Questions - Edexcel Paper 1

is hit from a point on the top of a platform of vertical height 3 m above the ground
reaches its maximum vertical height after travelling a horizontal distance of 90 m
is at a vertical height of 27 m above the ground after travelling a horizontal distance of 120 m

Given also that $H$ is modelled as a quadratic function in $x$

find $H$ in terms of $x$
Hence find, according to the model,
1. the maximum vertical height of the ball above the ground,
2. the horizontal distance travelled by the ball, from when it was hit to when it first hits the ground, giving your answer to the nearest metre.
The possible effects of wind or air resistance are two limitations of the model. Give one other limitation of this model.

Edexcel Paper 1 2021 October Q13

A curve $C$ has parametric equations

$$x = \frac { t ^ { 2 } + 5 } { t ^ { 2 } + 1 } \quad y = \frac { 4 t } { t ^ { 2 } + 1 } \quad t \in \mathbb { R }$$ Show that all points on $C$ satisfy $$( x - 3 ) ^ { 2 } + y ^ { 2 } = 4$$

Edexcel Paper 1 2021 October Q14

Given that

$$y = \frac { x - 4 } { 2 + \sqrt { x } } \quad x > 0$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \mathrm {~A} \sqrt { \mathrm { x } } } \quad x > 0$$ where $A$ is a constant to be found.

VIIV SIHI NI III HM IONOO

VIAV SIHI NI III M M I O N OO

VIUV SIHI NI IIIUM ION OC

Edexcel Paper 1 2021 October Q15

(i) Use proof by exhaustion to show that for $n \in \mathbb { N } , n \leqslant 4$

$$( n + 1 ) ^ { 3 } > 3 ^ { n }$$ (ii) Given that $m ^ { 3 } + 5$ is odd, use proof by contradiction to show, using algebra, that $m$ is even.

Edexcel Paper 1 Specimen Q1

The curve $C$ has equation

$$y = 3 x ^ { 4 } - 8 x ^ { 3 } - 3$$

Find (i) $\frac { \mathrm { d } y } { \mathrm {~d} x }$
(ii) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$
Verify that $C$ has a stationary point when $x = 2$
Determine the nature of this stationary point, giving a reason for your answer.

Edexcel Paper 1 Specimen Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f7994129-07ee-4f6d-9531-08a15a38b794-04_350_639_210_712} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The shape $A B C D O A$, as shown in Figure 1, consists of a sector $C O D$ of a circle centre $O$ joined to a sector $A O B$ of a different circle, also centre $O$. Given that arc length $C D = 3 \mathrm {~cm} , \angle C O D = 0.4$ radians and $A O D$ is a straight line of length 12 cm ,

find the length of $O D$,
find the area of the shaded sector $A O B$.

Edexcel Paper 1 Specimen Q3

3. A circle $C$ has equation $$x ^ { 2 } + y ^ { 2 } - 4 x + 10 y = k$$ where $k$ is a constant.

Find the coordinates of the centre of $C$.
State the range of possible values for $k$.

Edexcel Paper 1 Specimen Q4

4. Given that $a$ is a positive constant and $$\int _ { a } ^ { 2 a } \frac { t + 1 } { t } \mathrm {~d} t = \ln 7$$ show that $a = \ln k$, where $k$ is a constant to be found.

Edexcel Paper 1 Specimen Q5

5. A curve $C$ has parametric equations $$x = 2 t - 1 , \quad y = 4 t - 7 + \frac { 3 } { t } , \quad t \neq 0$$ Show that the Cartesian equation of the curve $C$ can be written in the form $$y = \frac { 2 x ^ { 2 } + a x + b } { x + 1 } , \quad x \neq - 1$$ where $a$ and $b$ are integers to be found.

