Questions - Edexcel - A-Level Maths

Edexcel C1 Q15

15. (a) By completing the square, find in terms of $k$ the roots of the equation $$x ^ { 2 } + 2 k x - 7 = 0$$ (b) Prove that, for all values of $k$, the roots of $x ^ { 2 } + 2 k x - 7 = 0$ are real and different.
(c) Given that $k = \sqrt { } 2$, find the exact roots of the equation.
[0pt] [P1 June 2002 Question 4]

Edexcel C1 Q16

16. \section*{Figure 3}

\includegraphics[max width=\textwidth, alt={}]{b85f4635-aa93-4c6a-9d1f-2ef5bac1b48c-08_581_575_395_609}

The points $A ( - 3 , - 2 )$ and $B ( 8,4 )$ are at the ends of a diameter of the circle shown in Fig. 3.

Find the coordinates of the centre of the circle.
Find an equation of the diameter $A B$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
Find an equation of tangent to the circle at $B$. The line $l$ passes through $A$ and the origin.
Find the coordinates of the point at which $l$ intersects the tangent to the circle at $B$, giving your answer as exact fractions.

Edexcel C1 Q17

17. (a) Solve the inequality $$3 x - 8 > x + 13$$ (b) Solve the inequality $$x ^ { 2 } - 5 x - 14 > 0$$

Edexcel C1 Q18

(a) An arithmetic series has first term $a$ and common difference $d$. Prove that the sum of the first $n$ terms of the series is

$$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ A company made a profit of $\pounds 54000$ in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference $\pounds d$. This model predicts total profits of $\pounds 619200$ for the 9 years 2001 to 2009 inclusive.
(b) Find the value of $d$. Using your value of $d$,
(c) find the predicted profit for the year 2011.

Edexcel C1 Q19

19. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$

Write down the maximum value of $\mathrm { f } ( x )$.
Sketch the graph of $y = \mathrm { f } ( x )$, showing the coordinates of the points at which the graph meets the coordinate axes. The points $A$ and $B$ on the graph of $y = \mathrm { f } ( x )$ have coordinates $( - 2 , - 7 )$ and $( 3,8 )$ respectively.
Find, in the form $y = m x + c$, an equation of the straight line through $A$ and $B$.
Find the coordinates of the point at which the line $A B$ crosses the $x$-axis. The mid-point of $A B$ lies on the line with equation $y = k x$, where $k$ is a constant.
Find the value of $k$.

Edexcel C1 Q20

20. The curve $C$ has equation $y = \mathrm { f } ( x )$. Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 29$$ and that $C$ passes through the point $P ( 2,6 )$,

