Questions P1 - A-Level Maths

find the value of $a$ and the value of $b$.
State, according to the model, the height of the tree when it was planted.

Edexcel P1 2019 October Q3

3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-06_583_588_395_680} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve $C$ with equation $y = x ^ { 2 } - 5 x + 13$ The point $M$ is the minimum point of $C$. The straight line $l$ passes through the origin $O$ and intersects $C$ at the points $M$ and $N$ as shown. Find, showing your working,

the coordinates of $M$,
the coordinates of $N$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-06_531_561_1793_680} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the curve $C$ and the line $l$. The finite region $R$, shown shaded in Figure 3, is bounded by $C , l$ and the $y$-axis.
Use inequalities to define the region $R$.

Edexcel P1 2019 October Q4

4. A parallelogram $A B C D$ has area $40 \mathrm {~cm} ^ { 2 }$ Given that $A B$ has length $10 \mathrm {~cm} , B C$ has length 6 cm and angle $D A B$ is obtuse, find

the size of angle $D A B$, in degrees, to 2 decimal places,
the length of diagonal $B D$, in cm , to one decimal place.

Edexcel P1 2019 October Q5

5. A curve has equation $$y = \frac { x ^ { 3 } } { 6 } + 4 \sqrt { x } - 15 \quad x \geqslant 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving the answer in simplest form. The point $P \left( 4 , \frac { 11 } { 3 } \right)$ lies on the curve.
Find the equation of the normal to the curve at $P$. Write your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers to be found.
VIIIV SIHI NI III M I I N OC VIIV SIHI NI IM IMM ION OC VI4V SIHI NI JIIYM IONOO

Edexcel P1 2019 October Q6

6. The curve $C$ has equation $y = \frac { 4 } { x } + k$, where $k$ is a positive constant.

Sketch a graph of $C$, stating the equation of the horizontal asymptote and the coordinates of the point of intersection with the $x$-axis. The line with equation $y = 10 - 2 x$ is a tangent to $C$.
Find the possible values for $k$.
$\_\_\_\_$ -

Edexcel P1 2019 October Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-16_648_822_296_561} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows part of the curve with equation $y = 2 x ^ { 2 } + 5$ The point $P ( 2,13 )$ lies on the curve.

Find the gradient of the tangent to the curve at $P$. The point $Q$ with $x$ coordinate $2 + h$ also lies on the curve.
Find, in terms of $h$, the gradient of the line $P Q$. Give your answer in simplest form.
Explain briefly the relationship between the answer to (b) and the answer to (a).

Edexcel P1 2019 October Q8

8. Solve, using algebra, the equation $$x - 6 x ^ { \frac { 1 } { 2 } } + 4 = 0$$ Fully simplify your answers, writing them in the form $a + b \sqrt { c }$, where $a , b$ and $c$ are integers to be found.
(5)

Edexcel P1 2019 October Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-20_671_856_303_548} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a sketch of part of the curve $C$ with equation $y = \sin \left( \frac { x } { 12 } \right)$, where $x$ is measured in radians. The point $M$ shown in Figure 5 is a minimum point on $C$.

State the period of $C$.
State the coordinates of $M$. The smallest positive solution of the equation $\sin \left( \frac { x } { 12 } \right) = k$, where $k$ is a constant, is $\alpha$. Find, in terms of $\alpha$,
1. the negative solution of the equation $\sin \left( \frac { x } { 12 } \right) = k$ that is closest to zero,
2. the smallest positive solution of the equation $\cos \left( \frac { x } { 12 } \right) = k$.

Edexcel P1 2019 October Q10

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-22_592_665_251_676} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} Figure 6 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$, where $$f ( x ) = ( 2 x + 5 ) ( x - 3 ) ^ { 2 }$$

Deduce the values of $x$ for which $\mathrm { f } ( x ) \leqslant 0$ The curve crosses the $y$-axis at the point $P$, as shown.
Expand $\mathrm { f } ( x )$ to the form $$a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where $a$, $b$, $c$ and $d$ are integers to be found.
Hence, or otherwise, find
1. the coordinates of $P$,
2. the gradient of the curve at $P$. The curve with equation $y = \mathrm { f } ( x )$ is translated two units in the positive $x$ direction to a curve with equation $y = \mathrm { g } ( x )$.
1. Find $\mathrm { g } ( x )$, giving your answer in a simplified factorised form.
2. Hence state the $y$ intercept of the curve with equation $y = \mathrm { g } ( x )$.