Edexcel Paper 1 Specimen Q6

6. A company plans to extract oil from an oil field. The daily volume of oil $V$, measured in barrels that the company will extract from this oil field depends upon the time, $t$ years, after the start of drilling. The company decides to use a model to estimate the daily volume of oil that will be extracted. The model includes the following assumptions:

The initial daily volume of oil extracted from the oil field will be 16000 barrels.
The daily volume of oil that will be extracted exactly 4 years after the start of drilling will be 9000 barrels.
The daily volume of oil extracted will decrease over time.

The diagram below shows the graphs of two possible models. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f7994129-07ee-4f6d-9531-08a15a38b794-08_629_716_918_292} \captionsetup{labelformat=empty} \caption{Model $A$}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f7994129-07ee-4f6d-9531-08a15a38b794-08_574_711_918_1064} \captionsetup{labelformat=empty} \caption{Model $B$}

\end{figure}

1. Use model $A$ to estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling.
2. Write down a limitation of using model $A$.
1. Using an exponential model and the information given in the question, find a possible equation for model $B$.
2. Using your answer to (b)(i) estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling.

Edexcel Paper 1 Specimen Q7

7. Figure 2 Figure 2 shows a sketch of a triangle $A B C$.
Given $\overrightarrow { A B } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ and $\overrightarrow { B C } = \mathbf { i } - 9 \mathbf { j } + 3 \mathbf { k }$,
show that $\angle B A C = 105.9 ^ { \circ }$ to one decimal place.

Edexcel Paper 1 Specimen Q8

8. $$f ( x ) = \ln ( 2 x - 5 ) + 2 x ^ { 2 } - 30 , \quad x > 2.5$$

Show that $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval [3.5,4] A student takes 4 as the first approximation to $\alpha$.
Given $\mathrm { f } ( 4 ) = 3.099$ and $\mathrm { f } ^ { \prime } ( 4 ) = 16.67$ to 4 significant figures,
apply the Newton-Raphson procedure once to obtain a second approximation for $\alpha$, giving your answer to 3 significant figures.
Show that $\alpha$ is the only root of $\mathrm { f } ( x ) = 0$

Edexcel Paper 1 Specimen Q9

(a) Prove that

$$\tan \theta + \cot \theta \equiv 2 \operatorname { cosec } 2 \theta , \quad \theta \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence explain why the equation $$\tan \theta + \cot \theta = 1$$ does not have any real solutions.

Edexcel Paper 1 Specimen Q10

10. Given that $\theta$ is measured in radians, prove, from first principles, that the derivative of $\sin \theta$ is $\cos \theta$ You may assume the formula for $\sin ( A \pm B )$ and that as $h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1$ and $\frac { \cos h - 1 } { h } \rightarrow 0$

Edexcel Paper 1 Specimen Q11

11. An archer shoots an arrow. The height, $H$ metres, of the arrow above the ground is modelled by the formula $$H = 1.8 + 0.4 d - 0.002 d ^ { 2 } , \quad d \geqslant 0$$ where $d$ is the horizontal distance of the arrow from the archer, measured in metres.
Given that the arrow travels in a vertical plane until it hits the ground,

find the horizontal distance travelled by the arrow, as given by this model.
With reference to the model, interpret the significance of the constant 1.8 in the formula.
Write $1.8 + 0.4 d - 0.002 d ^ { 2 }$ in the form $$A - B ( d - C ) ^ { 2 }$$ where $A , B$ and $C$ are constants to be found. It is decided that the model should be adapted for a different archer.
The adapted formula for this archer is $$H = 2.1 + 0.4 d - 0.002 d ^ { 2 } , \quad d \geqslant 0$$ Hence or otherwise, find, for the adapted model
1. the maximum height of the arrow above the ground.
2. the horizontal distance, from the archer, of the arrow when it is at its maximum height.

