Edexcel C1 (Core Mathematics 1)

Question 1

(a) Write down the value of $8 ^ { \frac { 1 } { 3 } }$.
(b) Find the value of $8 ^ { - \frac { 2 } { 3 } }$.
Given that $y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0$,
(a) find $\frac { \mathrm { d } y } { \mathrm {~d} x }$,
(b) find $\int y \mathrm {~d} x$.

$$x ^ { 2 } - 8 x - 29 \equiv ( x + a ) ^ { 2 } + b$$ where $a$ and $b$ are constants.
(a) Find the value of $a$ and the value of $b$.
(b) Hence, or otherwise, show that the roots of $$x ^ { 2 } - 8 x - 29 = 0$$ are $c \pm d \sqrt { } 5$, where $c$ and $d$ are integers to be found.
4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-006_689_920_292_511}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$. The curve passes through the origin $O$ and through the point $( 6,0 )$. The maximum point on the curve is $( 3,5 )$. On separate diagrams, sketch the curve with equation
(a) $y = 3 \mathrm { f } ( x )$,
(b) $y = \mathrm { f } ( x + 2 )$. On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the $x$-axis.
5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1
x ^ { 2 } + y ^ { 2 } = 29 \end{gathered}$$ 6. Find the set of values of $x$ for which
(a) $3 ( 2 x + 1 ) > 5 - 2 x$,
(b) $2 x ^ { 2 } - 7 x + 3 > 0$,
(c) both $3 ( 2 x + 1 ) > 5 - 2 x$ and $2 x ^ { 2 } - 7 x + 3 > 0$.
7. (a) Show that $\frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x }$ can be written as $9 x ^ { - \frac { 1 } { 2 } } - 6 + x ^ { \frac { 1 } { 2 } }$. Given that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , x > 0$, and that $y = \frac { 2 } { 3 }$ at $x = 1$,
(b) find $y$ in terms of $x$.
8. The line $l _ { 1 }$ passes through the point $( 9 , - 4 )$ and has gradient $\frac { 1 } { 3 }$.
(a) Find an equation for $l _ { 1 }$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. The line $l _ { 2 }$ passes through the origin $O$ and has gradient - 2 . The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $P$.
(b) Calculate the coordinates of $P$. Given that $l _ { 1 }$ crosses the $y$-axis at the point $C$,
(c) calculate the exact area of $\triangle O C P$.
\includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_59_2568_1882}
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9. An arithmetic series has first term $a$ and common difference $d$.
(a) Prove that the sum of the first $n$ terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of $n$ months. His monthly repayments form an arithmetic sequence. He repays $\pounds 149$ in the first month, $\pounds 147$ in the second month, $\pounds 145$ in the third month, and so on. He makes his final repayment in the $n$th month, where $n > 21$.
(b) Find the amount Sean repays in the 21st month. Over the $n$ months, he repays a total of $\pounds 5000$.
(c) Form an equation in $n$, and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0$$ (d) Solve the equation in part (c).
(e) State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.
10. The curve $C$ has equation $y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 8 x + 3$. The point $P$ has coordinates $( 3,0 )$.
(a) Show that $P$ lies on $C$.
(b) Find the equation of the tangent to $C$ at $P$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants. Another point $Q$ also lies on $C$. The tangent to $C$ at $Q$ is parallel to the tangent to $C$ at $P$.
(c) Find the coordinates of $Q$.

Question 2

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2. The sequence of positive numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is given by: $$u _ { n + 1 } = \left( u _ { n } - 3 \right) ^ { 2 } , \quad u _ { 1 } = 1$$

Find $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$.
Write down the value of $u _ { 20 }$.

Question 4

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4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-006_689_920_292_511}

$y = 3 \mathrm { f } ( x )$,
$y = \mathrm { f } ( x + 2 )$. On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the $x$-axis.

Question 5

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5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1
x ^ { 2 } + y ^ { 2 } = 29 \end{gathered}$$

Question 8

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8. The line $l _ { 1 }$ passes through the point $( 9 , - 4 )$ and has gradient $\frac { 1 } { 3 }$.

Find an equation for $l _ { 1 }$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. The line $l _ { 2 }$ passes through the origin $O$ and has gradient - 2 . The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $P$.
Calculate the coordinates of $P$. Given that $l _ { 1 }$ crosses the $y$-axis at the point $C$,
calculate the exact area of $\triangle O C P$.
\includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_59_2568_1882}
\includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_1829_2648_114}

Question 9

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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 $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of $n$ months. His monthly repayments form an arithmetic sequence. He repays $\pounds 149$ in the first month, $\pounds 147$ in the second month, $\pounds 145$ in the third month, and so on. He makes his final repayment in the $n$th month, where $n > 21$.
Find the amount Sean repays in the 21st month. Over the $n$ months, he repays a total of $\pounds 5000$.
Form an equation in $n$, and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0$$
Solve the equation in part (c).
State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.

