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Edexcel C1 2009 January Q9
11 marks Moderate -0.3
9. The first term of an arithmetic series is \(a\) and the common difference is \(d\). The 18th term of the series is 25 and the 21st term of the series is \(32 \frac { 1 } { 2 }\).
  1. Use this information to write down two equations for \(a\) and \(d\).
  2. Show that \(a = - 17.5\) and find the value of \(d\). The sum of the first \(n\) terms of the series is 2750 .
  3. Show that \(n\) is given by $$n ^ { 2 } - 15 n = 55 \times 40 .$$
  4. Hence find the value of \(n\).
Edexcel C1 2009 January Q10
11 marks Easy -1.2
  1. The line \(l _ { 1 }\) passes through the point \(A ( 2,5 )\) and has gradient \(- \frac { 1 } { 2 }\).
    1. Find an equation of \(l _ { 1 }\), giving your answer in the form \(y = m x + c\).
    The point \(B\) has coordinates (-2, 7).
  2. Show that \(B\) lies on \(l _ { 1 }\).
  3. Find the length of \(A B\), giving your answer in the form \(k \sqrt { } 5\), where \(k\) is an integer. The point \(C\) lies on \(l _ { 1 }\) and has \(x\)-coordinate equal to \(p\).
    The length of \(A C\) is 5 units.
  4. Show that \(p\) satisfies $$p ^ { 2 } - 4 p - 16 = 0 .$$
Edexcel C1 2009 January Q11
13 marks Moderate -0.3
  1. The curve \(C\) has equation
$$y = 9 - 4 x - \frac { 8 } { x } , \quad x > 0$$ The point \(P\) on \(C\) has \(x\)-coordinate equal to 2 .
  1. Show that the equation of the tangent to \(C\) at the point \(P\) is \(y = 1 - 2 x\).
  2. Find an equation of the normal to \(C\) at the point \(P\). The tangent at \(P\) meets the \(x\)-axis at \(A\) and the normal at \(P\) meets the \(x\)-axis at \(B\).
  3. Find the area of triangle \(A P B\).
Edexcel C1 2010 January Q1
3 marks Easy -1.8
Given that \(y = x ^ { 4 } + x ^ { \frac { 1 } { 3 } } + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
Edexcel C1 2010 January Q4
7 marks Moderate -0.8
4. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 x ^ { - \frac { 1 } { 2 } } + x \sqrt { } x , \quad x > 0$$ Given that \(y = 35\) at \(x = 4\), find \(y\) in terms of \(x\), giving each term in its simplest form.
Edexcel C1 2010 January Q5
7 marks Moderate -0.3
5. Solve the simultaneous equations $$\begin{array} { r } y - 3 x + 2 = 0 \\ y ^ { 2 } - x - 6 x ^ { 2 } = 0 \end{array}$$
Edexcel C1 2010 January Q6
8 marks Moderate -0.8
6. The curve \(C\) has equation $$y = \frac { ( x + 3 ) ( x - 8 ) } { x } , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form.
  2. Find an equation of the tangent to \(C\) at the point where \(x = 2\)
Edexcel C1 2010 January Q7
9 marks Moderate -0.8
7. Jill gave money to a charity over a 20 -year period, from Year 1 to Year 20 inclusive. She gave \(\pounds 150\) in Year \(1 , \pounds 160\) in Year 2, \(\pounds 170\) in Year 3, and so on, so that the amounts of money she gave each year formed an arithmetic sequence.
  1. Find the amount of money she gave in Year 10.
  2. Calculate the total amount of money she gave over the 20 -year period. Kevin also gave money to the charity over the same 20 -year period. He gave \(\pounds A\) in Year 1 and the amounts of money he gave each year increased, forming an arithmetic sequence with common difference \(\pounds 30\). The total amount of money that Kevin gave over the 20 -year period was twice the total amount of money that Jill gave.
  3. Calculate the value of \(A\).
Edexcel C1 2010 January Q8
7 marks Moderate -0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{280f0f3b-fdb5-4ac9-adc6-150819b03539-10_646_986_246_562} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve has a maximum point \(( - 2,5 )\) and an asymptote \(y = 1\), as shown in Figure 1. On separate diagrams, sketch the curve with equation
  1. \(y = \mathrm { f } ( x ) + 2\)
  2. \(y = 4 \mathrm { f } ( x )\)
  3. \(y = \mathrm { f } ( \mathrm { x } + 1 )\) On each diagram, show clearly the coordinates of the maximum point and the equation of the asymptote.
Edexcel C1 2010 January Q9
13 marks Moderate -0.8
  1. (a) Factorise completely \(x ^ { 3 } - 4 x\) (b) Sketch the curve \(C\) with equation
$$y = x ^ { 3 } - 4 x ,$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. The point \(A\) with \(x\)-coordinate - 1 and the point \(B\) with \(x\)-coordinate 3 lie on the curve \(C\).
(c) Find an equation of the line which passes through \(A\) and \(B\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
(d) Show that the length of \(A B\) is \(k \sqrt { } 10\), where \(k\) is a constant to be found.
Edexcel C1 2010 January Q10
10 marks Moderate -0.3
10. $$\mathrm { f } ( x ) = x ^ { 2 } + 4 k x + ( 3 + 11 k ) , \quad \text { where } k \text { is a constant. }$$
  1. Express \(\mathrm { f } ( x )\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found in terms of \(k\). Given that the equation \(\mathrm { f } ( x ) = 0\) has no real roots,
  2. find the set of possible values of \(k\). Given that \(k = 1\),
  3. sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any point at which the graph crosses a coordinate axis.
