Edexcel C2 (Core Mathematics 2)

1. 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)
2. 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 \(C\),
3. find an equation for \(C\).
   3. Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which
4. \(3 ^ { x } = 5\),
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 $$5 \sin ^ { 2 } x + 3 \sin x - 2 = 0 .$$
6. 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 .
7. Find the value of \(a\) and the value of \(b\).
8. 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
9. the common ratio,
10. the first term,
11. the sum of the first 50 terms, giving your answer to 3 decimal places,
12. 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
13. the length of the arc \(B D\),
14. the perimeter of \(R\), giving your answer to 3 significant figures,
15. 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.
16. 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.
17. 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 .
18. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the stage is given by $$A = 80 x - \left( 2 + \frac { \pi } { 2 } \right) x ^ { 2 }$$
19. Use calculus to find the value of \(x\) at which \(A\) has a stationary value.
20. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\).
21. 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}
22. \(5 ^ { x } = 8\), giving your answer to 3 significant figures,
23. \(\log _ { 2 } ( x + 1 ) - \log _ { 2 } x = \log _ { 2 } 7\).
    3. (a) Use the factor theorem to show that \(( x + 4 )\) is a factor of \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\).
24. 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 \(11 q\),
25. find the value of \(p\) and the value of \(q\).
    5. Solve, for \(0 \leq x \leq 180 ^ { \circ }\), the equation
26. \(\sin \left( x + 10 ^ { \circ } \right) = \frac { \sqrt { } 3 } { 2 }\),
27. \(\cos 2 x = - 0.9\), 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 $$y = \frac { 1 } { 10 } x \sqrt { } ( 20 - x ) , \quad 0 \leq x \leq 20 .$$
28. Complete the table below, giving values of \(y\) to 3 decimal places.

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

29. 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 \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
30. 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.
31. 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 \(x\),
32. 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
33. the coordinates of \(A\),
34. the radius of \(C\),
35. 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 \(T\),
36. 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.
37. 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.
38. 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 \(Q\).
39. Find the exact area of \(R\).
40. 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 \(\mathrm { f } ( 1 ) = 0\),
41. find the value of \(c\),
42. factorise \(\mathrm { f } ( x )\) completely,
43. find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x - 3 )\).

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.

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 .

1. Find the value of \(a\) and the value of \(b\).
2. 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

1. the common ratio,
2. the first term,
3. the sum of the first 50 terms, giving your answer to 3 decimal places,
4. 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] \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

1. the length of the arc \(B D\),
2. the perimeter of \(R\), giving your answer to 3 significant figures,
3. 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.

1. 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.
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 .

1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the stage is given by $$A = 80 x - \left( 2 + \frac { \pi } { 2 } \right) x ^ { 2 }$$
2. Use calculus to find the value of \(x\) at which \(A\) has a stationary value.
3. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\).
4. 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. \(5 ^ { x } = 8\), giving your answer to 3 significant figures,
6. \(\log _ { 2 } ( x + 1 ) - \log _ { 2 } x = \log _ { 2 } 7\).
   3. (a) Use the factor theorem to show that \(( x + 4 )\) is a factor of \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\).
7. 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 \(11 q\),
8. find the value of \(p\) and the value of \(q\).
   5. Solve, for \(0 \leq x \leq 180 ^ { \circ }\), the equation
9. \(\sin \left( x + 10 ^ { \circ } \right) = \frac { \sqrt { } 3 } { 2 }\),
10. \(\cos 2 x = - 0.9\), 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 $$y = \frac { 1 } { 10 } x \sqrt { } ( 20 - x ) , \quad 0 \leq x \leq 20 .$$
11. Complete the table below, giving values of \(y\) to 3 decimal places.
12. 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 $$s = \int _ { 0 } ^ { 30 } \sqrt { } \left( 1.2 ^ { t } - 1 \right) \mathrm { d } t$$
13. Use the trapezium rule, with all the values from your table, to estimate the value of \(s\).
    7. The curve \(C\) has equation $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 2 .$$
14. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
15. Using the result from part (a), find the coordinates of the turning points of \(C\).
16. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
17. 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 $$5 \sin \left( \theta + 30 ^ { \circ } \right) = 3 .$$
18. 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
19. the \(x\)-coordinates of the points \(A\) and \(B\),
20. 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 \(a\).
    $$\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 29 x - 60$$
21. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
22. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
23. 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.
24. 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