Question 11

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The line $l _ { 1 }$ passes through the points $P ( - 1,2 )$ and $Q ( 11,8 )$.
1. Find an equation for $l _ { 1 }$ in the form $y = m x + c$, where $m$ and $c$ are constants.
The line $l _ { 2 }$ passes through the point $R ( 10,0 )$ and is perpendicular to $l _ { 1 }$. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $S$.
Calculate the coordinates of $S$.
Show that the length of $R S$ is $3 \sqrt { } 5$.
Hence, or otherwise, find the exact area of triangle $P Q R$.
$$y = 4 x ^ { 3 } - 1 + 2 x ^ { \frac { 1 } { 2 } } , \quad x > 0$$ find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. (a) Express $\sqrt { } 108$ in the form $a \sqrt { } 3$, where $a$ is an integer.
Express $( 2 - \sqrt { 3 } ) ^ { 2 }$ in the form $b + c \sqrt { 3 }$, where $b$ and $c$ are integers to be found.
□
3. Given that $\quad \mathrm { f } ( x ) = \frac { 1 } { x } , \quad x \neq 0$,
sketch the graph of $y = \mathrm { f } ( x ) + 3$ and state the equations of the asymptotes.
Find the coordinates of the point where $y = \mathrm { f } ( x ) + 3$ crosses a coordinate axis.
4. Solve the simultaneous equations $$\begin{aligned} & y = x - 2
& y ^ { 2 } + x ^ { 2 } = 10 \end{aligned}$$ 5. The equation $2 x ^ { 2 } - 3 x - ( k + 1 ) = 0$, where $k$ is a constant, has no real roots. Find the set of possible values of $k$.
6. (a) Show that $( 4 + 3 \sqrt { } x ) ^ { 2 }$ can be written as $16 + k \sqrt { } x + 9 x$, where $k$ is a constant to be found.
Find $\int ( 4 + 3 \sqrt { } x ) ^ { 2 } \mathrm {~d} x$.
7. The curve $C$ has equation $y = \mathrm { f } ( x ) , x \neq 0$, and the point $P ( 2,1 )$ lies on $C$. Given that $$f ^ { \prime } ( x ) = 3 x ^ { 2 } - 6 - \frac { 8 } { x ^ { 2 } } ,$$
find $\mathrm { f } ( x )$.
Find an equation for the tangent to $C$ at the point $P$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are integers.
8. The curve $C$ has equation $y = 4 x + 3 x ^ { \frac { 3 } { 2 } } - 2 x ^ { 2 } , \quad x > 0$.
Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Show that the point $P ( 4,8 )$ lies on $C$.
Show that an equation of the normal to $C$ at the point $P$ is $$3 y = x + 20$$ The normal to $C$ at $P$ cuts the $x$-axis at the point $Q$.
Find the length $P Q$, giving your answer in a simplified surd form.
1. Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns:
Row 1 □ Row 2 □ I Row 3 $\square$ She notices that 4 sticks are required to make the single square in the first row, 7 sticks to make 2 squares in the second row and in the third row she needs 10 sticks to make 3 squares.
Find an expression, in terms of $n$, for the number of sticks required to make a similar arrangement of $n$ squares in the $n$th row. Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.
Find the total number of sticks Ann uses in making these 10 rows. Ann started with 1750 sticks. Given that Ann continues the pattern to complete $k$ rows but does not have sufficient sticks to complete the $( k + 1 )$ th row,
show that $k$ satisfies $( 3 k - 100 ) ( k + 35 ) < 0$.
Find the value of $k$.
1. (a) On the same axes sketch the graphs of the curves with equations
  1. $y = x ^ { 2 } ( x - 2 )$,
  2. $y = x ( 6 - x )$,
    and indicate on your sketches the coordinates of all the points where the curves cross the $x$-axis.
2. Use algebra to find the coordinates of the points where the graphs intersect.
\begin{table}[h]
\captionsetup{labelformat=empty} \caption{physicsandmathstutor.com}
\end{table} Paper Reference(s) \section*{6663/01} \section*{Edexcel GCE Core Mathematics C1 Advanced Subsidiary } Turn over
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1. Simplify $( 3 + \sqrt { } 5 ) ( 3 - \sqrt { } 5 )$.
2. (a) Find the value of $8 ^ { \frac { 4 } { 3 } }$.
3. Simplify $\frac { 15 x ^ { \frac { 4 } { 3 } } } { 3 x }$.
4. Given that $y = 3 x ^ { 2 } + 4 \sqrt { } x , x > 0$, find
5. $\frac { \mathrm { d } y } { \mathrm {~d} x }$,
6. $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$,
7. $\int y \mathrm {~d} x$.
8. A girl saves money over a period of 200 weeks. She saves 5 p in Week 1,7 p in Week 2, 9p in Week 3, and so on until Week 200. Her weekly savings form an arithmetic sequence.
9. Find the amount she saves in Week 200.
10. Calculate her total savings over the complete 200 week period.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-064_709_790_238_605} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation $y = \frac { 3 } { x } , x \neq 0$.
On a separate diagram, sketch the curve with equation $y = \frac { 3 } { x + 2 } , x \neq - 2$, showing the coordinates of any point at which the curve crosses a coordinate axis.
(3)
Write down the equations of the asymptotes of the curve in part (a).
6. (a) By eliminating $y$ from the equations $$\begin{gathered} y = x - 4
2 x ^ { 2 } - x y = 8 \end{gathered}$$ show that $$x ^ { 2 } + 4 x - 8 = 0$$
Hence, or otherwise, solve the simultaneous equations $$\begin{gathered} y = x - 4
2 x ^ { 2 } - x y = 8 \end{gathered}$$ giving your answers in the form $a \pm b \sqrt { } 3$, where $a$ and $b$ are integers.
7. The equation $x ^ { 2 } + k x + ( k + 3 ) = 0$, where $k$ is a constant, has different real roots.
Show that $k ^ { 2 } - 4 k - 12 > 0$.
Find the set of possible values of $k$.
8. A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by $$\begin{aligned} a _ { 1 } & = k
a _ { n + 1 } & = 3 a _ { n } + 5 , \quad n \geqslant 1 \end{aligned}$$ where $k$ is a positive integer.
Write down an expression for $a _ { 2 }$ in terms of $k$.
Show that $a _ { 3 } = 9 k + 20$.
1. Find $\sum _ { r = 1 } ^ { 4 } a _ { r }$ in terms of $k$.
2. Show that $\sum _ { r = 1 } ^ { 4 } a _ { r }$ is divisible by 10 .
  9. The curve $C$ with equation $y = \mathrm { f } ( x )$ passes through the point $( 5,65 )$. Given that $\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } - 10 x - 12$,
use integration to find $\mathrm { f } ( x )$.
Hence show that $\mathrm { f } ( x ) = x ( 2 x + 3 ) ( x - 4 )$.
In the space provided on page 17, sketch $C$, showing the coordinates of the points where $C$ crosses the $x$-axis. 10. The curve $C$ has equation $y = x ^ { 2 } ( x - 6 ) + \frac { 4 } { x } , x > 0$. The points $P$ and $Q$ lie on $C$ and have $x$-coordinates 1 and 2 respectively.
Show that the length of $P Q$ is $\sqrt { } 170$.
Show that the tangents to $C$ at $P$ and $Q$ are parallel.
Find an equation for the normal to $C$ at $P$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. 11. The line $l _ { 1 }$ has equation $y = 3 x + 2$ and the line $l _ { 2 }$ has equation $3 x + 2 y - 8 = 0$.
Find the gradient of the line $l _ { 2 }$. The point of intersection of $l _ { 1 }$ and $l _ { 2 }$ is $P$.
Find the coordinates of $P$. The lines $l _ { 1 }$ and $l _ { 2 }$ cross the line $y = 1$ at the points $A$ and $B$ respectively.
Find the area of triangle $A B P$.
\begin{table}[h]
\captionsetup{labelformat=empty} \caption{physicsandmathstutor.com}
\end{table} Paper Reference(s) \section*{6663/01} \section*{Edexcel GCE
Core Mathematics C1 Advanced Subsidiary }
\includegraphics[max width=\textwidth, alt={}]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-074_254_175_594_1272}
Wednesday 9 January 2008 - Afternoon \section*{Materials required for examination
Mathematical Formulae (Green)} \section*{Items included with question papers
Nil} Calculators may NOT be used in this examination. In the boxes above, write your centre number, candidate number, your surname, initials and signature.
Check that you have the correct question paper.
Answer ALL the questions.
You must write your answer for each question in the space following the question. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).
There are 11 questions in this question paper. The total mark for this paper is 75.
There are 24 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled.
You should show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Examiner's use only} \includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-074_97_307_495_1636}
\end{figure} \begin{table}[h]
\captionsetup{labelformat=empty} \caption{ $$\frac { 5 - \sqrt { 3 } } { 2 + \sqrt { 3 } }$$ giving your answer in the form $a + b \sqrt { } 3$, where $a$ and $b$ are integers.
4. The point $A ( - 6,4 )$ and the point $B ( 8 , - 3 )$ lie on the line $L$.
Find an equation for $L$ in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.
Find the distance $A B$, giving your answer in the form $k \sqrt { } 5$, where $k$ is an integer.
5. (a) Write $\frac { 2 \sqrt { } x + 3 } { x }$ in the form $2 x ^ { p } + 3 x ^ { q }$ where $p$ and $q$ are constants. Given that $y = 5 x - 7 + \frac { 2 \sqrt { } x + 3 } { x } , \quad x > 0$,
find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, simplifying the coefficient of each term.
6. \begin{figure}[h] \begin{center} \includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-080_695_678_370_630} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$. The curve crosses the $x$-axis at the points $( 1,0 )$ and $( 4,0 )$. The maximum point on the curve is $( 2,5 )$.
In separate diagrams sketch the curves with the following equations.
On each diagram show clearly the coordinates of the maximum point and of each point at which the curve crosses the $x$-axis.