Edexcel P1 2019 October Q11

A curve has equation $y = \mathrm { f } ( x )$.

The point $P \left( 4 , \frac { 32 } { 3 } \right)$ lies on the curve.
Given that

$\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 4 } { \sqrt { x } } - 3$
$\quad \mathrm { f } ^ { \prime } ( x ) = 5$ at $P$
find
1. the equation of the tangent to the curve at $P$, writing your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be found,
2. $\mathrm { f } ( x )$.

Edexcel P1 2020 October Q1

Given that

$$\left( 3 p q ^ { 2 } \right) ^ { 4 } \times 2 p \sqrt { q ^ { 8 } } \equiv a p ^ { b } q ^ { c }$$ find the values of the constants $a , b$ and $c$.

Edexcel P1 2020 October Q2

2. $$f ( x ) = 3 + 12 x - 2 x ^ { 2 }$$

Express $\mathrm { f } ( x )$ in the form
2. $\mathrm { f } ( x ) = 3 + 12 x - 2 x ^ { 2 }$
Express $\mathrm { f } ( x )$ in the form $$\begin{aligned} & \qquad a - b ( x + c ) ^ { 2 }
& \text { where } a , b \text { and } c \text { are integers to be found. }
& \text { he curve with equation } y = \mathrm { f } ( x ) - 7 \text { crosses the } x \text {-axis at the points } P \text { and } Q \text { and crosses }
& \text { te } y \text {-axis at the point } R \text {. }
& \text { F) Find the area of the triangle } P Q R \text {, giving your answer in the form } m \sqrt { n } \text { where } m \text { and }
& n \text { are integers to be found. } \end{aligned}$$ $\_\_\_\_$ "

Edexcel P1 2020 October Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-08_885_1388_260_287} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the design for a badge.
The design consists of two congruent triangles, $A O C$ and $B O C$, joined to a sector $A O B$ of a circle centre $O$.

Angle $A O B = \alpha$
$A O = O B = 3 \mathrm {~cm}$
$O C = 5 \mathrm {~cm}$

Given that the area of sector $A O B$ is $7.2 \mathrm {~cm} ^ { 2 }$

show that $\alpha = 1.6$ radians.
Hence find
1. the area of the badge, giving your answer in $\mathrm { cm } ^ { 2 }$ to 2 significant figures,
2. the perimeter of the badge, giving your answer in cm to one decimal place.
  VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel P1 2020 October Q4

4. Use algebra to solve the simultaneous equations $$\begin{array} { r } y - 3 x = 4
x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = 4 \end{array}$$ You must show all stages of your working.

Edexcel P1 2020 October Q5

5. (i) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-14_572_1025_212_463} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$.
The curve passes through the points $( - 5,0 )$ and $( 0 , - 3 )$ and touches the $x$-axis at the point $( 2,0 )$. On separate diagrams sketch the curve with equation

$y = \mathrm { f } ( x + 2 )$
$y = \mathrm { f } ( - x )$ On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.
(ii) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-14_415_814_1548_571} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve with equation $$y = k \cos \left( x + \frac { \pi } { 6 } \right) \quad 0 \leqslant x \leqslant 2 \pi$$ where $k$ is a constant.
The curve meets the $y$-axis at the point $( 0 , \sqrt { 3 } )$ and passes through the points $( p , 0 )$ and ( $q , 0$ ). Find
the value of $k$,
the exact value of $p$ and the exact value of $q$.

Edexcel P1 2020 October Q6

6. The point $A$ has coordinates $( - 4,11 )$ and the point $B$ has coordinates $( 8,2 )$.

Find the gradient of the line $A B$, giving your answer as a fully simplified fraction. The point $M$ is the midpoint of $A B$. The line $l$ passes through $M$ and is perpendicular to $A B$.
Find an equation for $l$, giving your answer in the form $p x + q y + r = 0$ where $p , q$ and $r$ are integers to be found. The point $C$ lies on $l$ such that the area of triangle $A B C$ is 37.5 square units.
Find the two possible pairs of coordinates of point $C$.

