Edexcel C1 — Question 11

Exam BoardEdexcel
ModuleC1 (Core Mathematics 1)
TopicStraight Lines & Coordinate Geometry
11. The line \(l _ { 1 }\) passes through the points \(P ( - 1,2 )\) and \(Q ( 11,8 )\).
  1. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants. The line \(l _ { 2 }\) passes through the point \(R ( 10,0 )\) and is perpendicular to \(l _ { 1 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(S\).
  2. Calculate the coordinates of \(S\).
  3. Show that the length of \(R S\) is \(3 \sqrt { 5 }\).
  4. Hence, or otherwise, find the exact area of triangle \(P Q R\). \section*{Edexcel GCE
    Core Mathematics C1
    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 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 . 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. Given that
    $$y = 4 x ^ { 3 } - 1 + 2 x ^ { \frac { 1 } { 2 } } , \quad x > 0 ,$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. (a) Express \(\sqrt { } 108\) in the form \(a \sqrt { } 3\), where \(a\) is an integer.
  5. Express \(( 2 - \sqrt { 3 } ) ^ { 2 }\) in the form \(b + c \sqrt { 3 }\), where \(b\) and \(c\) are integers to be found.
    3. Given that $$\mathrm { f } ( x ) = \frac { 1 } { x } , \quad x \neq 0 ,$$
  6. sketch the graph of \(y = \mathrm { f } ( x ) + 3\) and state the equations of the asymptotes.
  7. Find the coordinates of the point where \(y = \mathrm { f } ( x ) + 3\) crosses a coordinate axis.
    4. Solve the simultaneous equations $$\begin{aligned} & y = x - 2
    & y ^ { 2 } + x ^ { 2 } = 10 \end{aligned}$$
    1. The equation \(2 x ^ { 2 } - 3 x - ( k + 1 ) = 0\), where \(k\) is a constant, has no real roots.
    Find the set of possible values of \(k\).
    6. (a) Show that \(( 4 + 3 \sqrt { x } ) ^ { 2 }\) can be written as \(16 + k \sqrt { x } + 9 x\), where \(k\) is a constant to be found.
  8. Find \(\int ( 4 + 3 \sqrt { x } ) ^ { 2 } d x\).
    7. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x \neq 0\), and the point \(P ( 2,1 )\) lies on \(C\). Given that $$f ^ { \prime } ( x ) = 3 x ^ { 2 } - 6 - \frac { 8 } { x ^ { 2 } }$$
  9. find \(\mathrm { f } ( x )\).
  10. Find an equation for the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers.
    8. The curve \(C\) has equation \(y = 4 x + 3 x ^ { \frac { 3 } { 2 } } - 2 x ^ { 2 } , \quad x > 0\).
  11. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  12. Show that the point \(P ( 4,8 )\) lies on \(C\).
  13. Show that an equation of the normal to \(C\) at the point \(P\) is $$3 y = x + 20$$ The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(Q\).
  14. Find the length \(P Q\), giving your answer in a simplified surd form.
    9. Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns: Row 1 □ I Row 2 □ I
    □
    Row 3 □ I\_I □
    She notices that 4 sticks are required to make the single square in the first row, 7 sticks to make 2 squares in the second row and in the third row she needs 10 sticks to make 3 squares.
  15. Find an expression, in terms of \(n\), for the number of sticks required to make a similar arrangement of \(n\) squares in the \(n\)th row. Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.
  16. Find the total number of sticks Ann uses in making these 10 rows. Ann started with 1750 sticks. Given that Ann continues the pattern to complete \(k\) rows but does not have sufficient sticks to complete the \(( k + 1 )\) th row,
  17. show that \(k\) satisfies \(( 3 k - 100 ) ( k + 35 ) < 0\).
  18. Find the value of \(k\).
    10. (a) On the same axes sketch the graphs of the curves with equations
    1. \(y = x ^ { 2 } ( x - 2 )\),
    2. \(y = x ( 6 - x )\),
      and indicate on your sketches the coordinates of all the points where the curves cross the \(x\)-axis.