$y = 2 \mathrm { f } ( x )$,

$y = \mathrm { f } ( - x )$. The maximum point on the curve with equation $y = \mathrm { f } ( x + a )$ is on the $y$-axis.

Write down the value of the constant $a$.

A sequence is given by:

$$\begin{aligned} & x _ { 1 } = 1
& x _ { n + 1 } = x _ { n } \left( p + x _ { n } \right) \end{aligned}$$ where $p$ is a constant ( $p \neq 0$ ) .

Find $x _ { 2 }$ in terms of $p$.

Show that $x _ { 3 } = 1 + 3 p + 2 p ^ { 2 }$. Given that $x _ { 3 } = 1$,

find the value of $p$,

write down the value of $x _ { 2008 }$.
8. The equation $$x ^ { 2 } + k x + 8 = k$$ has no real solutions for $x$.

Show that $k$ satisfies $k ^ { 2 } + 4 k - 32 < 0$.

Hence find the set of possible values of $k$.
9. The curve $C$ has equation $y = \mathrm { f } ( x ) , x > 0$, and $\mathrm { f } ^ { \prime } ( x ) = 4 x - 6 \sqrt { } x + \frac { 8 } { x ^ { 2 } }$. Given that the point $P ( 4,1 )$ lies on $C$,

find $\mathrm { f } ( x )$ and simplify your answer.

Find an equation of the normal to $C$ at the point $P ( 4,1 )$.

The curve $C$ has equation

$$y = ( x + 3 ) ( x - 1 ) ^ { 2 }$$

Sketch $C$ showing clearly the coordinates of the points where the curve meets the coordinate axes.

Show that the equation of $C$ can be written in the form $$y = x ^ { 3 } + x ^ { 2 } - 5 x + k$$ where $k$ is a positive integer, and state the value of $k$. There are two points on $C$ where the gradient of the tangent to $C$ is equal to 3 .

Find the $x$-coordinates of these two points.

The first term of an arithmetic sequence is 30 and the common difference is - 1.5
Find the value of the 25th term.

The $r$ th term of the sequence is 0 .

Find the value of $r$. The sum of the first $n$ terms of the sequence is $S _ { n }$.

Find the largest positive value of $S _ { n }$.
\begin{table}[h]

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\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-094_467_707_274_587} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$. The curve passes through the point ( 0,7 ) and has a minimum point at ( 7,0 ). On separate diagrams, sketch the curve with equation

$y = \mathrm { f } ( x ) + 3$,

$y = \mathrm { f } ( 2 x )$. On each diagram, show clearly the coordinates of the minimum point and the coordinates of the point at which the curve crosses the $y$-axis.
4. $$\mathrm { f } ( x ) = 3 x + x ^ { 3 } , \quad x > 0$$

Differentiate to find $\mathrm { f } ^ { \prime } ( x )$. Given that $\mathrm { f } ^ { \prime } ( x ) = 15$,

find the value of $x$.
5. A sequence $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ is defined by $$\begin{gathered} x _ { 1 } = 1 ,
x _ { n + 1 } = a x _ { n } - 3 , n \geqslant 1 , \end{gathered}$$ where $a$ is a constant.

Find an expression for $x _ { 2 }$ in terms of $a$.

Show that $x _ { 3 } = a ^ { 2 } - 3 a - 3$. Given that $x _ { 3 } = 7$,

find the possible values of $a$.
6. The curve $C$ has equation $y = \frac { 3 } { x }$ and the line $l$ has equation $y = 2 x + 5$.

On the axes below, sketch the graphs of $C$ and $l$, indicating clearly the coordinates of any intersections with the axes.

Find the coordinates of the points of intersection of $C$ and $l$.
\includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-097_1137_1141_1046_397} 7. Sue is training for a marathon. Her training includes a run every Saturday starting with a run of 5 km on the first Saturday. Each Saturday she increases the length of her run from the previous Saturday by 2 km .

Show that on the 4th Saturday of training she runs 11 km .

Find an expression, in terms of $n$, for the length of her training run on the $n$th Saturday.

Show that the total distance she runs on Saturdays in $n$ weeks of training is $n ( n + 4 ) \mathrm { km }$. On the $n$th Saturday Sue runs 43 km .

Find the value of $n$.

Find the total distance, in km , Sue runs on Saturdays in $n$ weeks of training.
8. Given that the equation $2 q x ^ { 2 } + q x - 1 = 0$, where $q$ is a constant, has no real roots,

show that $q ^ { 2 } + 8 q < 0$.

Hence find the set of possible values of $q$.
9. The curve $C$ has equation $y = k x ^ { 3 } - x ^ { 2 } + x - 5$, where $k$ is a constant.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$. The point $A$ with $x$-coordinate $- \frac { 1 } { 2 }$ lies on $C$. The tangent to $C$ at $A$ is parallel to the line with equation $2 y - 7 x + 1 = 0$. Find

the value of $k$,

the value of the $y$-coordinate of $A$.
10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-104_543_865_287_539} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The points $Q ( 1,3 )$ and $R ( 7,0 )$ lie on the line $l _ { 1 }$, as shown in Figure 2.
The length of $Q R$ is $a \sqrt { } 5$.

Find the value of $a$. The line $l _ { 2 }$ is perpendicular to $l _ { 1 }$, passes through $Q$ and crosses the $y$-axis at the point $P$, as shown in Figure 2. Find

an equation for $l _ { 2 }$,

the coordinates of $P$,

the area of $\triangle P Q R$.

The gradient of a curve $C$ is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( x ^ { 2 } + 3 \right) ^ { 2 } } { x ^ { 2 } } , x \neq 0$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } + 6 + 9 x ^ { - 2 }$.

The point $( 3,20 )$ lies on $C$.

Find an equation for the curve $C$ in the form $y = \mathrm { f } ( x )$.
\begin{table}[h]

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\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-113_991_1160_285_388} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ with equation $y = \mathrm { f } ( x )$. There is a maximum at $( 0,0 )$, a minimum at $( 2 , - 1 )$ and $C$ passes through $( 3,0 )$. On separate diagrams sketch the curve with equation

$y = \mathrm { f } ( x + 3 )$,

$y = \mathrm { f } ( - x )$. On each diagram show clearly the coordinates of the maximum point, the minimum point and any points of intersection with the $x$-axis.

Given that $\frac { 2 x ^ { 2 } - x ^ { \frac { 3 } { 2 } } } { \sqrt { } x }$ can be written in the form $2 x ^ { p } - x ^ { q }$,
write down the value of $p$ and the value of $q$.