find $y$ in terms of $x$.
Verify that $C$ passes through the point ( 4,0 ).
Find an equation of the tangent to $C$ at $P$. The tangent to $C$ at the point $Q$ is parallel to the tangent at $P$.
Calculate the exact $x$-coordinate of $Q$.
21. $$y = 7 + 10 x ^ { \frac { 3 } { 2 } }$$
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Find $\int y \mathrm {~d} x$.
22. (a) Given that $3 ^ { x } = 9 ^ { y - 1 }$, show that $x = 2 y - 2$.
Solve the simultaneous equations $$\begin{gathered} x = 2 y - 2
x ^ { 2 } = y ^ { 2 } + 7 \end{gathered}$$
1. The straight line $l _ { 1 }$ with equation $y = \frac { 3 } { 2 } x - 2$ crosses the $y$-axis at the point $P$. The point $Q$ has coordinates $( 5 , - 3 )$.
The straight line $l _ { 2 }$ is perpendicular to $l _ { 1 }$ and passes through $Q$.
Calculate the coordinates of the mid-point of $P Q$.
Find an equation for $l _ { 2 }$ in the form $a x + b y = c$, where $a$, b and $c$ are integer constants. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $R$.
Calculate the exact coordinates of $R$.
24. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$
Use integration to find $y$ in terms of $x$.
Given that $y = 7$ when $x = 1$, find the value of $y$ at $x = 2$.
25. Find the set of values for $x$ for which
$6 x - 7 < 2 x + 3$,
$2 x ^ { 2 } - 11 x + 5 < 0$,
both $6 x - 7 < 2 x + 3$ and $2 x ^ { 2 } - 11 x + 5 < 0$.
[0pt] [P1 June 2003 Question 2]
26. In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by $x$ phones per month for the next 35 months, so that $( 280 + x )$ phones will be sold in the second month, $( 280 + 2 x )$ in the third month, and so on. Using this model with $x = 5$, calculate
1. the number of phones sold in the 36th month,
2. the total number of phones sold over the 36 months. The shop sets a sales target of 17000 phones to be sold over the 36 months.
  Using the same model,
find the least value of $x$ required to achieve this target.
[0pt] [P1 June 2003 Question 3]
27. The points $A$ and $B$ have coordinates $( 4,6 )$ and $( 12,2 )$ respectively. The straight line $l _ { 1 }$ passes through $A$ and $B$.
Find an equation for $l _ { 1 }$ in the form $a x + b y = c$, where $a$, b and $c$ are integers. The straight line $l _ { 2 }$ passes through the origin and has gradient - 4 .
Write down an equation for $l _ { 2 }$. The lines $l _ { 1 }$ and $l _ { 2 }$ intercept at the point $C$.
Find the exact coordinates of the mid-point of $A C$.
28. For the curve $C$ with equation $y = x ^ { 4 } - 8 x ^ { 2 } + 3$,
find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, The point $A$, on the curve $C$, has $x$-coordinate 1 .
Find an equation for the normal to $C$ at $A$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
[0pt] [P1 June 2003 Question 8*]
29. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r )$$
Write down the first two terms of the series.
Find the common difference of the series. Given that $n = 50$,
find the sum of the series.
30. (a) Solve the equation $4 x ^ { 2 } + 12 x = 0$. $$f ( x ) = 4 x ^ { 2 } + 12 x + c$$ where $c$ is a constant.
Given that $\mathrm { f } ( x ) = 0$ has equal roots, find the value of $c$ and hence solve $\mathrm { f } ( x ) = 0$.
31. Solve the simultaneous equations $$\begin{aligned} & x - 3 y + 1 = 0
& x ^ { 2 } - 3 x y + y ^ { 2 } = 11 \end{aligned}$$
1. A container made from thin metal is in the shape of a right circular cylinder with height $h \mathrm {~cm}$ and base radius $r \mathrm {~cm}$. The container has no lid. When full of water, the container holds $500 \mathrm {~cm} ^ { 3 }$ of water.
Show that the exterior surface area, $A \mathrm {~cm} ^ { 2 }$, of the container is given by $$A = \pi r ^ { 2 } + \frac { 1000 } { r } .$$ 33. \section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{b85f4635-aa93-4c6a-9d1f-2ef5bac1b48c-15_668_748_358_699}
The points $A$ and $B$ have coordinates $( 2 , - 3 )$ and $( 8,5 )$ respectively, and $A B$ is a chord of a circle with centre $C$, as shown in Fig. 1.
Find the gradient of $A B$. The point $M$ is the mid-point of $A B$.
Find an equation for the line through $C$ and $M$. Given that the $x$-coordinate of $C$ is 4 ,
find the $y$-coordinate of $C$,
show that the radius of the circle is $\frac { 5 \sqrt { } 17 } { 4 }$.
34. The first three terms of an arithmetic series are $p , 5 p - 8$, and $3 p + 8$ respectively.
Show that $p = 4$.
Find the value of the 40th term of this series.
Prove that the sum of the first $n$ terms of the series is a perfect square.
35. $$\mathrm { f } ( x ) = x ^ { 2 } - k x + 9 , \text { where } k \text { is a constant. }$$
Find the set of values of $k$ for which the equation $\mathrm { f } ( x ) = 0$ has no real solutions. Given that $k = 4$,
express $\mathrm { f } ( x )$ in the form $( x - p ) ^ { 2 } + q$, where $p$ and $q$ are constants to be found,
36. The curve $C$ with equation $y = \mathrm { f } ( x )$ is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0 .$$
Show that, when $x = 8$, the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ is $9 \sqrt { } 2$. The curve $C$ passes through the point $( 4,30 )$.
Using integration, find $\mathrm { f } ( x )$.
37. \section*{Figure 2}
\includegraphics[max width=\textwidth, alt={}]{b85f4635-aa93-4c6a-9d1f-2ef5bac1b48c-17_687_1074_351_539}
Figure 2 shows the curve with equation $y ^ { 2 } = 4 ( x - 2 )$ and the line with equation $2 x - 3 y = 12$.
The curve crosses the $x$-axis at the point $A$, and the line intersects the curve at the points $P$ and $Q$.
Write down the coordinates of $A$.
Find, using algebra, the coordinates of $P$ and $Q$.
Show that $\angle P A Q$ is a right angle.
38. A sequence is defined by the recurrence relation $$u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , \quad n = 1,2,3 , \ldots ,$$ where $a$ is a constant.
Given that $a = 20$ and $u _ { 1 } = 3$, find the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$, giving your answers to 2 decimal places.
Given instead that $u _ { 1 } = u _ { 2 } = 3$,
1. calculate the value of $a$,
2. write down the value of $u _ { 5 }$.
  [0pt] [P2 January 2004 Question 2]
  39. The points $A$ and $B$ have coordinates $( 1,2 )$ and $( 5,8 )$ respectively.
Find the coordinates of the mid-point of $A B$.
Find, in the form $y = m x + c$, an equation for the straight line through $A$ and $B$.
40. Giving your answers in the form $a + b \sqrt { 2 }$, where $a$ and $b$ are rational numbers, find
$( 3 - \sqrt { } 8 ) ^ { 2 }$,
$\frac { 1 } { 4 - \sqrt { 8 } }$.
41. The width of a rectangular sports pitch is $x$ metres, $x > 0$. The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m ,
form a linear inequality in $x$. Given that the area of the pitch must be greater than $4800 \mathrm {~m} ^ { 2 }$,
form a quadratic inequality in $x$.
by solving your inequalities, find the set of possible values of $x$.
42. The curve $C$ has equation $y = x ^ { 2 } - 4$ and the straight line $l$ has equation $y + 3 x = 0$.
In the space below, sketch $C$ and $l$ on the same axes.
Write down the coordinates of the points at which $C$ meets the coordinate axes.
Using algebra, find the coordinates of the points at which $l$ intersects $C$.
43. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$
Show that $\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }$.
Hence, or otherwise, differentiate $\mathrm { f } ( x )$ with respect to $x$.

