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.
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]
\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.
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
\(\cos 2 x = - 0.9\), 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]
\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 \(( 2 x - 3 )\).