Given that $y = 5 x ^ { 4 } - 3 + \frac { 2 x ^ { 2 } - x ^ { \frac { 3 } { 2 } } } { \sqrt { } x }$,

find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, simplifying the coefficient of each term.
7. The equation $k x ^ { 2 } + 4 x + ( 5 - k ) = 0$, where $k$ is a constant, has 2 different real solutions for $x$.

Show that $k$ satisfies $$k ^ { 2 } - 5 k + 4 > 0$$

Hence find the set of possible values of $k$.
8. The point $P ( 1 , a )$ lies on the curve with equation $y = ( x + 1 ) ^ { 2 } ( 2 - x )$.

Find the value of $a$.

On the axes below sketch the curves with the following equations:

$y = ( x + 1 ) ^ { 2 } ( 2 - x )$,
$y = \frac { 2 } { x }$. On your diagram show clearly the coordinates of any points at which the curves meet the axes.

With reference to your diagram in part (b) state the number of real solutions to the equation $$( x + 1 ) ^ { 2 } ( 2 - x ) = \frac { 2 } { x }$$

\includegraphics[max width=\textwidth, alt={}]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-117_1356_1353_1238_297}

9. The first term of an arithmetic series is $a$ and the common difference is $d$. The 18th term of the series is 25 and the 21st term of the series is $32 \frac { 1 } { 2 }$.

Use this information to write down two equations for $a$ and $d$.

Show that $a = - 17.5$ and find the value of $d$. The sum of the first $n$ terms of the series is 2750 .

Show that $n$ is given by $$n ^ { 2 } - 15 n = 55 \times 40$$

Hence find the value of $n$.

The line $l _ { 1 }$ passes through the point $A ( 2,5 )$ and has gradient $- \frac { 1 } { 2 }$.
Find an equation of $l _ { 1 }$, giving your answer in the form $y = m x + c$.

The point $B$ has coordinates ( $- 2,7$ ).

Show that $B$ lies on $l _ { 1 }$.

Find the length of $A B$, giving your answer in the form $k \sqrt { } 5$, where $k$ is an integer. The point $C$ lies on $l _ { 1 }$ and has $x$-coordinate equal to $p$.
The length of $A C$ is 5 units.

Show that $p$ satisfies $$p ^ { 2 } - 4 p - 16 = 0$$

The curve $C$ has equation

$$y = 9 - 4 x - \frac { 8 } { x } , \quad x > 0$$ The point $P$ on $C$ has $x$-coordinate equal to 2 .

Show that the equation of the tangent to $C$ at the point $P$ is $y = 1 - 2 x$.

Find an equation of the normal to $C$ at the point $P$. The tangent at $P$ meets the $x$-axis at $A$ and the normal at $P$ meets the $x$-axis at $B$.

Find the area of triangle $A P B$.
\begin{table}[h]

\captionsetup{labelformat=empty} \caption{physicsandmathstutor.com}

\end{table} Paper Reference(s) \section*{6663/01} \section*{Edexcel GCE
Core Mathematics C1 Advanced Subsidiary } Examiner's use only
\includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-125_95_309_495_1636} $\frac { \text { Items included with question papers } } { \text { Nil } }$ Calculators may NOT be used in this examination. In the boxes above, write your centre number, candidate number, your surname, initials and signature.
Check that you have the correct question paper.
Answer ALL the questions.
You must write your answer for each question in the space following the question. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).
There are 11 questions in this question paper. The total mark for this paper is 75.
There are 28 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled.
You should show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit. Turn over
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Simplify
$( 3 \sqrt { } 7 ) ^ { 2 }$
$( 8 + \sqrt { } 5 ) ( 2 - \sqrt { } 5 )$
Given that $32 \sqrt { } 2 = 2 ^ { a }$, find the value of $a$.
Given that $y = 2 x ^ { 3 } + \frac { 3 } { x ^ { 2 } } , x \neq 0$, find
$\frac { \mathrm { d } y } { \mathrm {~d} x }$
$\int y \mathrm {~d} x$, simplifying each term.
Find the set of values of $x$ for which
$4 x - 3 > 7 - x$
$2 x ^ { 2 } - 5 x - 12 < 0$
both $4 x - 3 > 7 - x$ and $2 x ^ { 2 } - 5 x - 12 < 0$
A 40-year building programme for new houses began in Oldtown in the year 1951 (Year 1) and finished in 1990 (Year 40).

The numbers of houses built each year form an arithmetic sequence with first term $a$ and common difference $d$. Given that 2400 new houses were built in 1960 and 600 new houses were built in 1990, find

the value of $d$,

the value of $a$,

the total number of houses built in Oldtown over the 40-year period.
6. The equation $x ^ { 2 } + 3 p x + p = 0$, where $p$ is a non-zero constant, has equal roots. Find the value of $p$.
(4)
7. A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by $$\begin{aligned} a _ { 1 } & = k
a _ { n + 1 } & = 2 a _ { n } - 7 , \quad n \geqslant 1 \end{aligned}$$ where $k$ is a constant.

Write down an expression for $a _ { 2 }$ in terms of $k$.

Show that $a _ { 3 } = 4 k - 21$. Given that $\sum _ { r = 1 } ^ { 4 } a _ { r } = 43$,

find the value of $k$.
8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-133_908_1046_201_495} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The points $A$ and $B$ have coordinates $( 6,7 )$ and $( 8,2 )$ respectively.
The line $l$ passes through the point $A$ and is perpendicular to the line $A B$, as shown in Figure 1.

Find an equation for $l$ in the form $a x + b y + c = 0$, where $a$, b and $c$ are integers. Given that $l$ intersects the $y$-axis at the point $C$, find

the coordinates of $C$,

the area of $\triangle O C B$, where $O$ is the origin.
9. $$f ( x ) = \frac { ( 3 - 4 \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , \quad x > 0$$

Show that $\mathrm { f } ( x ) = 9 x ^ { - \frac { 1 } { 2 } } + A x ^ { \frac { 1 } { 2 } } + B$, where $A$ and $B$ are constants to be found.

Find $\mathrm { f } ^ { \prime } ( x )$.

Evaluate f'(9).
10. (a) Factorise completely $x ^ { 3 } - 6 x ^ { 2 } + 9 x$

Sketch the curve with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x$$ showing the coordinates of the points at which the curve meets the $x$-axis. Using your answer to part (b), or otherwise,

sketch, on a separate diagram, the curve with equation $$y = ( x - 2 ) ^ { 3 } - 6 ( x - 2 ) ^ { 2 } + 9 ( x - 2 )$$ showing the coordinates of the points at which the curve meets the $x$-axis.

The curve $C$ has equation

$$y = x ^ { 3 } - 2 x ^ { 2 } - x + 9 , \quad x > 0$$ The point $P$ has coordinates (2, 7).

Show that $P$ lies on $C$.

