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Edexcel C2 2007 June Q5
9 marks Moderate -0.3
5. The curve \(C\) has equation $$y = x \sqrt { } \left( x ^ { 3 } + 1 \right) , \quad 0 \leqslant x \leqslant 2$$
  1. Complete the table below, giving the values of \(y\) to 3 decimal places at \(x = 1\) and \(x = 1.5\).
    \(x\)00.511.52
    \(y\)00.5306
  2. Use the trapezium rule, with all the \(y\) values from your table, to find an approximation for the value of \(\int _ { 0 } ^ { 2 } x \sqrt { } \left( x ^ { 3 } + 1 \right) \mathrm { d } x\), giving your answer to 3 significant figures. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{22ebc302-765c-4734-b312-b286ccb20be9-06_1110_644_1119_648} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the curve \(C\) with equation \(y = x \sqrt { } \left( x ^ { 3 } + 1 \right) , 0 \leqslant x \leqslant 2\), and the straight line segment \(l\), which joins the origin and the point \(( 2,6 )\). The finite region \(R\) is bounded by \(C\) and \(l\).
  3. Use your answer to part (b) to find an approximation for the area of \(R\), giving your answer to 3 significant figures.
    (3) \section*{LU}
Edexcel C2 2007 June Q6
6 marks Moderate -0.3
6. (a) Find, to 3 significant figures, the value of \(x\) for which \(8 ^ { x } = 0.8\).
(b) Solve the equation $$2 \log _ { 3 } x - \log _ { 3 } 7 x = 1$$
Edexcel C2 2007 June Q7
9 marks Moderate -0.5
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{22ebc302-765c-4734-b312-b286ccb20be9-09_778_988_223_500} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The points \(A\) and \(B\) lie on a circle with centre \(P\), as shown in Figure 3.
The point \(A\) has coordinates \(( 1 , - 2 )\) and the mid-point \(M\) of \(A B\) has coordinates \(( 3,1 )\). The line \(l\) passes through the points \(M\) and \(P\).
  1. Find an equation for \(l\). Given that the \(x\)-coordinate of \(P\) is 6 ,
  2. use your answer to part (a) to show that the \(y\)-coordinate of \(P\) is - 1 ,
  3. find an equation for the circle.
Edexcel C2 2007 June Q8
9 marks Moderate -0.3
8. A trading company made a profit of \(\pounds 50000\) in 2006 (Year 1). A model for future trading predicts that profits will increase year by year in a geometric sequence with common ratio \(r , r > 1\). The model therefore predicts that in 2007 (Year 2) a profit of \(\pounds 50000 r\) will be made.
  1. Write down an expression for the predicted profit in Year \(n\). The model predicts that in Year \(n\), the profit made will exceed \(\pounds 200000\).
  2. Show that \(n > \frac { \log 4 } { \log r } + 1\). Using the model with \(r = 1.09\),
  3. find the year in which the profit made will first exceed \(\pounds 200000\),
  4. find the total of the profits that will be made by the company over the 10 years from 2006 to 2015 inclusive, giving your answer to the nearest \(\pounds 10000\).
Edexcel C2 2007 June Q9
10 marks Moderate -0.8
9. (a) Sketch, for \(0 \leqslant x \leqslant 2 \pi\), the graph of \(y = \sin \left( x + \frac { \pi } { 6 } \right)\).
(b) Write down the exact coordinates of the points where the graph meets the coordinate axes.
(c) Solve, for \(0 \leqslant x \leqslant 2 \pi\), the equation $$\sin \left( x + \frac { \pi } { 6 } \right) = 0.65$$ giving your answers in radians to 2 decimal places.
Edexcel C2 2007 June Q10
11 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{22ebc302-765c-4734-b312-b286ccb20be9-15_538_529_205_744} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a solid brick in the shape of a cuboid measuring \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\). The total surface area of the brick is \(600 \mathrm {~cm} ^ { 2 }\).
  1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the brick is given by $$V = 200 x - \frac { 4 x ^ { 3 } } { 3 }$$ Given that \(x\) can vary,
  2. use calculus to find the maximum value of \(V\), giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
  3. Justify that the value of \(V\) you have found is a maximum.
Edexcel C2 2008 June Q1
6 marks Moderate -0.8
1. $$f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 39 x + 20$$
  1. Use the factor theorem to show that \(( x + 4 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Factorise f ( \(x\) ) completely.
Edexcel C2 2008 June Q2
6 marks Easy -1.2
2. $$y = \sqrt { } \left( 5 ^ { x } + 2 \right)$$
  1. Complete the table below, giving the values of \(y\) to 3 decimal places.
    \(x\)00.511.52
    \(y\)2.6463.630
  2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximation for the value of \(\int _ { 0 } ^ { 2 } \sqrt { } \left( 5 ^ { x } + 2 \right) \mathrm { d } x\).
Edexcel C2 2008 June Q3
6 marks Moderate -0.3
3. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + a x ) ^ { 10 }\), where \(a\) is a non-zero constant. Give each term in its simplest form. Given that, in this expansion, the coefficient of \(x ^ { 3 }\) is double the coefficient of \(x ^ { 2 }\),
(b) find the value of \(a\).
Edexcel C2 2008 June Q4
6 marks Moderate -0.3
4. (a) Find, to 3 significant figures, the value of \(x\) for which \(5 ^ { x } = 7\).
(b) Solve the equation \(5 ^ { 2 x } - 12 \left( 5 ^ { x } \right) + 35 = 0\).
