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Edexcel C1 2006 January Q10
11 marks Moderate -0.3
10. $$x ^ { 2 } + 2 x + 3 \equiv ( x + a ) ^ { 2 } + b .$$
  1. Find the values of the constants \(a\) and \(b\).
  2. In the space provided below, sketch the graph of \(y = x ^ { 2 } + 2 x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes.
  3. 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.
  4. Find the set of possible values of \(k\), giving your answer in surd form.
Edexcel C1 2007 January Q1
4 marks Easy -1.8
  1. Given that
$$y = 4 x ^ { 3 } - 1 + 2 x ^ { \frac { 1 } { 2 } } , \quad x > 0 ,$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). \includegraphics[max width=\textwidth, alt={}, center]{fff086fd-f5d8-45b7-8db1-8b22ba5aab31-02_29_45_2690_1852}
Edexcel C1 2007 January Q2
4 marks Easy -1.3
2. (a) Express \(\sqrt { } 108\) in the form \(a \sqrt { } 3\), where \(a\) is an integer.
(b) Express \(( 2 - \sqrt { 3 } ) ^ { 2 }\) in the form \(b + c \sqrt { 3 }\), where \(b\) and \(c\) are integers to be found.
Edexcel C1 2007 January Q3
6 marks Moderate -0.8
3. Given that \(\quad \mathrm { f } ( x ) = \frac { 1 } { x } , \quad x \neq 0\),
  1. sketch the graph of \(y = \mathrm { f } ( x ) + 3\) and state the equations of the asymptotes.
  2. Find the coordinates of the point where \(y = \mathrm { f } ( x ) + 3\) crosses a coordinate axis.
Edexcel C1 2007 January Q4
7 marks Moderate -0.8
4. Solve the simultaneous equations $$\begin{aligned} & y = x - 2 , \\ & y ^ { 2 } + x ^ { 2 } = 10 . \end{aligned}$$
Edexcel C1 2007 January Q5
4 marks Moderate -0.3
5. 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 \(k\).
Edexcel C1 2007 January Q6
5 marks Moderate -0.8
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.
(b) Find \(\int ( 4 + 3 \sqrt { } x ) ^ { 2 } \mathrm {~d} x\).
Edexcel C1 2007 January Q7
9 marks Moderate -0.3
7. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x \neq 0\), and the point \(P ( 2,1 )\) lies on \(C\). Given that $$f ^ { \prime } ( x ) = 3 x ^ { 2 } - 6 - \frac { 8 } { x ^ { 2 } } ,$$
  1. find \(\mathrm { f } ( x )\).
  2. 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.
Edexcel C1 2007 January Q8
11 marks Moderate -0.8
8. The curve \(C\) has equation \(y = 4 x + 3 x ^ { \frac { 3 } { 2 } } - 2 x ^ { 2 } , \quad x > 0\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that the point \(P ( 4,8 )\) lies on \(C\).
  3. 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 \(Q\).
  4. Find the length \(P Q\), giving your answer in a simplified surd form.
Edexcel C1 2007 January Q9
12 marks Moderate -0.3
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 â–¡ Row 2 â–¡ 1 Row 3 \includegraphics[max width=\textwidth, alt={}, center]{fff086fd-f5d8-45b7-8db1-8b22ba5aab31-11_40_104_566_479} 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.
  1. 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.
  2. 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,
  3. show that \(k\) satisfies \(( 3 k - 100 ) ( k + 35 ) < 0\).
  4. Find the value of \(k\).
Edexcel C1 2007 January Q10
13 marks Moderate -0.3
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.
    (b) Use algebra to find the coordinates of the points where the graphs intersect.
Edexcel C1 2008 January Q1
4 marks Easy -1.2
Find \(\int \left( 3 x ^ { 2 } + 4 x ^ { 5 } - 7 \right) d x\).
Edexcel C1 2008 January Q4
7 marks Moderate -0.8
4. The point \(A ( - 6,4 )\) and the point \(B ( 8 , - 3 )\) lie on the line \(L\).
  1. Find an equation for \(L\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
  2. Find the distance \(A B\), giving your answer in the form \(k \sqrt { 5 }\), where \(k\) is an integer.
Edexcel C1 2008 January Q5
6 marks Easy -1.2
5. (a) Write \(\frac { 2 \sqrt { } x + 3 } { x }\) in the form \(2 x ^ { p } + 3 x ^ { q }\) where \(p\) and \(q\) are constants. Given that \(y = 5 x - 7 + \frac { 2 \sqrt { } x + 3 } { x } , \quad x > 0\),
(b) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying the coefficient of each term.
Edexcel C1 2008 January Q6
7 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba0ee180-4c22-49f7-8a8e-a7268828b067-07_693_676_370_632} \captionsetup{labelformat=empty} \caption{Figure 1}
\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 \(( 1,0 )\) and \(( 4,0 )\). The maximum point on the curve is \(( 2,5 )\).
In separate diagrams sketch the curves with the following equations.
On each diagram show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.
  1. \(y = 2 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( - x )\). The maximum point on the curve with equation \(y = \mathrm { f } ( x + a )\) is on the \(y\)-axis.