Find the equation of the tangent to $C$ at $P$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants. The point $Q$ also lies on $C$.
Given that the tangent to $C$ at $Q$ is perpendicular to the tangent to $C$ at $P$,

show that the $x$-coordinate of $Q$ is $\frac { 1 } { 3 } ( 2 + \sqrt { 6 } )$.
\begin{table}[h]

\captionsetup{labelformat=empty} \caption{physicsandmathstutor.com}

\end{table} Paper Reference(s) \section*{6663/01} \section*{Edexcel GCE
Core Mathematics C1 Advanced Subsidiary } \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Examiner's use only} \includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-140_97_309_493_1636}

\end{figure} $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 x ^ { - \frac { 1 } { 2 } } + x \sqrt { } x , \quad x > 0$$ Given that $y = 35$ at $x = 4$, find $y$ in terms of $x$, giving each term in its simplest form.
5. Solve the simultaneous equations $$\begin{array} { r } y - 3 x + 2 = 0
y ^ { 2 } - x - 6 x ^ { 2 } = 0 \end{array}$$

The curve $C$ has equation

$$y = \frac { ( x + 3 ) ( x - 8 ) } { x } , \quad x > 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in its simplest form.

Find an equation of the tangent to $C$ at the point where $x = 2$
7. Jill gave money to a charity over a 20 -year period, from Year 1 to Year 20 inclusive. She gave $\pounds 150$ in Year 1, $\pounds 160$ in Year 2, $\pounds 170$ in Year 3, and so on, so that the amounts of money she gave each year formed an arithmetic sequence.

Find the amount of money she gave in Year 10.

Calculate the total amount of money she gave over the 20 -year period. Kevin also gave money to the charity over the same 20 -year period.
He gave $\pounds A$ in Year 1 and the amounts of money he gave each year increased, forming an arithmetic sequence with common difference $\pounds 30$. The total amount of money that Kevin gave over the 20 -year period was twice the total amount of money that Jill gave.

Calculate the value of $A$.
8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-149_646_991_246_559} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$.
The curve has a maximum point $( - 2,5 )$ and an asymptote $y = 1$, as shown in Figure 1.
On separate diagrams, sketch the curve with equation

$y = \mathrm { f } ( x ) + 2$

$y = 4 \mathrm { f } ( x )$

$y = \mathrm { f } ( \mathrm { x } + 1 )$ On each diagram, show clearly the coordinates of the maximum point and the equation of the asymptote.

(a) Factorise completely $x ^ { 3 } - 4 x$
Sketch the curve $C$ with equation

$$y = x ^ { 3 } - 4 x$$ showing the coordinates of the points at which the curve meets the $x$-axis. The point $A$ with $x$-coordinate - 1 and the point $B$ with $x$-coordinate 3 lie on the curve $C$.

Find an equation of the line which passes through $A$ and $B$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants.

Show that the length of $A B$ is $k \sqrt { } 10$, where $k$ is a constant to be found.
10.
$\mathrm { f } ( x ) = x ^ { 2 } + 4 k x + ( 3 + 11 k ) , \quad$ where $k$ is a constant.

Express $\mathrm { f } ( x )$ in the form $( x + p ) ^ { 2 } + q$, where $p$ and $q$ are constants to be found in terms of $k$. Given that the equation $\mathrm { f } ( x ) = 0$ has no real roots,

find the set of possible values of $k$. Given that $k = 1$,

sketch the graph of $y = \mathrm { f } ( x )$, showing the coordinates of any point at which the graph crosses a coordinate axis. Turn over
advancing learning, changing lives

Write

$$\sqrt { } ( 75 ) - \sqrt { } ( 27 )$$ in the form $k \sqrt { } x$, where $k$ and $x$ are integers.
2. Find $$\int \left( 8 x ^ { 3 } + 6 x ^ { \frac { 1 } { 2 } } - 5 \right) d x$$ giving each term in its simplest form.
3. Find the set of values of $x$ for which

$3 ( x - 2 ) < 8 - 2 x$

$( 2 x - 7 ) ( 1 + x ) < 0$

both $3 ( x - 2 ) < 8 - 2 x$ and $( 2 x - 7 ) ( 1 + x ) < 0$
4. (a) Show that $x ^ { 2 } + 6 x + 11$ can be written as $$( x + p ) ^ { 2 } + q$$ where $p$ and $q$ are integers to be found.

In the space at the top of page 7 , sketch the curve with equation $y = x ^ { 2 } + 6 x + 11$, showing clearly any intersections with the coordinate axes.

Find the value of the discriminant of $x ^ { 2 } + 6 x + 11$

A sequence of positive numbers is defined by

$$\begin{aligned} a _ { n + 1 } & = \sqrt { } \left( a _ { n } ^ { 2 } + 3 \right) , \quad n \geqslant 1
a _ { 1 } & = 2 \end{aligned}$$

Find $a _ { 2 }$ and $a _ { 3 }$, leaving your answers in surd form.

Show that $a _ { 5 } = 4$
6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-162_568_942_269_498} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$. The curve has a maximum point $A$ at $( - 2,3 )$ and a minimum point $B$ at $( 3 , - 5 )$. On separate diagrams sketch the curve with equation

$y = \mathrm { f } ( x + 3 )$

$y = 2 \mathrm { f } ( x )$ On each diagram show clearly the coordinates of the maximum and minimum points.
The graph of $y = \mathrm { f } ( x ) + a$ has a minimum at (3, 0), where $a$ is a constant.

Write down the value of $a$.

Given that

$$y = 8 x ^ { 3 } - 4 \sqrt { } x + \frac { 3 x ^ { 2 } + 2 } { x } , \quad x > 0$$ find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
8. (a) Find an equation of the line joining $A ( 7,4 )$ and $B ( 2,0 )$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

Find the length of $A B$, leaving your answer in surd form. The point $C$ has coordinates $( 2 , t )$, where $t > 0$, and $A C = A B$.

Find the value of $t$.

Find the area of triangle $A B C$.
9. A farmer has a pay scheme to keep fruit pickers working throughout the 30 day season. He pays $\pounds a$ for their first day, $\pounds ( a + d )$ for their second day, $\pounds ( a + 2 d )$ for their third day, and so on, thus increasing the daily payment by $\pounds d$ for each extra day they work. A picker who works for all 30 days will earn $\pounds 40.75$ on the final day.

Use this information to form an equation in $a$ and $d$. A picker who works for all 30 days will earn a total of $\pounds 1005$

Show that $15 ( a + 40.75 ) = 1005$

Hence find the value of $a$ and the value of $d$.
10. (a) On the axes below sketch the graphs of

$y = x ( 4 - x )$
$y = x ^ { 2 } ( 7 - x )$
showing clearly the coordinates of the points where the curves cross the coordinate axes.

Show that the $x$-coordinates of the points of intersection of $$y = x ( 4 - x ) \text { and } y = x ^ { 2 } ( 7 - x )$$ are given by the solutions to the equation $x \left( x ^ { 2 } - 8 x + 4 \right) = 0$ The point $A$ lies on both of the curves and the $x$ and $y$ coordinates of $A$ are both positive.