  19. Use algebra to find the coordinates of the points where the graphs intersect. END \section*{(2)}
    1. Simplify \(( 3 + \sqrt { 5 } ) ( 3 - \sqrt { 5 } )\).
    \section*{Edexcel GCE
    Core Mathematics C1 Advanced Subsidiary }
    \includegraphics[max width=\textwidth, alt={}]{466833b9-730d-424c-b33b-dd93a14ab21d-13_181_138_452_991}
    \section*{Monday 21 May 2007 - Morning
    Time: 1 hour 30 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 11 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.
    2. (a) Find the value of \(8 ^ { \frac { 4 } { 3 } }\).
  20. Simplify \(\frac { 15 x ^ { \frac { 4 } { 3 } } } { 3 x }\).
    3. Given that \(y = 3 x ^ { 2 } + 4 \sqrt { } x , x > 0\), find
  21. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  22. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
  23. \(\int y \mathrm {~d} x\).
    4. A girl saves money over a period of 200 weeks. She saves 5 p in Week 1, 7 p in Week 2, 9p in Week 3, and so on until Week 200. Her weekly savings form an arithmetic sequence.
  24. Find the amount she saves in Week 200.
  25. Calculate her total savings over the complete 200 week period.
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{466833b9-730d-424c-b33b-dd93a14ab21d-14_549_661_244_429} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { 3 } { x } , x \neq 0\).
  26. On a separate diagram, sketch the curve with equation \(y = \frac { 3 } { x + 2 } , x \neq - 2\), showing the coordinates of any point at which the curve crosses a coordinate axis.
  27. Write down the equations of the asymptotes of the curve in part (a).
    6. (a) By eliminating \(y\) from the equations $$\begin{gathered} y = x - 4
    2 x ^ { 2 } - x y = 8 \end{gathered}$$ show that $$x ^ { 2 } + 4 x - 8 = 0$$
  28. Hence, or otherwise, solve the simultaneous equations $$\begin{gathered} y = x - 4
    2 x ^ { 2 } - x y = 8 \end{gathered}$$ giving your answers in the form \(a \pm b \sqrt { } 3\), where \(a\) and \(b\) are integers.
    7. The equation \(x ^ { 2 } + k x + ( k + 3 ) = 0\), where \(k\) is a constant, has different real roots.
  29. Show that \(k ^ { 2 } - 4 k - 12 > 0\).
  30. Find the set of possible values of \(k\).
    8. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{gathered} a _ { 1 } = k ,
    a _ { n + 1 } = 3 a _ { n } + 5 , \quad n \geq 1 , \end{gathered}$$ where \(k\) is a positive integer.
  31. Write down an expression for \(a _ { 2 }\) in terms of \(k\).
  32. Show that \(a _ { 3 } = 9 k + 20\).
    1. Find \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) in terms of \(k\).
    2. Show that \(\sum _ { r = 1 } ^ { 4 } a _ { r }\) is divisible by 10 .
      9. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 5,65 )\). Given that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } - 10 x - 12\),
  34. use integration to find \(\mathrm { f } ( x )\).
  35. Hence show that \(\mathrm { f } ( x ) = x ( 2 x + 3 ) ( x - 4 )\).
  36. Sketch \(C\), showing the coordinates of the points where \(C\) crosses the \(x\)-axis.
    10. The curve \(C\) has equation \(y = x ^ { 2 } ( x - 6 ) + \frac { 4 } { x } , x > 0\). The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 2 respectively.
  37. Show that the length of \(P Q\) is \(\sqrt { } 170\).
  38. Show that the tangents to \(C\) at \(P\) and \(Q\) are parallel.
  39. Find an equation for the normal to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
    11. The line \(l _ { 1 }\) has equation \(y = 3 x + 2\) and the line \(l _ { 2 }\) has equation \(3 x + 2 y - 8 = 0\).
  40. Find the gradient of the line \(l _ { 2 }\). The point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\) is \(P\).
  41. Find the coordinates of \(P\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) cross the line \(y = 1\) at the points \(A\) and \(B\) respectively.
  42. Find the area of triangle \(A B P\).
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