Edexcel C2 Q1

Find the first three terms, in ascending powers of $x$, of the binomial expansion of $( 3 + 2 x ) ^ { 5 }$, giving each term in its simplest form.
(4)
The points $A$ and $B$ have coordinates $( 5 , - 1 )$ and $( 13,11 )$ respectively.
1. Find the coordinates of the mid-point of $A B$.
Given that $A B$ is a diameter of the circle $C$,
find an equation for $C$.
3. Find, giving your answer to 3 significant figures where appropriate, the value of $x$ for which
$3 ^ { x } = 5$,
$\log _ { 2 } ( 2 x + 1 ) - \log _ { 2 } x = 2$.
4. (a) Show that the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ can be written as $$5 \sin ^ { 2 } x + 3 \sin x - 2 = 0 .$$
Hence solve, for $0 \leq x < 360 ^ { \circ }$, the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x ) ,$$ giving your answers to 1 decimal place where appropriate.
5. $\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + a x + b$, where $a$ and $b$ are constants. When $\mathrm { f } ( x )$ is divided by $( x - 2 )$, the remainder is 1 .
When $\mathrm { f } ( x )$ is divided by $( x + 1 )$, the remainder is 28 .
Find the value of $a$ and the value of $b$.
Show that ( $x - 3$ ) is a factor of $\mathrm { f } ( x )$.
6. The second and fourth terms of a geometric series are 7.2 and 5.832 respectively. The common ratio of the series is positive.
For this series, find
the common ratio,
the first term,
the sum of the first 50 terms, giving your answer to 3 decimal places,
the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places.
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-02_446_688_251_1745}
\end{figure} Figure 1 shows the triangle $A B C$, with $A B = 8 \mathrm {~cm} , A C = 11 \mathrm {~cm}$ and $\angle B A C = 0.7$ radians. The $\operatorname { arc } B D$, where $D$ lies on $A C$, is an arc of a circle with centre $A$ and radius 8 cm . The region $R$, shown shaded in Figure 1, is bounded by the straight lines $B C$ and $C D$ and the $\operatorname { arc } B D$. Find
the length of the arc $B D$,
the perimeter of $R$, giving your answer to 3 significant figures,
the area of $R$, giving your answer to 3 significant figures.
8. Figure 2
\includegraphics[max width=\textwidth, alt={}, center]{de11ae90-8693-4346-b195-1d01f2b164f5-03_791_972_276_303} The line with equation $y = 3 x + 20$ cuts the curve with equation $y = x ^ { 2 } + 6 x + 10$ at the points $A$ and $B$, as shown in Figure 2.
Use algebra to find the coordinates of $A$ and the coordinates of $B$. The shaded region $S$ is bounded by the line and the curve, as shown in Figure 2.
Use calculus to find the exact area of $S$.
9. Figure 3
\includegraphics[max width=\textwidth, alt={}, center]{de11ae90-8693-4346-b195-1d01f2b164f5-03_670_782_258_1820} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the rectangular part is $2 x$ metres and the width is $y$ metres. The diameter of the semicircular part is $2 x$ metres. The perimeter of the stage is 80 m .
Show that the area, $A \mathrm {~m} ^ { 2 }$, of the stage is given by $$A = 80 x - \left( 2 + \frac { \pi } { 2 } \right) x ^ { 2 }$$
Use calculus to find the value of $x$ at which $A$ has a stationary value.
Prove that the value of $x$ you found in part (b) gives the maximum value of $A$.
Calculate, to the nearest $\mathrm { m } ^ { 2 }$, the maximum area of the stage. \section*{6664/01} \section*{Monday 20 June 2005 - Morning} Mathematical Formulae (Green) Items included with question papers Nil Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C2), the paper reference (6664), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
There are 10 questions in this question paper. The total mark for this paper is 75. You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. Find the coordinates of the stationary point on the curve with equation $y = 2 x ^ { 2 } - 12 x$.
\section*{2. Solve}
$5 ^ { x } = 8$, giving your answer to 3 significant figures,
$\log _ { 2 } ( x + 1 ) - \log _ { 2 } x = \log _ { 2 } 7$.
3. (a) Use the factor theorem to show that $( x + 4 )$ is a factor of $2 x ^ { 3 } + x ^ { 2 } - 25 x + 12$.
Factorise $2 x ^ { 3 } + x ^ { 2 } - 25 x + 12$ completely.
4. (a) Write down the first three terms, in ascending powers of $x$, of the binomial expansion of $( 1 + p x ) ^ { 12 }$, where $p$ is a non-zero constant. Given that, in the expansion of $( 1 + p x ) ^ { 12 }$, the coefficient of $x$ is $( - q )$ and the coefficient of $x ^ { 2 }$ is $11 q$,
find the value of $p$ and the value of $q$.
5. Solve, for $0 \leq x \leq 180 ^ { \circ }$, the equation
$\sin \left( x + 10 ^ { \circ } \right) = \frac { \sqrt { } 3 } { 2 }$,
$\cos 2 x = - 0.9$, giving your answers to 1 decimal place.
6. A river, running between parallel banks, is 20 m wide. The depth, $y$ metres, of the river measured at a point $x$ metres from one bank is given by the formula $$y = \frac { 1 } { 10 } x \sqrt { } ( 20 - x ) , \quad 0 \leq x \leq 20 .$$
Complete the table below, giving values of $y$ to 3 decimal places.
$x$ 0 4 8 12 16 20
$y$ 0 2.771 0
Use the trapezium rule with all the values in the table to estimate the cross-sectional area of the river. Given that the cross-sectional area is constant and that the river is flowing uniformly at $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$,
estimate, in $\mathrm { m } ^ { 3 }$, the volume of water flowing per minute, giving your answer to 3 significant figures.
7. In the triangle $A B C , A B = 8 \mathrm {~cm} , A C = 7 \mathrm {~cm} , \angle A B C = 0.5$ radians and $\angle A C B = x$ radians.
Use the sine rule to find the value of $\sin x$, giving your answer to 3 decimal places. Given that there are two possible values of $x$,
find these values of $x$, giving your answers to 2 decimal places.
8. The circle $C$, with centre at the point $A$, has equation $x ^ { 2 } + y ^ { 2 } - 10 x + 9 = 0$. Find
the coordinates of $A$,
the radius of $C$,
the coordinates of the points at which $C$ crosses the $x$-axis. Given that the line $l$ with gradient $\frac { 7 } { 2 }$ is a tangent to $C$, and that $l$ touches $C$ at the point $T$,
find an equation of the line which passes through $A$ and $T$.
9. (a) A geometric series has first term $a$ and common ratio $r$. Prove that the sum of the first $n$ terms of the series is $$\frac { a \left( 1 - r ^ { n } \right) } { 1 - r } .$$ Mr King will be paid a salary of $\pounds 35000$ in the year 2005. Mr King's contract promises a $4 \%$ increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence.
Find, to the nearest $\pounds 100$, Mr King's salary in the year 2008. Mr King will receive a salary each year from 2005 until he retires at the end of 2024.
Find, to the nearest $\pounds 1000$, the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-06_665_899_338_287}
\end{figure} Figure 1 shows part of a curve $C$ with equation $y = 2 x + \frac { 8 } { x ^ { 2 } } - 5 , x > 0$.
The points $P$ and $Q$ lie on $C$ and have $x$-coordinates 1 and 4 respectively. The region $R$, shaded in Figure 1, is bounded by $C$ and the straight line joining $P$ and $Q$.
Find the exact area of $R$.
Use calculus to show that $y$ is increasing for $x > 2$. Materials required for examination
Mathematical Formulae (Green) Items included with question papers Nil \section*{Tuesday 10 January 2006 - Afternoon} Time: 1 hour 30 minutes Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C2), the paper reference (6664), your surname, other name and signature. When a calculator is used, the answer should be given to an appropriate degree of accuracy. 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 9 questions on this paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1.
$\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 5 x + c$, where $c$ is a constant.
Given that $\mathrm { f } ( 1 ) = 0$,
find the value of $c$,
factorise $\mathrm { f } ( x )$ completely,
find the remainder when $\mathrm { f } ( x )$ is divided by $( 2 x - 3 )$.

