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.
- 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.
- 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.
- Use calculus to find the exact value of \(\int _ { 1 } ^ { 2 } \left( 3 x ^ { 2 } + 5 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x\).
- Write down the value of \(\log _ { 6 } 36\).
- 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
$$( 1 + x ) ( 1 - 2 x ) ^ { 5 } \approx 1 - 9 x .$$
- 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
$$x ^ { 3 } + 4 x ^ { 2 } + x - 6 = 0 .$$
- 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 \(P Q R S\).