Find the exact coordinates of $A$, leaving your answer in the form ( $p + q \sqrt { } 3 , r + s \sqrt { } 3$ ), where $p , q , r$ and $s$ are integers.
\includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-168_1178_1203_1407_379} 11. The curve $C$ has equation $y = \mathrm { f } ( x ) , x > 0$, where $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x - \frac { 5 } { \sqrt { } x } - 2$$ Given that the point $P ( 4,5 )$ lies on $C$, find

$\mathrm { f } ( x )$,

an equation of the tangent to $C$ at the point $P$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. Turn over
advancing learning, changing lives

(a) Find the value of $16 ^ { - \frac { 1 } { 4 } }$
Simplify $x \left( 2 x ^ { - \frac { 1 } { 4 } } \right) ^ { 4 }$
Find

$$\int \left( 12 x ^ { 5 } - 3 x ^ { 2 } + 4 x ^ { \frac { 1 } { 3 } } \right) \mathrm { d } x$$ giving each term in its simplest form.
3. Simplify $$\frac { 5 - 2 \sqrt { 3 } } { \sqrt { 3 } - 1 }$$ giving your answer in the form $p + q \sqrt { } 3$, where $p$ and $q$ are rational numbers.
4. A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by $$\begin{aligned} a _ { 1 } & = 2
a _ { n + 1 } & = 3 a _ { n } - c \end{aligned}$$ where $c$ is a constant.

Find an expression for $a _ { 2 }$ in terms of $c$. Given that $\sum _ { i = 1 } ^ { 3 } a _ { i } = 0$

find the value of $c$.
5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-177_646_1075_319_431} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = \frac { x } { x - 2 } , \quad x \neq 2$$ The curve passes through the origin and has two asymptotes, with equations $y = 1$ and $x = 2$, as shown in Figure 1.

In the space below, sketch the curve with equation $y = \mathrm { f } ( x - 1 )$ and state the equations of the asymptotes of this curve.
(3)

Find the coordinates of the points where the curve with equation $y = \mathrm { f } ( x - 1 )$ crosses the coordinate axes.
(4)

An arithmetic sequence has first term $a$ and common difference $d$. The sum of the first 10 terms of the sequence is 162 .
Show that $10 a + 45 d = 162$

Given also that the sixth term of the sequence is 17 ,

write down a second equation in $a$ and $d$,

find the value of $a$ and the value of $d$.
7. The curve with equation $y = \mathrm { f } ( x )$ passes through the point $( - 1,0 )$. Given that $$\mathrm { f } ^ { \prime } ( x ) = 12 x ^ { 2 } - 8 x + 1$$ find $\mathrm { f } ( x )$.
8. The equation $x ^ { 2 } + ( k - 3 ) x + ( 3 - 2 k ) = 0$, where $k$ is a constant, has two distinct real roots.

Show that $k$ satisfies $$k ^ { 2 } + 2 k - 3 > 0$$

Find the set of possible values of $k$.
9. The line $L _ { 1 }$ has equation $2 y - 3 x - k = 0$, where $k$ is a constant. Given that the point $A ( 1,4 )$ lies on $L _ { 1 }$, find

the value of $k$,

the gradient of $L _ { 1 }$. The line $L _ { 2 }$ passes through $A$ and is perpendicular to $L _ { 1 }$.

Find an equation of $L _ { 2 }$ giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. The line $L _ { 2 }$ crosses the $x$-axis at the point $B$.

Find the coordinates of $B$.

Find the exact length of $A B$.

(a) On the axes below, sketch the graphs of
1. $y = x ( x + 2 ) ( 3 - x )$
2. $y = - \frac { 2 } { x }$
  showing clearly the coordinates of all the points where the curves cross the coordinate axes.
Using your sketch state, giving a reason, the number of real solutions to the equation

$$x ( x + 2 ) ( 3 - x ) + \frac { 2 } { x } = 0$$ \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-184_988_992_1274_482}
11. The curve $C$ has equation $$y = \frac { 1 } { 2 } x ^ { 3 } - 9 x ^ { \frac { 3 } { 2 } } + \frac { 8 } { x } + 30 , \quad x > 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.

Show that the point $P ( 4 , - 8 )$ lies on $C$.

Find an equation of the normal to $C$ at the point $P$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
Turn over
advancing learning, changing lives

Find the value of
$25 ^ { \frac { 1 } { 2 } }$
$25 ^ { - \frac { 3 } { 2 } }$
Given that $y = 2 x ^ { 5 } + 7 + \frac { 1 } { x ^ { 3 } } , x \neq 0$, find, in their simplest form,
$\frac { \mathrm { d } y } { \mathrm {~d} x }$,
$\int y \mathrm {~d} x$.
The points $P$ and $Q$ have coordinates $( - 1,6 )$ and $( 9,0 )$ respectively.

The line $l$ is perpendicular to $P Q$ and passes through the mid-point of $P Q$.
Find an equation for $l$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
4. Solve the simultaneous equations $$\begin{aligned} x + y & = 2
4 y ^ { 2 } - x ^ { 2 } & = 11 \end{aligned}$$

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

$$\begin{aligned} a _ { 1 } & = k
a _ { n + 1 } & = 5 a _ { n } + 3 , \quad n \geqslant 1 , \end{aligned}$$ where $k$ is a positive integer.

Write down an expression for $a _ { 2 }$ in terms of $k$.

Show that $a _ { 3 } = 25 k + 18$.

Find $\sum _ { r = 1 } ^ { 4 } a _ { r }$ in terms of $k$, in its simplest form.
Show that $\sum _ { r = 1 } ^ { 4 } a _ { r }$ is divisible by 6 .
6. Given that $\frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }$ can be written in the form $6 x ^ { p } + 3 x ^ { q }$,

write down the value of $p$ and the value of $q$. Given that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x + 3 x ^ { \frac { 5 } { 2 } } } { \sqrt { } x }$, and that $y = 90$ when $x = 4$,

find $y$ in terms of $x$, simplifying the coefficient of each term.
7. $$\mathrm { f } ( x ) = x ^ { 2 } + ( k + 3 ) x + k$$ where $k$ is a real constant.

Find the discriminant of $\mathrm { f } ( x )$ in terms of $k$.

Show that the discriminant of $\mathrm { f } ( x )$ can be expressed in the form $( k + a ) ^ { 2 } + b$, where $a$ and $b$ are integers to be found.

Show that, for all values of $k$, the equation $\mathrm { f } ( x ) = 0$ has real roots.
8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-195_490_743_207_603} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ with equation $y = \mathrm { f } ( x )$.
The curve $C$ passes through the origin and through $( 6,0 )$.
The curve $C$ has a minimum at the point $( 3 , - 1 )$.
On separate diagrams, sketch the curve with equation

$y = \mathrm { f } ( 2 x )$,

$y = - \mathrm { f } ( x )$,

$y = \mathrm { f } ( x + p )$, where $p$ is a constant and $0 < p < 3$. On each diagram show the coordinates of any points where the curve intersects the $x$-axis and of any minimum or maximum points.

(a) Calculate the sum of all the even numbers from 2 to 100 inclusive,

$$2 + 4 + 6 + \ldots \ldots + 100$$

In the arithmetic series $$k + 2 k + 3 k + \ldots \ldots + 100$$ $k$ is a positive integer and $k$ is a factor of 100.