Edexcel C2 Q4

4. (a) Show that the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ can be written as $$5 \sin ^ { 2 } x + 3 \sin x - 2 = 0 .$$ (b) Hence solve, for $0 \leq x < 360 ^ { \circ }$, the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x ) ,$$ giving your answers to 1 decimal place where appropriate.

Edexcel C2 Q5

5. $\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + a x + b$, where $a$ and $b$ are constants. When $\mathrm { f } ( x )$ is divided by $( x - 2 )$, the remainder is 1 .
When $\mathrm { f } ( x )$ is divided by $( x + 1 )$, the remainder is 28 .

Find the value of $a$ and the value of $b$.
Show that ( $x - 3$ ) is a factor of $\mathrm { f } ( x )$.

Edexcel C2 Q6

6. The second and fourth terms of a geometric series are 7.2 and 5.832 respectively. The common ratio of the series is positive.
For this series, find

the common ratio,
the first term,
the sum of the first 50 terms, giving your answer to 3 decimal places,
the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places.

Edexcel C2 Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-02_446_688_251_1745}

\end{figure} Figure 1 shows the triangle $A B C$, with $A B = 8 \mathrm {~cm} , A C = 11 \mathrm {~cm}$ and $\angle B A C = 0.7$ radians. The $\operatorname { arc } B D$, where $D$ lies on $A C$, is an arc of a circle with centre $A$ and radius 8 cm . The region $R$, shown shaded in Figure 1, is bounded by the straight lines $B C$ and $C D$ and the $\operatorname { arc } B D$. Find

the length of the arc $B D$,
the perimeter of $R$, giving your answer to 3 significant figures,
the area of $R$, giving your answer to 3 significant figures.

Edexcel C2 Q8

8. Figure 2
\includegraphics[max width=\textwidth, alt={}, center]{de11ae90-8693-4346-b195-1d01f2b164f5-03_791_972_276_303} The line with equation $y = 3 x + 20$ cuts the curve with equation $y = x ^ { 2 } + 6 x + 10$ at the points $A$ and $B$, as shown in Figure 2.

Use algebra to find the coordinates of $A$ and the coordinates of $B$. The shaded region $S$ is bounded by the line and the curve, as shown in Figure 2.
Use calculus to find the exact area of $S$.

Edexcel C2 Q9

9. Figure 3
\includegraphics[max width=\textwidth, alt={}, center]{de11ae90-8693-4346-b195-1d01f2b164f5-03_670_782_258_1820} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the rectangular part is $2 x$ metres and the width is $y$ metres. The diameter of the semicircular part is $2 x$ metres. The perimeter of the stage is 80 m .