25. 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 \(\tan \theta\).
26. 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 \(C\).
27. Find an equation of the straight line through \(P\) and \(Q\). Given that \(Q\) lies on the line \(y = 1\),
28. show that the \(x\)-coordinate of \(Q\) is 5 ,
29. 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
30. the length of the arc \(B C\), in m , to 2 decimal places,
31. the area of the sector \(B A C\), in \(\mathrm { m } ^ { 2 }\), to 2 decimal places,
32. the size of \(\angle C A D\), in radians, to 2 decimal places,
33. 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 .
34. Show that \(25 r ^ { 2 } - 25 r + 4 = 0\).
35. Find the two possible values of \(r\).
36. Find the corresponding two possible values of \(a\).
37. 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,
38. 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}
39. 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 \(N\).
40. Find \(\int \left( x ^ { 3 } - 8 x ^ { 2 } + 20 x \right) \mathrm { d } x\).
41. 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
42. \(\mathrm { f } ^ { \prime \prime } ( x )\),
43. \(\int _ { 1 } ^ { 2 } f ( x ) d x\).
    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.
44. 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. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + x - 6$$
45. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
46. Factorise \(\mathrm { f } ( x )\) completely.
47. 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 $$C = \frac { 1400 } { v } + \frac { 2 v } { 7 } .$$
48. Find the value of \(v\) for which \(C\) is a minimum.
49. Find \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) and hence verify that \(C\) is a minimum for this value of \(v\).
50. 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 \(6 \sqrt { } 3 \mathrm {~m}\),
51. find the exact size of angle \(P Q R\) in radians.
52. Show that the area of the patio \(P Q R S\) is \(12 \pi \mathrm {~m} ^ { 2 }\).
53. Find the exact area of the triangle \(P Q R\).
54. Find, in \(\mathrm { m } ^ { 2 }\) to 1 decimal place, the area of the segment PRS.
55. Find, in m to 1 decimal place, the perimeter of the patio \(P Q R S\).

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

1. Prove that the sum of the first \(n\) terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r } .$$
2. Find $$\sum _ { k = 1 } ^ { 10 } 100 \left( 2 ^ { k } \right) .$$
3. Find the sum to infinity of the geometric series $$\frac { 5 } { 6 } + \frac { 5 } { 18 } + \frac { 5 } { 54 } + \ldots$$
4. 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.
   $$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 16 x + 12$$
5. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ). Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
6. 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
7. the value of \(k\),
8. 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 \(C A = 5 \mathrm {~cm}\).
9. Show that \(\cos A = \frac { 3 } { 4 }\).
10. Hence, or otherwise, find the exact value of \(\sin A\).
    5. The curve \(C\) has equation $$\left. y = x \sqrt { ( } x ^ { 3 } + 1 \right) , \quad 0 \leq x \leq 2 .$$
11. Copy and complete the table below, giving the values of \(y\) to 3 decimal places at \(x = 1\) and \(x = 1.5\).
12. 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 \(x ^ { 2 }\),
13. find the value of \(a\).
    4. (a) Find, to 3 significant figures, the value of \(x\) for which \(5 ^ { x } = 7\).
14. 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 \(P ( 8,3 )\).
15. Find an equation for \(C\).
16. 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
17. the 20th term of the series, to 3 decimal places,
18. the sum to infinity of the series. Given that the sum to \(k\) terms of the series is greater than 24.95,
19. show that \(k > \frac { \log 0.002 } { \log 0.8 }\),
20. 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
21. the length of the \(\operatorname { arc } B C\),
22. 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
23. the perimeter of \(R\), giving your answer to 3 significant figures,
24. 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 \(A\).
25. 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.
26. Using calculus, find the exact area of \(R\).
    (8)
    9. Solve, for \(0 \leq x < 360 ^ { \circ }\),
27. \(\quad \sin \left( x - 20 ^ { \circ } \right) = \frac { 1 } { \sqrt { 2 } }\),
28. \(\quad \cos 3 x = - \frac { 1 } { 2 }\). \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. $$y = \sqrt { } \left( 10 x - x ^ { 2 } \right) .$$
29. 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 \(x = 3\)
30. Use the trapezium rule, with all the values from your table, to find an approximation for the area of \(R\).
31. 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 .
32. Show that the common ratio of the sequence is \(\frac { 2 } { 3 }\).
33. Find the first term of the sequence.
34. Find the sum of the first 15 terms of the sequence.
35. Find the sum to infinity of the sequence.
    6. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 4 y = 12$$
36. Find the centre and the radius of \(C\). The point \(P ( - 1,1 )\) and the point \(Q ( 7 , - 5 )\) both lie on \(C\).
37. Show that \(P Q\) is a diameter of \(C\). The point \(R\) lies on the positive \(y\)-axis and the angle \(P R Q = 90 ^ { \circ }\).
38. 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
    $$\log _ { 2 } y = - 3 .$$
39. 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 \(300 \mathrm {~cm} ^ { 3 }\).
40. Show that the surface area of the box, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = r ^ { 2 } + \frac { 1800 } { r } .$$
41. 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.
42. Prove that this value of \(r\) gives a minimum value of \(S\).
43. Find, to the nearest \(\mathrm { cm } ^ { 2 }\), this minimum value of \(S\).