  3. Write down the value of the constant \(a\).
Edexcel C1 2008 January Q7
8 marks Moderate -0.8
  1. A sequence is given by:
$$\begin{aligned} & x _ { 1 } = 1 \\ & x _ { n + 1 } = x _ { n } \left( p + x _ { n } \right) \end{aligned}$$ where \(p\) is a constant ( \(p \neq 0\) ) .
  1. Find \(x _ { 2 }\) in terms of \(p\).
  2. Show that \(x _ { 3 } = 1 + 3 p + 2 p ^ { 2 }\). Given that \(x _ { 3 } = 1\),
  3. find the value of \(p\),
  4. write down the value of \(x _ { 2008 }\).
Edexcel C1 2008 January Q8
7 marks Moderate -0.3
8. The equation $$x ^ { 2 } + k x + 8 = k$$ has no real solutions for \(x\).
  1. Show that \(k\) satisfies \(k ^ { 2 } + 4 k - 32 < 0\).
  2. Hence find the set of possible values of \(k\).
Edexcel C1 2008 January Q9
10 marks Moderate -0.3
9. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), and \(\mathrm { f } ^ { \prime } ( x ) = 4 x - 6 \sqrt { } x + \frac { 8 } { x ^ { 2 } }\). Given that the point \(P ( 4,1 )\) lies on \(C\),
  1. find \(\mathrm { f } ( x )\) and simplify your answer.
  2. Find an equation of the normal to \(C\) at the point \(P ( 4,1 )\).
Edexcel C1 2008 January Q10
12 marks Moderate -0.8
  1. The curve \(C\) has equation
$$y = ( x + 3 ) ( x - 1 ) ^ { 2 }$$
  1. Sketch \(C\) showing clearly the coordinates of the points where the curve meets the coordinate axes.
  2. Show that the equation of \(C\) can be written in the form $$y = x ^ { 3 } + x ^ { 2 } - 5 x + k ,$$ where \(k\) is a positive integer, and state the value of \(k\). There are two points on \(C\) where the gradient of the tangent to \(C\) is equal to 3 .
  3. Find the \(x\)-coordinates of these two points.
Edexcel C1 2008 January Q11
7 marks Standard +0.3
11. The first term of an arithmetic sequence is 30 and the common difference is - 1.5
  1. Find the value of the 25th term. The \(r\) th term of the sequence is 0 .
  2. Find the value of \(r\). The sum of the first \(n\) terms of the sequence is \(S _ { n }\).
  3. Find the largest positive value of \(S _ { n }\).
Edexcel C1 2009 January Q1
3 marks Easy -1.8
  1. Write down the value of \(125 ^ { \frac { 1 } { 3 } }\).
  2. Find the value of \(125 ^ { - \frac { 2 } { 3 } }\).
Edexcel C1 2009 January Q5
6 marks Moderate -0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{871f5957-180d-4379-88ce-186432f57bad-06_988_1158_285_390} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\). There is a maximum at \(( 0,0 )\), a minimum at \(( 2 , - 1 )\) and \(C\) passes through \(( 3,0 )\). On separate diagrams sketch the curve with equation
  1. \(y = \mathrm { f } ( x + 3 )\),
  2. \(y = \mathrm { f } ( - x )\). On each diagram show clearly the coordinates of the maximum point, the minimum point and any points of intersection with the \(x\)-axis.
Edexcel C1 2009 January Q6
6 marks Easy -1.3
  1. Given that \(\frac { 2 x ^ { 2 } - x ^ { \frac { 3 } { 2 } } } { \sqrt { } x }\) can be written in the form \(2 x ^ { p } - x ^ { q }\),
    1. write down the value of \(p\) and the value of \(q\).
    Given that \(y = 5 x ^ { 4 } - 3 + \frac { 2 x ^ { 2 } - x ^ { \frac { 3 } { 2 } } } { \sqrt { } x }\),
  2. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying the coefficient of each term.
Edexcel C1 2009 January Q7
7 marks Moderate -0.8
7. The equation \(k x ^ { 2 } + 4 x + ( 5 - k ) = 0\), where \(k\) is a constant, has 2 different real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 5 k + 4 > 0 .$$
  2. Hence find the set of possible values of \(k\).
Edexcel C1 2009 January Q8
7 marks Moderate -0.3
8. The point \(P ( 1 , a )\) lies on the curve with equation \(y = ( x + 1 ) ^ { 2 } ( 2 - x )\).
  1. Find the value of \(a\).
  2. On the axes below sketch the curves with the following equations:
    1. \(y = ( x + 1 ) ^ { 2 } ( 2 - x )\),
    2. \(y = \frac { 2 } { x }\). On your diagram show clearly the coordinates of any points at which the curves meet the axes.
  3. With reference to your diagram in part (b) state the number of real solutions to the equation $$( x + 1 ) ^ { 2 } ( 2 - x ) = \frac { 2 } { x } .$$
    \includegraphics[max width=\textwidth, alt={}]{871f5957-180d-4379-88ce-186432f57bad-10_1347_1344_1245_297}