Show that the area, $A \mathrm {~m} ^ { 2 }$, of the stage is given by $$A = 80 x - \left( 2 + \frac { \pi } { 2 } \right) x ^ { 2 }$$
Use calculus to find the value of $x$ at which $A$ has a stationary value.
Prove that the value of $x$ you found in part (b) gives the maximum value of $A$.
Calculate, to the nearest $\mathrm { m } ^ { 2 }$, the maximum area of the stage. \section*{6664/01} \section*{Monday 20 June 2005 - Morning} Mathematical Formulae (Green) Items included with question papers Nil Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C2), the paper reference (6664), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
There are 10 questions in this question paper. The total mark for this paper is 75. You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. Find the coordinates of the stationary point on the curve with equation $y = 2 x ^ { 2 } - 12 x$.
\section*{2. Solve}
$5 ^ { x } = 8$, giving your answer to 3 significant figures,
$\log _ { 2 } ( x + 1 ) - \log _ { 2 } x = \log _ { 2 } 7$.
3. (a) Use the factor theorem to show that $( x + 4 )$ is a factor of $2 x ^ { 3 } + x ^ { 2 } - 25 x + 12$.
Factorise $2 x ^ { 3 } + x ^ { 2 } - 25 x + 12$ completely.
4. (a) Write down the first three terms, in ascending powers of $x$, of the binomial expansion of $( 1 + p x ) ^ { 12 }$, where $p$ is a non-zero constant. Given that, in the expansion of $( 1 + p x ) ^ { 12 }$, the coefficient of $x$ is $( - q )$ and the coefficient of $x ^ { 2 }$ is $11 q$,
find the value of $p$ and the value of $q$.
5. Solve, for $0 \leq x \leq 180 ^ { \circ }$, the equation
$\sin \left( x + 10 ^ { \circ } \right) = \frac { \sqrt { } 3 } { 2 }$,
$\cos 2 x = - 0.9$, giving your answers to 1 decimal place.
6. A river, running between parallel banks, is 20 m wide. The depth, $y$ metres, of the river measured at a point $x$ metres from one bank is given by the formula $$y = \frac { 1 } { 10 } x \sqrt { } ( 20 - x ) , \quad 0 \leq x \leq 20 .$$
Complete the table below, giving values of $y$ to 3 decimal places.
Complete the table, giving the values of $v$ to 2 decimal places. The distance, $s$ metres, travelled by the train in 30 seconds is given by $$s = \int _ { 0 } ^ { 30 } \sqrt { } \left( 1.2 ^ { t } - 1 \right) \mathrm { d } t$$
Use the trapezium rule, with all the values from your table, to estimate the value of $s$.
7. The curve $C$ has equation $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 2 .$$
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Using the result from part (a), find the coordinates of the turning points of $C$.
Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
Hence, or otherwise, determine the nature of the turning points of $C$.
8. (a) Find all the values of $\theta$, to 1 decimal place, in the interval $0 ^ { \circ } \leq \theta < 360 ^ { \circ }$ for which $$5 \sin \left( \theta + 30 ^ { \circ } \right) = 3 .$$
Find all the values of $\theta$, to 1 decimal place, in the interval $0 ^ { \circ } \leq \theta < 360 ^ { \circ }$ for which $$\tan ^ { 2 } \theta = 4 .$$ 9. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-09_410_835_223_310}
\end{figure} Figure 3 shows the shaded region $R$ which is bounded by the curve $y = - 2 x ^ { 2 } + 4 x$ and the line $y = \frac { 3 } { 2 }$. The points $A$ and $B$ are the points of intersection of the line and the curve. Find
the $x$-coordinates of the points $A$ and $B$,
the exact area of $R$. Materials required for examination
Mathematical Formulae (Green)
Items included with question papers
Nil Reference(s)
6664/01 Core Mathematics C2
Advanced Subsidiary
Monday 22 May 2006 - Morning
Time: 1 hour 30 minutes Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C2), the paper reference (6664), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
There are 10 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of $( 2 + x ) ^ { 6 }$, giving each term in its simplest form.
2. Use calculus to find the exact value of $\int _ { 1 } ^ { 2 } \left( 3 x ^ { 2 } + 5 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x$.
3. 1. Write down the value of $\log _ { 6 } 36$.
  2. Express $2 \log _ { a } 3 + \log _ { a } 11$ as a single logarithm to base $a$.
$$\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 29 x - 60$$
Find the remainder when $\mathrm { f } ( x )$ is divided by ( $x + 2$ ).
Use the factor theorem to show that $( x + 3 )$ is a factor of $\mathrm { f } ( x )$.
Factorise $\mathrm { f } ( x )$ completely.
5. (a) Sketch the graph of $y = 3 ^ { x } , x \in \mathbb { R }$, showing the coordinates of the point at which the graph meets the $y$-axis.
Copy and complete the table, giving the values of $3 ^ { x }$ to 3 decimal places.
$x$ 0 0.2 0.4 0.6 0.8 1
$3 ^ { x }$ 1.246 1.552 3
Use the trapezium rule, with all the values from your tables, to find an approximation for the value of $\int _ { 0 } ^ { 1 } 3 ^ { x } \mathrm {~d} x$.
6. (a) Given that $\sin \theta = 5 \cos \theta$, find the value of $\tan \theta$.
Hence, or otherwise, find the values of $\theta$ in the interval $0 \leq \theta < 360 ^ { \circ }$ for which $\sin \theta = 5 \cos \theta$, giving your answers to 1 decimal place.
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-10_486_615_625_1866}
\end{figure} The line $y = 3 x - 4$ is a tangent to the circle $C$, touching $C$ at the point $\mathrm { P } ( 2,2 )$, as shown in Figure 1. The point $Q$ is the centre of $C$.