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel P1 2020 October Q7

7. The curve $C$ has equation $$y = \frac { 1 } { 2 - x }$$

Sketch the graph of $C$. On your sketch you should show the coordinates of any points of intersection with the coordinate axes and state clearly the equations of any asymptotes. The line $l$ has equation $y = 4 x + k$, where $k$ is a constant. Given that $l$ meets $C$ at two distinct points,
show that $$k ^ { 2 } + 16 k + 48 > 0$$
Hence find the range of possible values for $k$.

Edexcel P1 2020 October Q8

8. The curve $C$ has equation $$y = ( x - 2 ) ( x - 4 ) ^ { 2 }$$

Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 32$$ The line $l _ { 1 }$ is the tangent to $C$ at the point where $x = 6$
Find the equation of $l _ { 1 }$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be found. The line $l _ { 2 }$ is the tangent to $C$ at the point where $x = \alpha$
Given that $l _ { 1 }$ and $l _ { 2 }$ are parallel and distinct,
find the value of $\alpha$

Edexcel P1 2020 October Q9

9. A curve with equation $y = \mathrm { f } ( x )$ passes through the point $( 9,10 )$. Given that $$f ^ { \prime } ( x ) = 27 x ^ { 2 } - \frac { 21 x ^ { 3 } - 5 x } { 2 \sqrt { x } } \quad x > 0$$ find $\mathrm { f } ( x )$, fully simplifying each term.

Edexcel P1 2021 October Q1

Find

$$\int 12 x ^ { 3 } + \frac { 1 } { 6 \sqrt { x } } - \frac { 3 } { 2 x ^ { 4 } } \mathrm {~d} x$$ giving each term in simplest form.

Edexcel P1 2021 October Q2

2. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$y = 3 x ^ { 5 } + 4 x ^ { 3 } - x + 5$$ The points $P$ and $Q$ lie on the curve.
The gradient of the curve at both point $P$ and point $Q$ is 2
Find the $x$ coordinates of $P$ and $Q$.

Edexcel P1 2021 October Q3

3. (i) Solve
(ii) $$\frac { 3 } { x } > 4$$ Figure 1 shows a sketch of the curve $C$ and the straight line $l$.
The infinite region $R$, shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by $C$ and $l$ only. Given that

$\quad l$ has a gradient of 3
$C$ has equation $y = 2 x ^ { 2 } - 50$
$\quad C$ and $l$ intersect on the negative $x$-axis
use inequalities to define the region $R$.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-06_643_652_575_648} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ and the straight line $l$.
The infinite region $R$, shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by $C$ and $l$ only.
Given that

$l$ has a gradient of 3
$C$ has equation $y = 2 x ^ { 2 } - 50$
$C$ and $l$ intersect on the negative $x$-axis
use inequalities to define the region $R$.

Edexcel P1 2021 October Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-08_721_855_214_550} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$, where $$f ( x ) = \cos 2 x ^ { \circ } \quad 0 \leqslant x \leqslant k$$ The point $Q$ and the point $R ( k , 0 )$ lie on the curve and are shown in Figure 2.

State
1. the coordinates of $Q$,
2. the value of $k$.
Given that there are exactly two solutions to the equation $$\cos 2 x ^ { \circ } = p \quad \text { in the region } 0 \leqslant x \leqslant k$$ find the range of possible values for $p$.

Edexcel P1 2021 October Q5

5. The line $l _ { 1 }$ has equation $3 y - 2 x = 30$ The line $l _ { 2 }$ passes through the point $A ( 24,0 )$ and is perpendicular to $l _ { 1 }$
Lines $l _ { 1 }$ and $l _ { 2 }$ meet at the point $P$

Find, using algebra and showing your working, the coordinates of $P$. Given that $l _ { 1 }$ meets the $x$-axis at the point $B$,
find the area of triangle $B P A$.

Edexcel P1 2021 October Q6

6. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve $C$ has equation $y = \mathrm { f } ( x )$ where $$f ( x ) = 2 ( x + 1 ) ( x - 3 ) ^ { 2 }$$

Sketch a graph of $C$. Show on your graph the coordinates of the points where $C$ cuts or meets the coordinate axes.
Write $\mathrm { f } ( x )$ in the form $a x ^ { 3 } + b x ^ { 2 } + c x + d$, where $a , b , c$ and $d$ are constants to be found.
Hence, find the equation of the tangent to $C$ at the point where $x = \frac { 1 } { 3 }$

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Edexcel P1 2019 October Q2

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