Find, in terms of $k$, an expression for the number of terms in this series.
Show that the sum of this series is $$50 + \frac { 5000 } { k }$$

Find, in terms of $k$, the 50th term of the arithmetic sequence $$( 2 k + 1 ) , ( 4 k + 4 ) , ( 6 k + 7 ) , \ldots \ldots ,$$ giving your answer in its simplest form.

The curve $C$ has equation

$$y = ( x + 1 ) ( x + 3 ) ^ { 2 }$$

Sketch $C$, showing the coordinates of the points at which $C$ meets the axes.

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 14 x + 15$. The point $A$, with $x$-coordinate - 5 , lies on $C$.

Find the equation of the tangent to $C$ at $A$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants. Another point $B$ also lies on $C$. The tangents to $C$ at $A$ and $B$ are parallel.

Find the $x$-coordinate of $B$.

\includegraphics[max width=\textwidth, alt={}]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-201_255_180_593_1436}

Friday 13 January 2012 - Morning
Time: 1 hour 30 minutes Calculators may NOT be used in this examination. In the boxes above, write your centre number, candidate number, your surname, initials and signature.
Check that you have the correct question paper.
Answer ALL the questions.
You must write your answer to each question in the space following the question. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).
There are 10 questions in this question paper. The total mark for this paper is 75.
There are 28 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled.
You should show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit. $$\sqrt { } 32 + \sqrt { } 18$$ giving your answer in the form $a \sqrt { } 2$, where $a$ is an integer.

Simplify $$\frac { \sqrt { } 32 + \sqrt { } 18 } { 3 + \sqrt { } 2 }$$ giving your answer in the form $b \sqrt { } 2 + c$, where $b$ and $c$ are integers.
3. Find the set of values of $x$ for which

$4 x - 5 > 15 - x$

$x ( x - 4 ) > 12$
4. A sequence $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ is defined by $$\begin{aligned} x _ { 1 } & = 1
x _ { n + 1 } & = a x _ { n } + 5 , \quad n \geqslant 1 \end{aligned}$$ where $a$ is a constant.

Write down an expression for $x _ { 2 }$ in terms of $a$.

Show that $x _ { 3 } = a ^ { 2 } + 5 a + 5$ Given that $x _ { 3 } = 41$

find the possible values of $a$.
5. The curve $C$ has equation $y = x ( 5 - x )$ and the line $L$ has equation $2 y = 5 x + 4$

Use algebra to show that $C$ and $L$ do not intersect.

In the space on page 11, sketch $C$ and $L$ on the same diagram, showing the coordinates of the points at which $C$ and $L$ meet the axes.
6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-207_647_929_274_511} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The line $l _ { 1 }$ has equation $2 x - 3 y + 12 = 0$

Find the gradient of $l _ { 1 }$. The line $l _ { 1 }$ crosses the $x$-axis at the point $A$ and the $y$-axis at the point $B$, as shown in Figure 1. The line $l _ { 2 }$ is perpendicular to $l _ { 1 }$ and passes through $B$.

Find an equation of $l _ { 2 }$. The line $l _ { 2 }$ crosses the $x$-axis at the point $C$.

Find the area of triangle $A B C$.

A curve with equation $y = \mathrm { f } ( x )$ passes through the point $( 2,10 )$. Given that

$$f ^ { \prime } ( x ) = 3 x ^ { 2 } - 3 x + 5$$ find the value of $f ( 1 )$.
8. The curve $C _ { 1 }$ has equation $$y = x ^ { 2 } ( x + 2 )$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$

Sketch $C _ { 1 }$, showing the coordinates of the points where $C _ { 1 }$ meets the $x$-axis.

Find the gradient of $C _ { 1 }$ at each point where $C _ { 1 }$ meets the $x$-axis. The curve $C _ { 2 }$ has equation $$y = ( x - k ) ^ { 2 } ( x - k + 2 )$$ where $k$ is a constant and $k > 2$

Sketch $C _ { 2 }$, showing the coordinates of the points where $C _ { 2 }$ meets the $x$ and $y$ axes.

A company offers two salary schemes for a 10 -year period, Year 1 to Year 10 inclusive.

Scheme 1: Salary in Year 1 is $\pounds P$.
Salary increases by $\pounds ( 2 T )$ each year, forming an arithmetic sequence. Scheme 2: Salary in Year 1 is $\pounds ( P + 1800 )$.
Salary increases by $\pounds T$ each year, forming an arithmetic sequence.

Show that the total earned under Salary Scheme 1 for the 10-year period is $$\pounds ( 10 P + 90 T )$$ For the 10-year period, the total earned is the same for both salary schemes.

Find the value of $T$. For this value of $T$, the salary in Year 10 under Salary Scheme 2 is $\pounds 29850$

Find the value of $P$.
10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-214_780_949_278_406} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve $C$ with equation $$y = 2 - \frac { 1 } { x } , \quad x \neq 0$$ The curve crosses the $x$-axis at the point $A$.

Find the coordinates of $A$.

Show that the equation of the normal to $C$ at $A$ can be written as $$2 x + 8 y - 1 = 0$$ The normal to $C$ at $A$ meets $C$ again at the point $B$, as shown in Figure 2 .

Find the coordinates of $B$. Turn over

Find
giving each term in its simplest form.
Find

$$\int \left( 6 x ^ { 2 } + \frac { 2 } { x ^ { 2 } } + 5 \right) \mathrm { d } x$$ giving each term in its simplest form.
2. (a) Evaluate $( 32 ) ^ { \frac { 3 } { 5 } }$, giving your answer as an integer.

Simplify fully $\left( \frac { 25 x ^ { 4 } } { 4 } \right) ^ { - \frac { 1 } { 2 } }$
\includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-218_104_97_2613_1784}
3. Show that $\frac { 2 } { \sqrt { } ( 12 ) - \sqrt { } ( 8 ) }$ can be written in the form $\sqrt { } a + \sqrt { } b$, where $a$ and $b$ are integers.
4. $$y = 5 x ^ { 3 } - 6 x ^ { \frac { 4 } { 3 } } + 2 x - 3$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ giving each term in its simplest form.

Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$
5. A sequence of numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots$ is defined by $$\begin{aligned} & a _ { 1 } = 3
& a _ { n + 1 } = 2 a _ { n } - c \quad ( n \geqslant 1 ) \end{aligned}$$ where $c$ is a constant.

Write down an expression, in terms of $c$, for $a _ { 2 }$

Show that $a _ { 3 } = 12 - 3 c$ Given that $\sum _ { i = 1 } ^ { 4 } a _ { i } \geqslant 23$

find the range of values of $c$.
6. A boy saves some money over a period of 60 weeks. He saves 10 p in week 1, 15 p in week 2, 20p in week 3 and so on until week 60 . His weekly savings form an arithmetic sequence.

Find how much he saves in week 15

Calculate the total amount he saves over the 60 week period. The boy's sister also saves some money each week over a period of $m$ weeks. She saves 10 p in week $1,20 \mathrm { p }$ in week $2,30 \mathrm { p }$ in week 3 and so on so that her weekly savings form an arithmetic sequence. She saves a total of $\pounds 63$ in the $m$ weeks.

Show that $$m ( m + 1 ) = 35 \times 36$$

Hence write down the value of $m$.