Find an equation of the straight line through $P$ and $Q$. Given that $Q$ lies on the line $y = 1$,
show that the $x$-coordinate of $Q$ is 5 ,
find an equation for $C$.
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-11_440_474_294_548}
\end{figure} Figure 2 shows the cross-section $A B C D$ of a small shed.
The straight line $A B$ is vertical and has length 2.12 m .
The straight line $A D$ is horizontal and has length 1.86 m .
The curve $B C$ is an arc of a circle with centre $A$, and $C D$ is a straight line.
Given that the size of $\angle B A C$ is 0.65 radians, find
the length of the arc $B C$, in m , to 2 decimal places,
the area of the sector $B A C$, in $\mathrm { m } ^ { 2 }$, to 2 decimal places,
the size of $\angle C A D$, in radians, to 2 decimal places,
the area of the cross-section $A B C D$ of the shed, in $\mathrm { m } ^ { 2 }$, to 2 decimal places.
9. A geometric series has first term $a$ and common ratio $r$. The second term of the series is 4 and the sum to infinity of the series is 25 .
Show that $25 r ^ { 2 } - 25 r + 4 = 0$.
Find the two possible values of $r$.
Find the corresponding two possible values of $a$.
Show that the sum, $S _ { n }$, of the first $n$ terms of the series is given by $$S _ { n } = 25 \left( 1 - r ^ { n } \right)$$ Given that $r$ takes the larger of its two possible values,
find the smallest value of $n$ for which $S _ { n }$ exceeds 24 .
\includegraphics[max width=\textwidth, alt={}, center]{de11ae90-8693-4346-b195-1d01f2b164f5-12_442_899_342_310} \section*{Edexcel GCE
Core Mathematics C2 Advanced Subsidiary} Materials required for examination
Mathematical Formulae (Green) \section*{Wednesday 10 January 2007 - Afternoon Time: 1 hour 30 minutes} Items included with question papers Nil Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP $\mathbf { 4 8 G }$. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C2), the paper reference (6664), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided. Full marks may be obtained for answers to ALL questions. There are 10 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit. \footnotetext{N24322A
This publication may only be reproduced in accordance with Edexcel Limited copyright policy. ©2007 Edexcel Limited. } HP 48G. . .
新新 \section*{TOTAL FOR PAPER: 75 MARKS} \section*{END}
Hence calculate the exact area of $R$. The line through $B$ parallel to the $y$-axis meets the $x$-axis at the point $N$. The region $R$, shown shaded in Figure 3, is bounded by the curve, the $x$-axis and the line from $A$ to $N$.
Find $\int \left( x ^ { 3 } - 8 x ^ { 2 } + 20 x \right) \mathrm { d } x$.
Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at $A$, and hence verify that $A$ is a maximum. \includegraphics[max width=\textwidth, alt={}]{de11ae90-8693-4346-b195-1d01f2b164f5-12_74_49_929_1283} 2
"
1. $$\mathrm { f } ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 5 .$$ Find
$\mathrm { f } ^ { \prime \prime } ( x )$,
$\int _ { 1 } ^ { 2 } f ( x ) d x$.
2. (a) Find the first 4 terms, in ascending powers of $x$, of the binomial expansion of $( 1 - 2 x ) ^ { 5 }$. Give each term in its simplest form.
If $x$ is small, so that $x ^ { 2 }$ and higher powers can be ignored, show that $$( 1 + x ) ( 1 - 2 x ) ^ { 5 } \approx 1 - 9 x .$$
1. The line joining points $( - 1,4 )$ and $( 3,6 )$ is a diameter of the circle $C$.
Find an equation for $C$.
4. Solve the equation $5 ^ { x } = 17$, giving your answer to 3 significant figures.
5. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + x - 6$$
Use the factor theorem to show that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$.
Factorise $\mathrm { f } ( x )$ completely.
Write down all the solutions to the equation $$x ^ { 3 } + 4 x ^ { 2 } + x - 6 = 0 .$$
1. Find all the solutions, in the interval $0 \leq x < 2 \pi$, of the equation
$$2 \cos ^ { 2 } x + 1 = 5 \sin x ,$$ giving each solution in terms of $\pi$.
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-13_768_826_523_1731}
\end{figure} Figure 1 shows a sketch of part of the curve $C$ with equation $$y = x ( x - 1 ) ( x - 5 ) .$$ Use calculus to find the total area of the finite region, shown shaded in Figure 1, that is between $x = 0$ and $x = 2$ and is bounded by $C$, the $x$-axis and the line $x = 2$.
(9)
8. A diesel lorry is driven from Birmingham to Bury at a steady speed of $v$ kilometres per hour. The total cost of the journey, $\pounds C$, is given by $$C = \frac { 1400 } { v } + \frac { 2 v } { 7 } .$$
Find the value of $v$ for which $C$ is a minimum.
Find $\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }$ and hence verify that $C$ is a minimum for this value of $v$.
Calculate the minimum total cost of the journey.
9. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-14_433_725_769_397}
\end{figure} Figure 2 shows a plan of a patio. The patio $P Q R S$ is in the shape of a sector of a circle with centre $Q$ and radius 6 m . Given that the length of the straight line $P R$ is $6 \sqrt { } 3 \mathrm {~m}$,
find the exact size of angle $P Q R$ in radians.
Show that the area of the patio $P Q R S$ is $12 \pi \mathrm {~m} ^ { 2 }$.
Find the exact area of the triangle $P Q R$.
Find, in $\mathrm { m } ^ { 2 }$ to 1 decimal place, the area of the segment PRS.
Find, in m to 1 decimal place, the perimeter of the patio $P Q R S$.