Edexcel Paper 1 Specimen Q12

In a controlled experiment, the number of microbes, $N$, present in a culture $T$ days after the start of the experiment were counted.
$N$ and $T$ are expected to satisfy a relationship of the form

$$N = a T ^ { b } , \quad \text { where } a \text { and } b \text { are constants }$$

Show that this relationship can be expressed in the form $$\log _ { 10 } N = m \log _ { 10 } T + c$$ giving $m$ and $c$ in terms of the constants $a$ and/or $b$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7994129-07ee-4f6d-9531-08a15a38b794-18_1232_1046_804_513} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the line of best fit for values of $\log _ { 10 } N$ plotted against values of $\log _ { 10 } T$
Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment.
Explain why the information provided could not reliably be used to estimate the day when the number of microbes in the culture first exceeds 1000000 .
With reference to the model, interpret the value of the constant $a$.

Edexcel Paper 1 Specimen Q13

The curve $C$ has parametric equations

$$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$. The point $P$ lies on $C$ where $t = \frac { 2 \pi } { 3 }$
The line $l$ is the normal to $C$ at $P$.
Show that an equation for $l$ is $$2 x - 2 \sqrt { 3 } y - 1 = 0$$ The line $l$ intersects the curve $C$ again at the point $Q$.
Find the exact coordinates of $Q$. You must show clearly how you obtained your answers.

Edexcel Paper 1 Specimen Q14

14. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f7994129-07ee-4f6d-9531-08a15a38b794-26_567_412_212_824} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of part of the curve $C$ with equation $$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 5 , \quad x > 0$$ The finite region $S$, shown shaded in Figure 4, is bounded by the curve $C$, the line with equation $x = 1$, the $x$-axis and the line with equation $x = 3$ The table below shows corresponding values of $x$ and $y$ with the values of $y$ given to 4 decimal places as appropriate.

$x$	1	1.5	2	2.5	3
$y$	3	2.3041	1.9242	1.9089	2.2958

Use the trapezium rule, with all the values of $y$ in the table, to obtain an estimate for the area of $S$, giving your answer to 3 decimal places.
Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of $S$.
Show that the exact area of $S$ can be written in the form $\frac { a } { b } + \ln c$, where $a , b$ and $c$ are integers to be found.
(In part c, solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel Paper 1 Specimen Q15

15. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f7994129-07ee-4f6d-9531-08a15a38b794-30_551_1026_219_523} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = \frac { 4 \sin 2 x } { \mathrm { e } ^ { \sqrt { 2 } x - 1 } } , \quad 0 \leqslant x \leqslant \pi$$ The curve has a maximum turning point at $P$ and a minimum turning point at $Q$ as shown in Figure 5.

Show that the $x$ coordinates of point $P$ and point $Q$ are solutions of the equation $$\tan 2 x = \sqrt { 2 }$$
Using your answer to part (a), find the $x$-coordinate of the minimum turning point on the curve with equation
1. $y = \mathrm { f } ( 2 x )$.
2. $y = 3 - 2 \mathrm { f } ( x )$.

Edexcel Paper 1 Specimen Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-02_659_853_349_607} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \frac { x } { 1 + \sqrt { x } } , x \geqslant 0$
The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the line with equation $x = 1$, the $x$-axis and the line with equation $x = 3$ The table below shows corresponding values of $x$ and $y$ for $y = \frac { x } { 1 + \sqrt { } x }$

$x$	1	1.5	2	2.5	3
$y$	0.5	0.6742	0.8284	0.9686	1.0981

Use the trapezium rule, with all the values of $y$ in the table, to find an estimate for the area of $R$, giving your answer to 3 decimal places.
Explain how the trapezium rule can be used to give a better approximation for the area of $R$.
Giving your answer to 3 decimal places in each case, use your answer to part (a) to deduce an estimate for
1. $\int _ { 1 } ^ { 3 } \frac { 5 x } { 1 + \sqrt { x } } \mathrm {~d} x$
2. $\int _ { 1 } ^ { 3 } \left( 6 + \frac { x } { 1 + \sqrt { x } } \right) \mathrm { d } x$

Edexcel Paper 1 Specimen Q2

(a) Show that the binomial expansion of

$$( 4 + 5 x ) ^ { \frac { 1 } { 2 } }$$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$ is $$2 + \frac { 5 } { 4 } x + k x ^ { 2 }$$ giving the value of the constant $k$ as a simplified fraction.
(b) (i) Use the expansion from part (a), with $x = \frac { 1 } { 10 }$, to find an approximate value for $\sqrt { 2 }$ Give your answer in the form $\frac { p } { q }$ where $p$ and $q$ are integers.
(ii) Explain why substituting $x = \frac { 1 } { 10 }$ into this binomial expansion leads to a valid approximation.

