- (a) Write down the value of \(16 ^ { \frac { 1 } { 2 } }\).
(b) Find the value of \(16 ^ { - \frac { 3 } { 2 } }\). - (i) Given that \(y = 5 x ^ { 3 } + 7 x + 3\), find
(a) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
(b) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
(ii) Find \(\int \left( 1 + 3 \sqrt { x } - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\). - Given that the equation \(k x ^ { 2 } + 12 x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\).
- Solve the simultaneous equations
$$\begin{gathered}
x + y = 2
x ^ { 2 } + 2 y = 12
\end{gathered}$$
- The \(r\) th term of an arithmetic series is \(( 2 r - 5 )\).
(a) Write down the first three terms of this series.
(b) State the value of the common difference.
(c) Show that \(\sum _ { r = 1 } ^ { n } ( 2 r - 5 ) = n ( n - 4 )\).
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{466833b9-730d-424c-b33b-dd93a14ab21d-02_326_618_294_429}
\end{figure}
Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the points \(( 2,0 )\) and \(( 4,0 )\). The minimum point on the curve is \(P ( 3 , - 2 )\).
In separate diagrams sketch the curve with equation
(a) \(y = - \mathrm { f } ( x )\),
(b) \(y = \mathrm { f } ( 2 x )\).
On each diagram, give the coordinates of the points at which the curve crosses the \(x\)-axis, and the coordinates of the image of \(P\) under the given transformation.
7. The curve \(C\) has equation \(y = 4 x ^ { 2 } + \frac { 5 - x } { x } , x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate 1 .
(a) Show that the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(P\) is 3 .
(b) Find an equation of the tangent to \(C\) at \(P\).
This tangent meets the \(x\)-axis at the point \(( k , 0 )\).
(c) Find the value of \(k\).
8.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{466833b9-730d-424c-b33b-dd93a14ab21d-02_483_974_280_1644}
\end{figure}
The points \(A ( 1,7 ) , B ( 20,7 )\) and \(C ( p , q )\) form the vertices of a triangle \(A B C\), as shown in Figure 2. The point \(D ( 8,2 )\) is the mid-point of \(A C\).
(a) Find the value of \(p\) and the value of \(q\).
The line \(l\), which passes through \(D\) and is perpendicular to \(A C\), intersects \(A B\) at \(E\).
(b) Find an equation for \(l\), in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers.
(c) Find the exact \(x\)-coordinate of \(E\).
9. The gradient of the curve \(C\) is given by
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 }$$
The point \(P ( 1,4 )\) lies on \(C\).
(a) Find an equation of the normal to \(C\) at \(P\).
(b) Find an equation for the curve \(C\) in the form \(y = \mathrm { f } ( x )\).
(c) Using \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 }\), show that there is no point on \(C\) at which the tangent is parallel to the line \(y = 1 - 2 x\).
10. Given that
$$\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 18 , \quad x \geq 0 ,$$
(a) express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers.
The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \geq 0\), meets the \(y\)-axis at \(P\) and has a minimum point at \(Q\).
(b) Sketch the graph of \(C\), showing the coordinates of \(P\) and \(Q\).
The line \(y = 41\) meets \(C\) at the point \(R\).
(c) Find the \(x\)-coordinate of \(R\), giving your answer in the form \(p + q \sqrt { } 2\), where \(p\) and \(q\) are integers.
Materials required for examination
Mathematical Formulae (Green)
Paper Reference(s)
6663/01
Core Mathematics C1
Advanced Subsidiary
Monday 23 May 2005 - Morning
Time: \(\mathbf { 1 }\) hour \(\mathbf { 3 0 }\) minutes
Calculators may NOT be used in this examination.
Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C1), the paper reference (6663), 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 .
Advice to Candidates
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.
N23491A
- (a) Write down the value of \(8 ^ { \frac { 1 } { 3 } }\).
(b) Find the value of \(8 ^ { - \frac { 2 } { 3 } }\). - Given that \(y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0\),
(a) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
(b) find \(\int y \mathrm {~d} x\).
$$x ^ { 2 } - 8 x - 29 \equiv ( x + a ) ^ { 2 } + b$$
where \(a\) and \(b\) are constants.
(a) Find the value of \(a\) and the value of \(b\).
(b) Hence, or otherwise, show that the roots of
$$x ^ { 2 } - 8 x - 29 = 0$$
are \(c \pm d \sqrt { } 5\), where \(c\) and \(d\) are integers to be found.
4.
Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{466833b9-730d-424c-b33b-dd93a14ab21d-04_552_796_328_1720}
Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve passes through the origin \(O\) and through the point \(( 6,0 )\). The maximum point on the curve is \(( 3,5 )\).
On separate diagrams, sketch the curve with equation
(a) \(y = 3 \mathrm { f } ( x )\),
(b) \(y = \mathrm { f } ( x + 2 )\).
On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.
5. Solve the simultaneous equations
$$\begin{gathered}
x - 2 y = 1
x ^ { 2 } + y ^ { 2 } = 29
\end{gathered}$$
- Find the set of values of \(x\) for which
(a) \(3 ( 2 x + 1 ) > 5 - 2 x\),
(b) \(2 x ^ { 2 } - 7 x + 3 > 0\),
(c) both \(3 ( 2 x + 1 ) > 5 - 2 x\) and \(2 x ^ { 2 } - 7 x + 3 > 0\). - (a) Show that \(\frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x }\) can be written as \(9 x ^ { - \frac { 1 } { 2 } } - 6 + x ^ { \frac { 1 } { 2 } }\).
Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 3 - \sqrt { } x ) ^ { 2 } } { \sqrt { } x } , x > 0\), and that \(y = \frac { 2 } { 3 }\) at \(x = 1\),
(b) find \(y\) in terms of \(x\).
8. The line \(l _ { 1 }\) passes through the point \(( 9 , - 4 )\) and has gradient \(\frac { 1 } { 3 }\).
(a) Find an equation for \(l _ { 1 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers.
The line \(l _ { 2 }\) passes through the origin \(O\) and has gradient - 2 . The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).
(b) Calculate the coordinates of \(P\).
Given that \(l _ { 1 }\) crosses the \(y\)-axis at the point \(C\),
(c) calculate the exact area of \(\triangle O C P\).
9. An arithmetic series has first term \(a\) and common difference \(d\).
(a) Prove that the sum of the first \(n\) terms of the series is
$$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$
Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence.
He repays \(\pounds 149\) in the first month, \(\pounds 147\) in the second month, \(\pounds 145\) in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).
(b) Find the amount Sean repays in the 21st month.
Over the \(n\) months, he repays a total of \(\pounds 5000\).
(c) Form an equation in \(n\), and show that your equation may be written as
$$n ^ { 2 } - 150 n + 5000 = 0$$
(d) Solve the equation in part (c).
(e) State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.
10. The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 8 x + 3\).
The point \(P\) has coordinates \(( 3,0 )\).
(a) Show that \(P\) lies on \(C\).
(b) Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Another point \(Q\) also lies on \(C\). The tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).
(c) Find the coordinates of \(Q\).
\section*{Tuesday 10 January 2006 - Afternoon}
\section*{Materials required for examination
Mathematical Formulae (Green)}
Nil
Calculators may NOT be used in this examination.
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 C1), the paper reference (6663), 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 individual questions and the parts of questions are shown in round brackets: e.g. (2).
There are 10 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.
- Factorise completely
$$x ^ { 3 } - 4 x ^ { 2 } + 3 x$$