Edexcel C2 Q10

10. A geometric series is $a + a r + a r ^ { 2 } + \ldots$

Prove that the sum of the first $n$ terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r } .$$
Find $$\sum _ { k = 1 } ^ { 10 } 100 \left( 2 ^ { k } \right) .$$
Find the sum to infinity of the geometric series $$\frac { 5 } { 6 } + \frac { 5 } { 18 } + \frac { 5 } { 54 } + \ldots$$
State the condition for an infinite geometric series with common ratio $r$ to be convergent. \section*{6664/01} \section*{Advanced Subsidiary Level} \section*{Monday 21 May 2007 - Morning} Mathematical Formulae (Green) Nil Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C2), the paper reference (6664), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
There are 10 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
1. Evaluate $\int _ { 1 } ^ { 8 } \frac { 1 } { \sqrt { } x } \mathrm {~d} x$, giving your answer in the form $a + b \sqrt { } 2$, where $a$ and $b$ are integers.
$$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 16 x + 12$$
Find the remainder when $\mathrm { f } ( x )$ is divided by ( $x - 2$ ). Given that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$,
factorise $\mathrm { f } ( x )$ completely.
3. (a) Find the first four terms, in ascending powers of $x$, in the bionomial expansion of $( 1 + k x ) ^ { 6 }$, where $k$ is a non-zero constant. Given that, in this expansion, the coefficients of $x$ and $x ^ { 2 }$ are equal, find
the value of $k$,
the coefficient of $x ^ { 3 }$.
4. Figure 1 Figure 1 shows the triangle $A B C$, with $A B = 6 \mathrm {~cm} , B C = 4 \mathrm {~cm}$ and $C A = 5 \mathrm {~cm}$.
Show that $\cos A = \frac { 3 } { 4 }$.
Hence, or otherwise, find the exact value of $\sin A$.
5. The curve $C$ has equation $$\left. y = x \sqrt { ( } x ^ { 3 } + 1 \right) , \quad 0 \leq x \leq 2 .$$
Copy and complete the table below, giving the values of $y$ to 3 decimal places at $x = 1$ and $x = 1.5$.
Use the trapezium rule, with all the values of $y$ from your table, to find an approximation for the value of $\int _ { 0 } ^ { 2 } \sqrt { } \left( 5 ^ { x } + 2 \right) d x$.
3. (a) Find the first 4 terms, in ascending powers of $x$, of the binomial expansion of $( 1 + a x ) ^ { 10 }$, where $a$ is a non-zero constant. Give each term in its simplest form. Given that, in this expansion, the coefficient of $x ^ { 3 }$ is double the coefficient of $x ^ { 2 }$,
find the value of $a$.
4. (a) Find, to 3 significant figures, the value of $x$ for which $5 ^ { x } = 7$.
Solve the equation $5 ^ { 2 x } - 12 \left( 5 ^ { x } \right) + 35 = 0$.
5. The circle $C$ has centre $( 3,1 )$ and passes through the point $P ( 8,3 )$.
Find an equation for $C$.
Find an equation for the tangent to $C$ at $P$, giving your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.
(5)
6. A geometric series has first term 5 and common ratio $\frac { 4 } { 5 }$. Calculate
the 20th term of the series, to 3 decimal places,
the sum to infinity of the series. Given that the sum to $k$ terms of the series is greater than 24.95,
show that $k > \frac { \log 0.002 } { \log 0.8 }$,
find the smallest possible value of $k$.
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-23_535_673_260_415} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows $A B C$, a sector of a circle with centre $A$ and radius 7 cm .
Given that the size of $\angle B A C$ is exactly 0.8 radians, find
the length of the $\operatorname { arc } B C$,
the area of the sector $A B C$. The point $D$ is the mid-point of $A C$. The region $R$, shown shaded in Figure 1, is bounded by CD, $D B$ and the arc $B C$. Find
the perimeter of $R$, giving your answer to 3 significant figures,
the area of $R$, giving your answer to 3 significant figures.
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-23_431_817_312_1763} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation $y = 10 + 8 x + x ^ { 2 } - x ^ { 3 }$.
The curve has a maximum turning point $A$.
Using calculus, show that the $x$-coordinate of $A$ is 2 .
(3) The region $R$, shown shaded in Figure 2, is bounded by the curve, the $y$-axis and the line from $O$ to $A$, where $O$ is the origin.
Using calculus, find the exact area of $R$.
(8)
9. Solve, for $0 \leq x < 360 ^ { \circ }$,
$\quad \sin \left( x - 20 ^ { \circ } \right) = \frac { 1 } { \sqrt { 2 } }$,
$\quad \cos 3 x = - \frac { 1 } { 2 }$. \section*{Friday 9 January 2009 - Morning} \section*{Materials required for examination
Mathematical Formulae (Green)} \section*{Items included with question papers
Nil} Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation or integration, or have retrievable mathematical formulae stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C2), the paper reference (6664), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
The marks for 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 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit.
1. Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of $( 3 - 2 x ) ^ { 5 }$, giving each term in its simplest form.
  (4)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-24_565_670_466_1841} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve $C$ with equation $y = ( 1 + x ) ( 4 - x )$.
The curve intersects the $x$-axis at $x = - 1$ and $x = 4$. The region $R$, shown shaded in Figure 1, is bounded by $C$ and the $x$-axis. Use calculus to find the exact area of $R$.
3. $$y = \sqrt { } \left( 10 x - x ^ { 2 } \right) .$$
Copy and complete the table below, giving the values of $y$ to 2 decimal places. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-28_433_401_525_536} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the region $R$ which is bounded by the curve with equation $y = \sqrt { } \left( 2 ^ { x } + 1 \right)$, the $x$-axis and the lines $x = 0$ and $x = 3$
Use the trapezium rule, with all the values from your table, to find an approximation for the area of $R$.
By reference to the curve in Figure 1 state, giving a reason, whether your approximation in part (b) is an overestimate or an underestimate for the area of $R$.
5. The third term of a geometric sequence is 324 and the sixth term is 96 .
Show that the common ratio of the sequence is $\frac { 2 } { 3 }$.
Find the first term of the sequence.
Find the sum of the first 15 terms of the sequence.
Find the sum to infinity of the sequence.
6. The circle $C$ has equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 4 y = 12$$
Find the centre and the radius of $C$. The point $P ( - 1,1 )$ and the point $Q ( 7 , - 5 )$ both lie on $C$.
Show that $P Q$ is a diameter of $C$. The point $R$ lies on the positive $y$-axis and the angle $P R Q = 90 ^ { \circ }$.
Find the coordinates of $R$.
7. (i) Solve, for $- 180 ^ { \circ } \leq \theta \leq 180 ^ { \circ }$, $$( 1 + \tan \theta ) ( 5 \sin \theta - 2 ) = 0 .$$ (ii) Solve, for $0 \leq x < 360 ^ { \circ }$, $$4 \sin x = 3 \tan x .$$
1. (a) Find the value of $y$ such that
$$\log _ { 2 } y = - 3 .$$
Find the values of $x$ such that $$\frac { \log _ { 2 } 32 + \log _ { 2 } 16 } { \log _ { 2 } x } = \log _ { 2 } x .$$ 9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-29_362_296_646_598} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height $h \mathrm {~cm}$. The cross section is a sector of a circle. The sector has radius $r \mathrm {~cm}$ and angle 1 radian. The volume of the box is $300 \mathrm {~cm} ^ { 3 }$.
Show that the surface area of the box, $S \mathrm {~cm} ^ { 2 }$, is given by $$S = r ^ { 2 } + \frac { 1800 } { r } .$$
Use calculus to find the value of $r$ for which $S$ is stationary. Use calculus to find the value of $r$ for which $S$ is stationary.
Prove that this value of $r$ gives a minimum value of $S$.
Find, to the nearest $\mathrm { cm } ^ { 2 }$, this minimum value of $S$.

Edexcel C2 Q1

Find all values of $\theta$ in the interval $0 \leq \theta < 360$ for which
1. $\cos ( \theta + 75 ) ^ { \circ } = 0$.
2. $\sin 2 \theta ^ { \circ } = 0.7$, giving your answers to one decima1 place.
  [0pt] [P1 January 2001 Question 3]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9f1194cd-cc8a-4f8d-8010-c62fea344c4e-01_645_1408_1096_262} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows the curve with equation $y = 5 + 2 x - x ^ { 2 }$ and the line with equation $y = 2$. The curve and the line intersect at the points $A$ and $B$.
Find the $x$-coordinates of $A$ and $B$.
(3) The shaded region $R$ is bounded by the curve and the line.
Find the area of $R$.
(6)

Edexcel C2 Q3

3. The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find

the common ratio of the series,
the first term of the series,
the sum to infinity of the series.
Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series.
[0pt] [P1 January 2001 Question 5]

Edexcel C2 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9f1194cd-cc8a-4f8d-8010-c62fea344c4e-03_681_1237_258_287} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Triangle $A B C$ has $A B = 9 \mathrm {~cm} , B C 10 \mathrm {~cm}$ and $C A = 5 \mathrm {~cm}$. A circle, centre $A$ and radius 3 cm , intersects $A B$ and $A C$ at $P$ and $Q$ respectively, as shown in Fig. 3.