Edexcel C1 2011 January Q1
4 marks Easy -1.3
  1. Find the value of \(16 ^ { - \frac { 1 } { 4 } }\)
  2. Simplify \(x \left( 2 x ^ { - \frac { 1 } { 4 } } \right) ^ { 4 }\)
Edexcel C1 2011 January Q3
4 marks Easy -1.2
3. Simplify $$\frac { 5 - 2 \sqrt { 3 } } { \sqrt { 3 } - 1 }$$ giving your answer in the form \(p + q \sqrt { } 3\), where \(p\) and \(q\) are rational numbers.
Edexcel C1 2011 January Q4
5 marks Moderate -0.5
4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 2 \\ a _ { n + 1 } & = 3 a _ { n } - c \end{aligned}$$ where \(c\) is a constant.
  1. Find an expression for \(a _ { 2 }\) in terms of \(c\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 0\)
  2. find the value of \(c\).
Edexcel C1 2011 January Q5
7 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{95e11fd7-765c-477d-800b-7574bc1af81f-06_640_1063_322_438} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { x } { x - 2 } , \quad x \neq 2$$ The curve passes through the origin and has two asymptotes, with equations \(y = 1\) and \(x = 2\), as shown in Figure 1.
  1. In the space below, sketch the curve with equation \(y = \mathrm { f } ( x - 1 )\) and state the equations of the asymptotes of this curve.
  2. Find the coordinates of the points where the curve with equation \(y = \mathrm { f } ( x - 1 )\) crosses the coordinate axes.
Edexcel C1 2011 January Q6
7 marks Moderate -0.8
6. An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first 10 terms of the sequence is 162 .
  1. Show that \(10 a + 45 d = 162\) Given also that the sixth term of the sequence is 17 ,
  2. write down a second equation in \(a\) and \(d\),
  3. find the value of \(a\) and the value of \(d\).
Edexcel C1 2011 January Q7
5 marks Moderate -0.8
7. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( - 1,0 )\). Given that $$\mathrm { f } ^ { \prime } ( x ) = 12 x ^ { 2 } - 8 x + 1$$ find \(\mathrm { f } ( x )\).
Edexcel C1 2011 January Q8
7 marks Moderate -0.8
8. The equation \(x ^ { 2 } + ( k - 3 ) x + ( 3 - 2 k ) = 0\), where \(k\) is a constant, has two distinct real roots.
  1. Show that \(k\) satisfies $$k ^ { 2 } + 2 k - 3 > 0$$
  2. Find the set of possible values of \(k\).
Edexcel C1 2011 January Q9
11 marks Moderate -0.8
9. The line \(L _ { 1 }\) has equation \(2 y - 3 x - k = 0\), where \(k\) is a constant. Given that the point \(A ( 1,4 )\) lies on \(L _ { 1 }\), find
  1. the value of \(k\),
  2. the gradient of \(L _ { 1 }\). The line \(L _ { 2 }\) passes through \(A\) and is perpendicular to \(L _ { 1 }\).
  3. Find an equation of \(L _ { 2 }\) giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The line \(L _ { 2 }\) crosses the \(x\)-axis at the point \(B\).
  4. Find the coordinates of \(B\).
  5. Find the exact length of \(A B\).
Edexcel C1 2011 January Q10
8 marks Moderate -0.3
10. (a) On the axes below, sketch the graphs of
  1. \(y = x ( x + 2 ) ( 3 - x )\)
  2. \(y = - \frac { 2 } { x }\) showing clearly the coordinates of all the points where the curves cross the coordinate axes.
    (b) Using your sketch state, giving a reason, the number of real solutions to the equation $$x ( x + 2 ) ( 3 - x ) + \frac { 2 } { x } = 0$$ \includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-13_994_997_1270_479}
Edexcel C1 2011 January Q11
12 marks Moderate -0.8
11. The curve \(C\) has equation $$y = \frac { 1 } { 2 } x ^ { 3 } - 9 x ^ { \frac { 3 } { 2 } } + \frac { 8 } { x } + 30 , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that the point \(P ( 4 , - 8 )\) lies on \(C\).
  3. Find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-15_113_129_2405_1816}
Edexcel C1 2012 January Q1
6 marks Easy -1.2
Given that \(y = x ^ { 4 } + 6 x ^ { \frac { 1 } { 2 } }\), find in their simplest form
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. \(\int y \mathrm {~d} x\)
Edexcel C1 2012 January Q3
6 marks Moderate -0.8
3. Find the set of values of \(x\) for which
  1. \(4 x - 5 > 15 - x\)
  2. \(x ( x - 4 ) > 12\)
Edexcel C1 2012 January Q4
6 marks Moderate -0.8
4. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) is defined by $$\begin{aligned} x _ { 1 } & = 1 \\ x _ { n + 1 } & = a x _ { n } + 5 , \quad n \geqslant 1 \end{aligned}$$ where \(a\) is a constant.
  1. Write down an expression for \(x _ { 2 }\) in terms of \(a\).
  2. Show that \(x _ { 3 } = a ^ { 2 } + 5 a + 5\) Given that \(x _ { 3 } = 41\)
  3. find the possible values of \(a\).
Edexcel C1 2012 January Q5
8 marks Moderate -0.8
5. The curve \(C\) has equation \(y = x ( 5 - x )\) and the line \(L\) has equation \(2 y = 5 x + 4\)
  1. Use algebra to show that \(C\) and \(L\) do not intersect.
  2. In the space on page 11, sketch \(C\) and \(L\) on the same diagram, showing the coordinates of the points at which \(C\) and \(L\) meet the axes.