Edexcel C1 — Question 10

Exam BoardEdexcel
ModuleC1 (Core Mathematics 1)
TopicQuadratic Functions
10. Given that $$\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 18 , \quad x \geq 0 ,$$
  1. 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 \(Q\).
  2. Sketch the graph of \(C\), showing the coordinates of \(P\) and \(Q\). The line \(y = 41\) meets \(C\) at the point \(R\).
  3. 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
    1. (a) Write down the value of \(8 ^ { \frac { 1 } { 3 } }\).
    2. Find the value of \(8 ^ { - \frac { 2 } { 3 } }\).
    3. Given that \(y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0\),
    4. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
    5. find \(\int y \mathrm {~d} x\).
    $$x ^ { 2 } - 8 x - 29 \equiv ( x + a ) ^ { 2 } + b$$ where \(a\) and \(b\) are constants.
  4. Find the value of \(a\) and the value of \(b\).
  5. 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
  6. \(y = 3 \mathrm { f } ( x )\),
  7. \(y = \mathrm { f } ( x + 2 )\). 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 $$\begin{gathered} x - 2 y = 1
    x ^ { 2 } + y ^ { 2 } = 29 \end{gathered}$$
    1. Find the set of values of \(x\) for which
    2. \(3 ( 2 x + 1 ) > 5 - 2 x\),
    3. \(2 x ^ { 2 } - 7 x + 3 > 0\),
    4. both \(3 ( 2 x + 1 ) > 5 - 2 x\) and \(2 x ^ { 2 } - 7 x + 3 > 0\).
    5. (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 \(x = 1\),
  8. find \(y\) in terms of \(x\).
    8. The line \(l _ { 1 }\) passes through the point \(( 9 , - 4 )\) and has gradient \(\frac { 1 } { 3 }\).
  9. 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 \(P\).
  10. Calculate the coordinates of \(P\). Given that \(l _ { 1 }\) crosses the \(y\)-axis at the point \(C\),
  11. calculate the exact area of \(\triangle O C P\).
    9. An arithmetic series has first term \(a\) and common difference \(d\).
  12. 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 \(n > 21\).
  13. Find the amount Sean repays in the 21st month. Over the \(n\) months, he repays a total of \(\pounds 5000\).
  14. Form an equation in \(n\), and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0$$
  15. Solve the equation in part (c).
  16. 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 \(( 3,0 )\).
  17. Show that \(P\) lies on \(C\).
  18. 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 \(P\).
  19. 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.
    1. Factorise completely
    $$x ^ { 3 } - 4 x ^ { 2 } + 3 x$$
    1. The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), is given by
    $$u _ { n + 1 } = \left( u _ { n } - 3 \right) ^ { 2 } , \quad u _ { 1 } = 1 .$$
  20. Find \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
  21. Write down the value of \(u _ { 20 }\).
    3. The line \(L\) has equation \(y = 5 - 2 x\).
  22. Show that the point \(P ( 3 , - 1 )\) lies on \(L\).
  23. Find an equation of the line perpendicular to \(L\), which passes through \(P\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
    4. Given that \(y = 2 x ^ { 2 } - \frac { 6 } { x ^ { 3 } } , x \neq 0\),
  24. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  25. find \(\int y \mathrm {~d} x\).
    5. (a) Write \(\sqrt { } 45\) in the form \(a \sqrt { } 5\), where \(a\) is an integer.
  26. Express \(\frac { 2 ( 3 + \sqrt { 5 } ) } { ( 3 - \sqrt { 5 } ) }\) in the form \(b + c \sqrt { 5 }\), where \(b\) and \(c\) are integers.
    6. Figure 1
    \includegraphics[max width=\textwidth, alt={}, center]{466833b9-730d-424c-b33b-dd93a14ab21d-07_453_613_292_427} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve passes through the points \(( 0,3 )\) and \(( 4,0 )\) and touches the \(x\)-axis at the point \(( 1,0 )\). On separate diagrams, sketch the curve with equation
  27. \(y = \mathrm { f } ( x + 1 )\),
  28. \(y = 2 \mathrm { f } ( x )\),
  29. \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). On each diagram show clearly the coordinates of all the points at which the curve meets the axes.
    7. On Alice's 11th birthday she started to receive an annual allowance. The first annual allowance was \(\pounds 500\) and on each following birthday the allowance was increased by \(\pounds 200\).
  30. Show that, immediately after her 12th birthday, the total of the allowances that Alice had received was \(\pounds 1200\).
  31. Find the amount of Alice's annual allowance on her 18th birthday.
  32. Find the total of the allowances that Alice had received up to and including her 18th birthday. When the total of the allowances that Alice had received reached \(\pounds 32000\) the allowance stopped.