The point $P ( 4 , - 1 )$ lies on the curve $C$ with equation $y = \mathrm { f } ( x ) , x > 0$, and

$$\mathrm { f } ^ { \prime } ( x ) = \frac { 1 } { 2 } x - \frac { 6 } { \sqrt { } x } + 3$$

Find the equation of the tangent to $C$ at the point $P$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are integers.

Find f(x).
8. $$4 x - 5 - x ^ { 2 } = q - ( x + p ) ^ { 2 }$$ where $p$ and $q$ are integers.

Find the value of $p$ and the value of $q$.

Calculate the discriminant of $4 x - 5 - x ^ { 2 }$

On the axes on page 17, sketch the curve with equation $y = 4 x - 5 - x ^ { 2 }$ showing clearly the coordinates of any points where the curve crosses the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-226_1145_1143_258_388}
9. The line $L _ { 1 }$ has equation $4 y + 3 = 2 x$ The point $A ( p , 4 )$ lies on $L _ { 1 }$

Find the value of the constant $p$. The line $L _ { 2 }$ passes through the point $C ( 2,4 )$ and is perpendicular to $L _ { 1 }$

Find an equation for $L _ { 2 }$ giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. The line $L _ { 1 }$ and the line $L _ { 2 }$ intersect at the point $D$.

Find the coordinates of the point $D$.

Show that the length of $C D$ is $\frac { 3 } { 2 } \sqrt { } 5$ A point $B$ lies on $L _ { 1 }$ and the length of $A B = \sqrt { } ( 80 )$
The point $E$ lies on $L _ { 2 }$ such that the length of the line $C D E = 3$ times the length of $C D$.

Find the area of the quadrilateral $A C B E$.
10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-229_527_844_248_548} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ with equation $y = \mathrm { f } ( x )$ where $$f ( x ) = x ^ { 2 } ( 9 - 2 x )$$ There is a minimum at the origin, a maximum at the point $( 3,27 )$ and $C$ cuts the $x$-axis at the point $A$.

Write down the coordinates of the point $A$.

On separate diagrams sketch the curve with equation

$y = \mathrm { f } ( x + 3 )$
$y = \mathrm { f } ( 3 x )$ On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. The curve with equation $y = \mathrm { f } ( x ) + k$, where $k$ is a constant, has a maximum point at $( 3,10 )$.

Write down the value of $k$. Turn over

Factorise completely $x - 4 x ^ { 3 }$
Express $8 ^ { 2 x + 3 }$ in the form $2 ^ { y }$, stating $y$ in terms of $x$.
1. Express

$$( 5 - \sqrt { } 8 ) ( 1 + \sqrt { } 2 )$$ in the form $a + b \sqrt { } 2$, where $a$ and $b$ are integers.
(ii) Express $$\sqrt { } 80 + \frac { 30 } { \sqrt { } 5 }$$ in the form $c \sqrt { } 5$, where $c$ is an integer.
4. A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ satisfies $$u _ { n + 1 } = 2 u _ { n } - 1 , n \geqslant 1$$ Given that $u _ { 2 } = 9$,

find the value of $u _ { 3 }$ and the value of $u _ { 4 }$,

evaluate $\sum _ { r = 1 } ^ { 4 } u _ { r }$.
5. The line $l _ { 1 }$ has equation $y = - 2 x + 3$ The line $l _ { 2 }$ is perpendicular to $l _ { 1 }$ and passes through the point $( 5,6 )$.

Find an equation for $l _ { 2 }$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. The line $l _ { 2 }$ crosses the $x$-axis at the point $A$ and the $y$-axis at the point $B$.

Find the $x$-coordinate of $A$ and the $y$-coordinate of $B$. Given that $O$ is the origin,

find the area of the triangle $O A B$.
6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-238_919_1136_210_395} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \frac { 2 } { x } , x \neq 0$ The curve $C$ has equation $y = \frac { 2 } { x } - 5 , x \neq 0$, and the line $l$ has equation $y = 4 x + 2$

Sketch and clearly label the graphs of $C$ and $l$ on a single diagram. On your diagram, show clearly the coordinates of the points where $C$ and $l$ cross the coordinate axes.

Write down the equations of the asymptotes of the curve $C$.

Find the coordinates of the points of intersection of $y = \frac { 2 } { x } - 5$ and $y = 4 x + 2$

Lewis played a game of space invaders. He scored points for each spaceship that he captured.

Lewis scored 140 points for capturing his first spaceship.
He scored 160 points for capturing his second spaceship, 180 points for capturing his third spaceship, and so on. The number of points scored for capturing each successive spaceship formed an arithmetic sequence.

Find the number of points that Lewis scored for capturing his 20th spaceship.

Find the total number of points Lewis scored for capturing his first 20 spaceships. Sian played an adventure game. She scored points for each dragon that she captured. The number of points that Sian scored for capturing each successive dragon formed an arithmetic sequence. Sian captured $n$ dragons and the total number of points that she scored for capturing all $n$ dragons was 8500 . Given that Sian scored 300 points for capturing her first dragon and then 700 points for capturing her $n$th dragon,

find the value of $n$.
8. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 3 } + \frac { 4 x - 5 } { 2 x ^ { 3 } } , \quad x \neq 0$$ Given that $y = 7$ at $x = 1$, find $y$ in terms of $x$, giving each term in its simplest form.
9. The equation $$( k + 3 ) x ^ { 2 } + 6 x + k = 5 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for $x$.

Show that $k$ satisfies $$k ^ { 2 } - 2 k - 24 < 0$$

Hence find the set of possible values of $k$.
10. $$4 x ^ { 2 } + 8 x + 3 \equiv a ( x + b ) ^ { 2 } + c$$

Find the values of the constants $a , b$ and $c$.

On the axes on page 27, sketch the curve with equation $y = 4 x ^ { 2 } + 8 x + 3$, showing clearly the coordinates of any points where the curve crosses the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-245_1285_1284_317_322}
11. The curve $C$ has equation $$y = 2 x - 8 \sqrt { } x + 5 , \quad x \geqslant 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving each term in its simplest form. The point $P$ on $C$ has $x$-coordinate equal to $\frac { 1 } { 4 }$

Find the equation of the tangent to $C$ at the point $P$, giving your answer in the form $y = a x + b$, where $a$ and $b$ are constants. The tangent to $C$ at the point $Q$ is parallel to the line with equation $2 x - 3 y + 18 = 0$

Find the coordinates of $Q$.

Paper Reference(s) \section*{6663/01R} \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Examiner's use only} \includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-248_95_307_502_1640}

\end{figure} Advanced Subsidiary
\includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-248_246_168_593_1443} Monday 13 May 2013 - Afternoon Time: 1 hour 30 minutes Materials required for examination
Mathematical Formulae (Pink) Items included with question papers
Nil Calculators may NOT be used in this examination. In the boxes above, write your centre number, candidate number, your surname, initials and signature.
Check that you have the correct question paper.
Answer ALL the questions.
You must write your answer for each question in the space following the question. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2). There are 11 questions in this question paper. The total mark for this paper is 75.
There are 32 pages in this question paper. Any blank pages are indicated. You must ensure that your answers to parts of questions are clearly labelled.
You should show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit.