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Edexcel C1 2018 June Q6

7 marks Moderate -0.8

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

$$\begin{aligned} a _ { 1 } & = 4 \\ a _ { n + 1 } & = \frac { a _ { n } } { a _ { n } + 1 } , \quad n \geqslant 1 , n \in \mathbb { N } \end{aligned}$$

Find the values of $a _ { 2 } , a _ { 3 }$ and $a _ { 4 }$ Write your answers as simplified fractions. Given that $$a _ { n } = \frac { 4 } { p n + q } , \text { where } p \text { and } q \text { are constants }$$
state the value of $p$ and the value of $q$.
Hence calculate the value of $N$ such that $a _ { N } = \frac { 4 } { 321 }$

Edexcel C1 2018 June Q7

8 marks Moderate -0.3

The equation $20 x ^ { 2 } = 4 k x - 13 k x ^ { 2 } + 2$, where $k$ is a constant, has no real roots.

Show that $k$ satisfies the inequality $$2 k ^ { 2 } + 13 k + 20 < 0$$
Find the set of possible values for $k$.

Edexcel C1 2018 June Q8

8 marks Moderate -0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{937246f9-2b6a-48df-b919-c6db3d6f863b-20_1063_1319_251_365} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the straight line $l _ { 1 }$ with equation $4 y = 5 x + 12$

State the gradient of $l _ { 1 }$ The line $l _ { 2 }$ is parallel to $l _ { 1 }$ and passes through the point $E ( 12,5 )$, as shown in Figure 2.
Find the equation of $l _ { 2 }$. Write your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be determined. The line $l _ { 2 }$ cuts the $x$-axis at the point $C$ and the $y$-axis at the point $B$.
Find the coordinates of
1. the point $B$,
2. the point $C$. The line $l _ { 1 }$ cuts the $y$-axis at the point $A$.
  The point $D$ lies on $l _ { 1 }$ such that $A B C D$ is a parallelogram, as shown in Figure 2.
Find the area of $A B C D$.

Edexcel C1 2018 June Q9

12 marks Moderate -0.3

The curve $C$ has equation $y = \mathrm { f } ( x )$, where

$$f ^ { \prime } ( x ) = ( x - 3 ) ( 3 x + 5 )$$ Given that the point $P ( 1,20 )$ lies on $C$,

find $\mathrm { f } ( x )$, simplifying each term.
Show that $$f ( x ) = ( x - 3 ) ^ { 2 } ( x + A )$$ where $A$ is a constant to be found.
Sketch the graph of $C$. Show clearly the coordinates of the points where $C$ cuts or meets the $x$-axis and where $C$ cuts the $y$-axis.

Edexcel C1 2018 June Q10

10 marks Standard +0.3

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{937246f9-2b6a-48df-b919-c6db3d6f863b-28_643_1171_260_518} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of part of the curve $C$ with equation $$y = \frac { 1 } { 2 } x + \frac { 27 } { x } - 12 , \quad x > 0$$ The point $A$ lies on $C$ and has coordinates $\left( 3 , - \frac { 3 } { 2 } \right)$.

Show that the equation of the normal to $C$ at $A$ can be written as $10 y = 4 x - 27$ The normal to $C$ at $A$ meets $C$ again at the point $B$, as shown in Figure 3.
Use algebra to find the coordinates of $B$.

Edexcel C1 Q1

3 marks Easy -1.2

Solve the inequality $10 + x ^ { 2 } > x ( x - 2 )$.
(3)

Edexcel C1 Q2

4 marks Easy -1.2

2. Find $\int \left( x ^ { 2 } - \frac { 1 } { x ^ { 2 } } + \sqrt [ 3 ] { x } \right) \mathrm { d } x$

Edexcel C1 Q3

4 marks Easy -1.8

Find the value of

$81 ^ { \frac { 1 } { 2 } }$,
$81 ^ { \frac { 3 } { 4 } }$,
$81 ^ { - \frac { 3 } { 4 } }$.

Edexcel C1 Q5

7 marks Moderate -0.8

Show that eliminating $y$ from the equations $$\begin{gathered} 2 x + y = 8 \\ 3 x ^ { 2 } + x y = 1 \end{gathered}$$ produces the equation $$x ^ { 2 } + 8 x - 1 = 0$$
Hence solve the simultaneous equations $$\begin{gathered} 2 x + y = 8 \\ 3 x ^ { 2 } + x y = 1 \end{gathered}$$ giving your answers in the form $a + b \sqrt { } 17$, where $a$ and $b$ are integers.

5. continued Leave blank

Edexcel C1 Q6

9 marks Easy -1.2

6. $$f ( x ) = \frac { ( 2 x + 1 ) ( x + 4 ) } { \sqrt { x } } , \quad x > 0$$

Show that $\mathrm { f } ( x )$ can be written in the form $P x ^ { \frac { 3 } { 2 } } + Q x ^ { \frac { 1 } { 2 } } + R x ^ { - \frac { 1 } { 2 } }$, stating the values of the constants $P , Q$ and $R$.
Find f ${ } ^ { \prime } ( x )$.
Show that the tangent to the curve with equation $y = \mathrm { f } ( x )$ at the point where $x = 1$ is parallel to the line with equation $2 y = 11 x + 3$.
(3)

6. continued Leave blank

Edexcel C1 Q7

9 marks Moderate -0.8

Factorise completely $x ^ { 3 } - 4 x$.
(3)
Sketch the curve with equation $y = x ^ { 3 } - 4 x$, showing the coordinates of the points where the curve crosses the $x$-axis.
(3)
On a separate diagram, sketch the curve with equation $y = ( x - 1 ) ^ { 3 } - 4 ( x - 1 ) ,$ showing the coordinates of the points where the curve crosses the $x$-axis.
(3)
\end{tabular} & Leave blank
\hline \end{tabular} \end{center}
\includegraphics[max width=\textwidth, alt={}]{6400bb0c-f199-45f2-a4b1-55534e2c63b0-11_2608_1924_141_75}
\begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

Edexcel C1 Q8

10 marks Moderate -0.8

8. The straight line $l _ { 1 }$ has equation $y = 3 x - 6$.
The straight line $l _ { 2 }$ is perpendicular to $l _ { 1 }$ and passes through the point (6, 2).

Find an equation for $l _ { 2 }$ in the form $y = m x + c$, where $m$ and $c$ are constants.
(3)
The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $C$.
Use algebra to find the coordinates of $C$.
(3)
The lines $l _ { 1 }$ and $l _ { 2 }$ cross the $x$-axis at the points $A$ and $B$ respectively.
Calculate the exact area of triangle $A B C$.
(4) $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ \end{tabular} & Leave blank
\hline \end{tabular} \end{center}
8. continued Leave blank
\begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

Edexcel C1 Q9

11 marks Moderate -0.8

9. An arithmetic series has first term $a$ and common difference $d$.

Prove that the sum of the first $n$ terms of the series is $\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$ (4)
A polygon has 16 sides. The lengths of the sides of the polygon, starting with the shortest side, form an arithmetic sequence with common difference $d \mathrm {~cm}$.
The longest side of the polygon has length 6 cm and the perimeter of the polygon is 72 cm .
Find
the length of the shortest side of the polygon,
(5)
the value of $d$.
(2) $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ \end{tabular} & Leave blank
\hline \end{tabular} \end{center}
Leave blank
\begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

Edexcel C1 Q10

13 marks Moderate -0.8

10. For the curve $C$ with equation $y = \mathrm { f } ( x )$, $\frac { d y } { d x } = x ^ { 3 } + 2 x - 7 .$

Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
(2)
Show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \geq 2$ for all values of $x$.
(1)
Given that the point $P ( 2,4 )$ lies on $C$,
find $y$ in terms of $x$,
(5)
find an equation for the normal to $C$ at $P$ in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.
(5)
1. continued

Edexcel P2 2020 January Q1

7 marks Standard +0.3

The table below shows corresponding values of $x$ and $y$ for $y = \log _ { 2 } ( 2 x )$

The values of $y$ are given to 2 decimal places as appropriate. Using the trapezium rule with all the values of $y$ in the given table,

obtain an estimate for $\int _ { 2 } ^ { 14 } \log _ { 2 } ( 2 x ) \mathrm { d } x$, giving your answer to one decimal place. Using your answer to part (a) and making your method clear, estimate
1. $\quad \int _ { 2 } ^ { 14 } \frac { \log _ { 2 } \left( 4 x ^ { 2 } \right) } { 5 } \mathrm {~d} x$
2. $\int _ { 2 } ^ { 14 } \log _ { 2 } \left( \frac { 2 } { x } \right) \mathrm { d } x$
  $x$ 2 5 8 11 14
  $y$ 2 3.32 4 4.46 4.81

Edexcel P2 2020 January Q2

7 marks Standard +0.3

2. One of the terms in the binomial expansion of $( 3 + a x ) ^ { 6 }$, where $a$ is a constant, is $540 x ^ { 4 }$

Find the possible values of $a$.
Hence find the term independent of $x$ in the expansion of $$\left( \frac { 1 } { 81 } + \frac { 1 } { x ^ { 6 } } \right) ( 3 + a x ) ^ { 6 }$$

Edexcel P2 2020 January Q3

8 marks Standard +0.3

3. $$f ( x ) = 6 x ^ { 3 } + 17 x ^ { 2 } + 4 x - 12$$

Use the factor theorem to show that ( $2 x + 3$ ) is a factor of $\mathrm { f } ( x )$.
Hence, using algebra, write $\mathrm { f } ( x )$ as a product of three linear factors.
Solve, for $\frac { \pi } { 2 } < \theta < \pi$, the equation $$6 \tan ^ { 3 } \theta + 17 \tan ^ { 2 } \theta + 4 \tan \theta - 12 = 0$$ giving your answers to 3 significant figures.

Edexcel P2 2020 January Q4

6 marks Moderate -0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{08aac50c-7317-4510-927a-7f5f2e00f485-08_858_654_118_671} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $$y = 2 x ^ { 2 } + 7 \quad x \geqslant 0$$ The finite region $R$, shown shaded in Figure 1, is bounded by the curve, the $y$-axis and the line with equation $y = 17$ Find the exact area of $R$.

Edexcel P2 2020 January Q5

8 marks Moderate -0.3

5. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A colony of bees is being studied. The number of bees in the colony at the start of the study was 30000 Three years after the start of the study, the number of bees in the colony is 34000 A model predicts that the number of bees in the colony will increase by $p \%$ each year, so that the number of bees in the colony at the end of each year of study forms a geometric sequence. Assuming the model,

find the value of $p$, giving your answer to 2 decimal places. According to the model, at the end of $N$ years of study the number of bees in the colony exceeds 75000
Find, showing all steps in your working, the smallest integer value of $N$.

Edexcel P2 2020 January Q6

8 marks Standard +0.3

6. The circle $C$ has equation $$x ^ { 2 } + y ^ { 2 } + 6 x - 4 y - 14 = 0$$

Find
1. the coordinates of the centre of $C$,
2. the exact radius of $C$. The line with equation $y = k$, where $k$ is a constant, is a tangent to $C$.
Find the possible values of $k$. The line with equation $y = p$, where $p$ is a negative constant, is a chord of $C$.
Given that the length of this chord is 4 units,
find the value of $p$.

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

Edexcel P2 2020 January Q7

7 marks Standard +0.3

Show that the equation $$8 \tan \theta = 3 \cos \theta$$ may be rewritten in the form $$3 \sin ^ { 2 } \theta + 8 \sin \theta - 3 = 0$$
Hence solve, for $0 \leqslant x \leqslant 90 ^ { \circ }$, the equation $$8 \tan 2 x = 3 \cos 2 x$$ giving your answers to 2 decimal places.

Edexcel P2 2020 January Q8

7 marks Moderate -0.8

An arithmetic series has first term $a$ and common difference $d$. Prove that the sum to $n$ terms of this series is $$\frac { n } { 2 } \{ 2 a + ( n - 1 ) d \}$$
A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is given by $$u _ { n } = 5 n + 3 ( - 1 ) ^ { n }$$ Find the value of
1. $u _ { 5 }$
2. $\sum _ { n = 1 } ^ { 59 } u _ { n }$

Edexcel P2 2020 January Q9

7 marks Moderate -0.8

Sketch the curve with equation $$y = 3 \times 4 ^ { x }$$ showing the coordinates of any points of intersection with the coordinate axes. The curve with equation $y = 6 ^ { 1 - x }$ meets the curve with equation $y = 3 \times 4 ^ { x }$ at the point $P$.
Show that the $x$ coordinate of $P$ is $\frac { \log _ { 10 } 2 } { \log _ { 10 } 24 }$

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

Edexcel P2 2020 January Q10

10 marks Standard +0.3

10. A curve $C$ has equation $$y = 4 x ^ { 3 } - 9 x + \frac { k } { x } \quad x > 0$$ where $k$ is a constant.
The point $P$ with $x$ coordinate $\frac { 1 } { 2 }$ lies on $C$.
Given that $P$ is a stationary point of $C$,

show that $k = - \frac { 3 } { 2 }$
Determine the nature of the stationary point at $P$, justifying your answer. The curve $C$ has a second stationary point.
Using algebra, find the $x$ coordinate of this second stationary point. \includegraphics[max width=\textwidth, alt={}, center]{08aac50c-7317-4510-927a-7f5f2e00f485-26_2255_50_312_1980}

Edexcel P2 2021 January Q1

6 marks Moderate -0.3

1. $$f ( x ) = x ^ { 4 } + a x ^ { 3 } - 3 x ^ { 2 } + b x + 5$$ where $a$ and $b$ are constants.
When $\mathrm { f } ( x )$ is divided by ( $x + 1$ ), the remainder is 4

Show that $a + b = - 1$ When $\mathrm { f } ( x )$ is divided by ( $x - 2$ ), the remainder is - 23
Find the value of $a$ and the value of $b$.

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5. continued	Leave blank

8. continued	Leave blank

\(x\)	2	5	8	11	14
\(y\)	2	3.32	4	4.46	4.81

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Edexcel C1 2018 June Q6

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Edexcel C1 2018 June Q8

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Edexcel C1 2018 June Q10

Edexcel C1 Q1

Edexcel C1 Q2

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Edexcel P2 2020 January Q1

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Edexcel P2 2020 January Q9

Edexcel P2 2020 January Q10

Edexcel P2 2021 January Q1