2019 June Q2

7 marks Standard +0.8

(i) Determine all the possible integers $a$, where $a > 3$, such that

$$15 \equiv 3 \bmod a$$ (ii) Show that if $p$ is prime, $x$ is an integer and $x ^ { 2 } \equiv 1 \bmod p$ then either $$x \equiv 1 \bmod p \quad \text { or } \quad x \equiv - 1 \bmod p$$ (iii) A company has $\pounds 13940220$ to share between 11 charities. Without performing any division and showing all your working, decide if it is possible to share this money equally between the 11 charities.

\multirow{2}{*}{}		Second transformation
	*	I	\(X\)	\(Y\)	\(Z\)	\(R _ { 1 }\)	\(R _ { 2 }\)
\multirow{6}{*}{First Transformation}	I
	\(X\)		I			\(Z\)
	\(Y\)
	\(Z\)
	\(R _ { 1 }\)		\(Y\)
	\(R _ { 2 }\)

\multirow{2}{*}{}		Second transformation
	*	I	\(X\)	\(Y\)	Z	\(R _ { 1 }\)	\(R _ { 2 }\)
\multirow{6}{*}{First Transformation}	I
	X		I			Z
	Y
	Z
	\(R _ { 1 }\)		\(Y\)
	\(R _ { 2 }\)

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Edexcel FP1 AS 2022 June Q4

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