  33. Find how old Alice was when she received her last allowance.
    8. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 1,6 )\). Given that $$f ^ { \prime } ( x ) = 3 + \frac { 5 x ^ { 2 } + 2 } { x ^ { \frac { 1 } { 2 } } } , \quad x > 0$$ find \(\mathrm { f } ( x )\) and simplify your answer. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{466833b9-730d-424c-b33b-dd93a14ab21d-08_469_785_310_328}
    \end{figure} $$x ^ { 2 } + 2 x + 3 \equiv ( x + a ) ^ { 2 } + b .$$
  34. Find the values of the constants \(a\) and \(b\).
  35. Sketch the graph of \(y = x ^ { 2 } + 2 x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes.
  36. Find the value of the discriminant of \(x ^ { 2 } + 2 x + 3\). Explain how the sign of the discriminant relates to your sketch in part (b). The equation \(x ^ { 2 } + k x + 3 = 0\), where \(k\) is a constant, has no real roots.
  37. Find the set of possible values of \(k\), giving your answer in surd form. Figure 2 shows part of the curve \(C\) with equation $$y = ( x - 1 ) \left( x ^ { 2 } - 4 \right) .$$ The curve cuts the \(x\)-axis at the points \(P , ( 1,0 )\) and \(Q\), as shown in Figure 2.
  38. Write down the \(x\)-coordinate of \(P\) and the \(x\)-coordinate of \(Q\).
  39. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 2 x - 4\).
  40. Show that \(y = x + 7\) is an equation of the tangent to \(C\) at the point ( \(- 1,6\) ). The tangent to \(C\) at the point \(R\) is parallel to the tangent at the point \(( - 1,6 )\).
  41. Find the exact coordinates of \(R\). \section*{Edexcel GCE
    Core Mathematics C1
    \textbackslash section*\{Advanced Subsidiary\} } Materials required for examination
    Mathematical Formulae (Green) Calculators may NOT be used in this examination.
    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 11 questions in this question paper. The total mark for this paper is 75 . \section*{Items included with question papers Nil
    Nil
    Nil} 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. Monday 22 May 2006 - Morning
    Time: 1 hour 30 minutes
    4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$a _ { 1 } = 3$$ $$a _ { n + 1 } = 3 a _ { n } - 5 , \quad n \geq 1$$
  42. Find the value \(a _ { 2 }\) and the value of \(a _ { 3 }\).
  43. Calculate the value of \(\sum _ { r = 1 } ^ { 5 } a _ { r }\).
    5. Differentiate with respect to \(x\)
  44. \(x ^ { 4 } + 6 \sqrt { } x\),
  45. \(\frac { ( x + 4 ) ^ { 2 } } { x }\).
    1. Find \(\int \left( 6 x ^ { 2 } + 2 + x ^ { - \frac { 1 } { 2 } } \right) \mathrm { d } x\), giving each term in its simplest form.
    2. Find the set of values of \(x\) for which
    $$x ^ { 2 } - 7 x - 18 > 0$$
    1. On separate diagrams, sketch the graphs of
    2. \(y = ( x + 3 ) ^ { 2 }\),
    3. \(y = ( x + 3 ) ^ { 2 } + k\), where \(k\) is a positive constant.
    Show on each sketch the coordinates of each point at which the graph meets the axes.
    4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by 的 \(a _ { 1 } = 3\),保
    \(\_\_\_\_\) - \section*{-} (3)
  46. \(y = ( x + 3 ) ^ { 2 } + k\), where \(k\) is a positive constant.
  47. \(y = ( x + 3 ) ^ { 2 } + k\), where \(k\) is a positive constant.
    \(\_\_\_\_\) "
    Differentiate with respect to \(x\)
  48. ◯
    6. (a) Expand and simplify \(( 4 + \sqrt { } 3 ) ( 4 - \sqrt { } 3 )\).
  49. Express \(\frac { 26 } { 4 + \sqrt { 3 } }\) in the form \(a + b \sqrt { } 3\), where \(a\) and \(b\) are integers.
    7. An athlete prepares for a race by completing a practice run on each of 11 consecutive days. On each day after the first day he runs further than he ran on the previous day. The lengths of his 11 practice runs form an arithmetic sequence with first term \(a \mathrm {~km}\) and common difference \(d \mathrm {~km}\). He runs 9 km on the 11th day, and he runs a total of 77 km over the 11 day period.
    Find the value of \(a\) and the value of \(d\).
    8. The equation \(x ^ { 2 } + 2 p x + ( 3 p + 4 ) = 0\), where \(p\) is a positive constant, has equal roots.
  50. Find the value of \(p\).
  51. For this value of \(p\), solve the equation \(x ^ { 2 } + 2 p x + ( 3 p + 4 ) = 0\).
    9. Given that \(\mathrm { f } ( x ) = \left( x ^ { 2 } - 6 x \right) ( x - 2 ) + 3 x\),
  52. express \(\mathrm { f } ( x )\) in the form \(x \left( a x ^ { 2 } + b x + c \right)\), where \(a\), \(b\) and \(c\) are constants.
  53. Hence factorise \(\mathrm { f } ( x )\) completely.
  54. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of each point at which the graph meets the axes.
    10. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \neq 0\), passes through the point \(\left( 3,7 \frac { 1 } { 2 } \right)\). Given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x + \frac { 3 } { x ^ { 2 } }\),
  55. find \(\mathrm { f } ( x )\).
  56. Verify that \(\mathrm { f } ( - 2 ) = 5\).
  57. Find an equation for the tangent to \(C\) at the point ( \(- 2,5\) ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
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