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.
- 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.
- 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 \(0 \leq x < 360 ^ { \circ }\),
$$4 \sin x = 3 \tan x .$$
- (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 \(S\).