Edexcel Paper 1 Specimen Q3

A sequence of numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by

$$\begin{aligned} a _ { 1 } & = 3
a _ { n + 1 } & = \frac { a _ { n } - 3 } { a _ { n } - 2 } , \quad n \in \mathbb { N } \end{aligned}$$

Find $\sum _ { r = 1 } ^ { 100 } a _ { r }$
Hence find $\sum _ { r = 1 } ^ { 100 } a _ { r } + \sum _ { r = 1 } ^ { 99 } a _ { r }$

Edexcel Paper 1 Specimen Q4

Relative to a fixed origin $O$,
the point $A$ has position vector $\mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }$,
the point $B$ has position vector $4 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }$,
and the point $C$ has position vector $2 \mathbf { i } + 10 \mathbf { j } + 9 \mathbf { k }$.
Given that $A B C D$ is a parallelogram,
1. find the position vector of point $D$.
The vector $\overrightarrow { A X }$ has the same direction as $\overrightarrow { A B }$.
Given that $| \overrightarrow { A X } | = 10 \sqrt { 2 }$,
find the position vector of $X$.

Edexcel Paper 1 Specimen Q5

$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 48$, where $a$ is a constant

Given that $\mathrm { f } ( - 6 ) = 0$

1. show that $a = 4$
2. express $\mathrm { f } ( x )$ as a product of two algebraic factors. Given that $2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3$
show that $x ^ { 3 } + 4 x ^ { 2 } - 4 x + 48 = 0$
hence explain why $$2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3$$ has no real roots.

Edexcel Paper 1 Specimen Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-12_554_780_246_223} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-12_554_706_246_1133} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 2 shows the entrance to a road tunnel. The maximum height of the tunnel is measured as 5 metres and the width of the base of the tunnel is measured as 6 metres. Figure 3 shows a quadratic curve $B C A$ used to model this entrance.
The points $A , O , B$ and $C$ are assumed to lie in the same vertical plane and the ground $A O B$ is assumed to be horizontal.

Find an equation for curve $B C A$. A coach has height 4.1 m and width 2.4 m .
Determine whether or not it is possible for the coach to enter the tunnel.
Suggest a reason why this model may not be suitable to determine whether or not the coach can pass through the tunnel.

Questions — Edexcel Paper 1 (133 questions)

Browse by module

Browse by board

Edexcel Paper 1 2021 October Q12

Edexcel Paper 1 2021 October Q13

Edexcel Paper 1 2021 October Q14

Edexcel Paper 1 2021 October Q15

Edexcel Paper 1 Specimen Q1

Edexcel Paper 1 Specimen Q2

Edexcel Paper 1 Specimen Q3

Edexcel Paper 1 Specimen Q4

Edexcel Paper 1 Specimen Q5

Edexcel Paper 1 Specimen Q6

Edexcel Paper 1 Specimen Q7

Edexcel Paper 1 Specimen Q8

Edexcel Paper 1 Specimen Q9

Edexcel Paper 1 Specimen Q10

Edexcel Paper 1 Specimen Q11

Edexcel Paper 1 Specimen Q12

Edexcel Paper 1 Specimen Q13

Edexcel Paper 1 Specimen Q14

Edexcel Paper 1 Specimen Q15

Edexcel Paper 1 Specimen Q1

Edexcel Paper 1 Specimen Q2

Edexcel Paper 1 Specimen Q3

Edexcel Paper 1 Specimen Q4

Edexcel Paper 1 Specimen Q5

Edexcel Paper 1 Specimen Q6