Show that, to 3 decimal places, $\angle B A C = 1.504$ radians. Calculate,
the area, in $\mathrm { cm } ^ { 2 }$, of the sector $A P Q$,
the area, in $\mathrm { cm } ^ { 2 }$, of the shaded region $B P Q C$,
the perimeter, in cm , of the shaded region $B P Q C$.

Edexcel C2 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9f1194cd-cc8a-4f8d-8010-c62fea344c4e-04_723_556_310_571} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions $2 x \mathrm {~cm}$ by $x \mathrm {~cm}$ and height $h \mathrm {~cm}$, as shown in Fig. 4. Given that the capacity of a carton has to be $1030 \mathrm {~cm} ^ { 3 }$,

express $h$ in terms of $x$,
show that the surface area, $A \mathrm {~cm} ^ { 2 }$, of a carton is given by $$A = 4 x ^ { 2 } + \frac { 3090 } { x } .$$ The manufacturer needs to minimise the surface area of a carton.
Use calculus to find the value of $x$ for which $A$ is a minimum.
Calculate the minimum value of $A$.
Prove that this value of $A$ is a minimum.

Edexcel C2 Q6

6. Given that $2 \sin 2 \theta = \cos 2 \theta$,

show that $\tan 2 \theta = 0.5$.
Hence find the values of $\theta$, to one decimal place, in the interval $0 \leq \theta < 360$ for which $2 \sin 2 \theta ^ { \circ } = \cos 2 \theta ^ { \circ }$.
(5)
[0pt] [P1 June 2001 Question 2]

Edexcel C2 Q7

7. Figure 1 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9f1194cd-cc8a-4f8d-8010-c62fea344c4e-06_497_499_397_392} \captionsetup{labelformat=empty} \caption{Shape $X$}

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

\includegraphics[alt={},max width=\textwidth]{9f1194cd-cc8a-4f8d-8010-c62fea344c4e-06_604_478_349_1069} \captionsetup{labelformat=empty} \caption{Shape $Y$}

\end{figure} Figure 1 shows the cross-sections of two drawer handles. Shape $X$ is a rectangle $A B C D$ joined to a semicircle with $B C$ as diameter. The length $A B = d \mathrm {~cm}$ and $B C = 2 d \mathrm {~cm}$. Shape $Y$ is a sector $O P Q$ of a circle with centre $O$ and radius $2 d \mathrm {~cm}$.
Angle $P O Q$ is $\theta$ radians. Given that the areas of the shapes $X$ and $Y$ are equal,

prove that $\theta = 1 + \frac { 1 } { 4 } \pi$. Using this value of $\theta$, and given that $d = 3$, find in terms of $\pi$,
the perimeter of shape $X$,
the perimeter of shape $Y$.
Hence find the difference, in mm, between the perimeters of shapes $X$ and $Y$. \section*{8.} \section*{Figure 2}
\includegraphics[max width=\textwidth, alt={}]{9f1194cd-cc8a-4f8d-8010-c62fea344c4e-07_757_1148_354_356}
Figure 2 shows part of the curve with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x .$$ The curve touches the $x$-axis at $A$ and has a maximum turning point at $B$.
Show that the equation of the curve may be written as $$y = x ( x - 3 ) ^ { 2 } ,$$ and hence write down the coordinates of $A$.
Find the coordinates of $B$. The shaded region $R$ is bounded by the curve and the $x$-axis.
Find the area of $R$.

Edexcel C2 Q9

9. A population of deer is introduced into a park. The population $P$ at $t$ years after the deer have been introduced is modelled by $$P = \frac { 2000 a ^ { t } } { 4 + a ^ { t } }$$ where $a$ is a constant. Given that there are 800 deer in the park after 6 years,

calculate, to 4 decimal places, the value of $a$,
use the model to predict the number of years needed for the population of deer to increase from 800 to 1800.
With reference to this model, give a reason why the population of deer cannot exceed 2000.
[0pt] [P2 June 2001 Question 6]

Edexcel C2 Q10

10. (a) Given that $$( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = A + B x ^ { 2 } + C x ^ { 4 }$$ find the values of the constants $A , B$ and $C$.
(b) Using the substitution $y = x ^ { 2 }$ and your answers to part (a), solve, $$( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = 349$$ [P2 June 2001 Question 8]

Edexcel C2 Q11

11. A circle $C$ has equation $$x ^ { 2 } + y ^ { 2 } - 10 x + 6 y - 15 = 0$$

Find the coordinates of the centre of $C$.
Find the radius of $C$.
[0pt] [P3 June 2001 Question 1]

Edexcel C2 Q12

12. $\mathrm { f } ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } - 7 x + 14$, where $a$ and $b$ are constants. Given that when $\mathrm { f } ( x )$ is divided by ( $x - 1$ ) the remainder is 9 ,

write down an equation connecting $a$ and $b$. Given also that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$,
find the values of $a$ and $b$.
[0pt] [P3 June 2001 Question 2]

Edexcel C2 Q13

13. $\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 7 x + c$, where $c$ is a constant. Given that $\mathrm { f } ( 4 ) = 0$,

find the value of $c$,
factorise $\mathrm { f } ( x )$ as the product of a linear factor and a quadratic factor.
Hence show that, apart from $x = 4$, there are no real values of $x$ for which $\mathrm { f } ( x ) = 0$.

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\(x\)	0	4	8	12	16	20
\(y\)	0		2.771			0

\(x\)	0	0.2	0.4	0.6	0.8	1
\(3 ^ { x }\)		1.246	1.552			3