Edexcel C2 2008 June Q5
9 marks Moderate -0.3
5. The circle \(C\) has centre \(( 3,1 )\) and passes through the point \(P ( 8,3 )\).
  1. Find an equation for \(C\).
  2. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C2 2008 June Q6
9 marks Standard +0.8
6. A geometric series has first term 5 and common ratio \(\frac { 4 } { 5 }\). Calculate
  1. the 20th term of the series, to 3 decimal places,
  2. the sum to infinity of the series. Given that the sum to \(k\) terms of the series is greater than 24.95,
  3. show that \(k > \frac { \log 0.002 } { \log 0.8 }\),
  4. find the smallest possible value of \(k\).
Edexcel C2 2008 June Q7
12 marks Moderate -0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0454f5f6-b5ee-40b1-bc6a-ff8aeb06a455-09_817_1029_205_484} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows \(A B C\), a sector of a circle with centre \(A\) and radius 7 cm .
Given that the size of \(\angle B A C\) is exactly 0.8 radians, find
  1. the length of the arc \(B C\),
  2. the area of the sector \(A B C\). The point \(D\) is the mid-point of \(A C\). The region \(R\), shown shaded in Figure 1, is bounded by \(C D , D B\) and the arc \(B C\). Find
  3. the perimeter of \(R\), giving your answer to 3 significant figures,
  4. the area of \(R\), giving your answer to 3 significant figures.
Edexcel C2 2008 June Q8
11 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0454f5f6-b5ee-40b1-bc6a-ff8aeb06a455-11_668_1267_292_367} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = 10 + 8 x + x ^ { 2 } - x ^ { 3 }\).
The curve has a maximum turning point \(A\).
  1. Using calculus, show that the \(x\)-coordinate of \(A\) is 2 . The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(y\)-axis and the line from \(O\) to \(A\), where \(O\) is the origin.
  2. Using calculus, find the exact area of \(R\).
Edexcel C2 2008 June Q9
10 marks Moderate -0.8
9. Solve, for \(0 \leqslant x < 360 ^ { \circ }\),
  1. \(\quad \sin \left( x - 20 ^ { \circ } \right) = \frac { 1 } { \sqrt { 2 } }\)
  2. \(\cos 3 x = - \frac { 1 } { 2 }\)
Edexcel C2 2009 June Q1
5 marks Moderate -0.8
  1. Use calculus to find the value of
$$\int _ { 1 } ^ { 4 } ( 2 x + 3 \sqrt { } x ) d x$$
Edexcel C2 2009 June Q2
6 marks Moderate -0.8
2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 + k x ) ^ { 7 }$$ where \(k\) is a constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 2 }\) is 6 times the coefficient of \(x\),
(b) find the value of \(k\).
Edexcel C2 2009 June Q3
6 marks Moderate -0.5
3. $$f ( x ) = ( 3 x - 2 ) ( x - k ) - 8$$ where \(k\) is a constant.
  1. Write down the value of \(\mathrm { f } ( k )\). When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is 4
  2. Find the value of \(k\).
  3. Factorise f(x) completely.
Edexcel C2 2009 June Q4
8 marks Moderate -0.8
4. (a) Complete the table below, giving values of \(\sqrt { } \left( 2 ^ { x } + 1 \right)\) to 3 decimal places.
\(x\)00.511.522.53
\(\sqrt { } \left( 2 ^ { x } + 1 \right)\)1.4141.5541.7321.9573
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{78a994ba-50c5-434f-a060-9596edb505cd-05_653_595_616_676} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the region \(R\) which is bounded by the curve with equation \(y = \sqrt { } \left( 2 ^ { x } + 1 \right)\), the \(x\)-axis and the lines \(x = 0\) and \(x = 3\) (b) Use the trapezium rule, with all the values from your table, to find an approximation for the area of \(R\).
(c) By reference to the curve in Figure 1 state, giving a reason, whether your approximation in part (b) is an overestimate or an underestimate for the area of \(R\).
Edexcel C2 2009 June Q5
9 marks Moderate -0.3
  1. The third term of a geometric sequence is 324 and the sixth term is 96
    1. Show that the common ratio of the sequence is \(\frac { 2 } { 3 }\)
    2. Find the first term of the sequence.
    3. Find the sum of the first 15 terms of the sequence.
    4. Find the sum to infinity of the sequence.
Edexcel C2 2009 June Q6
11 marks Standard +0.3
6. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 4 y = 12$$
  1. Find the centre and the radius of \(C\). The point \(P ( - 1,1 )\) and the point \(Q ( 7 , - 5 )\) both lie on \(C\).
  2. Show that \(P Q\) is a diameter of \(C\). The point \(R\) lies on the positive \(y\)-axis and the angle \(P R Q = 90 ^ { \circ }\).
  3. Find the coordinates of \(R\).
Edexcel C2 2009 June Q7
10 marks Moderate -0.3
7. (i) Solve, for \(- 180 ^ { \circ } \leqslant \theta < 180 ^ { \circ }\), $$( 1 + \tan \theta ) ( 5 \sin \theta - 2 ) = 0$$ (ii) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \sin x = 3 \tan x .$$
Edexcel C2 2009 June Q8
7 marks Moderate -0.3
8. (a) Find the value of \(y\) such that $$\log _ { 2 } y = - 3$$ (b) Find the values of \(x\) such that $$\frac { \log _ { 2 } 32 + \log _ { 2 } 16 } { \log _ { 2 } x } = \log _ { 2 } x$$
Edexcel C2 2009 June Q9
13 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{78a994ba-50c5-434f-a060-9596edb505cd-14_554_454_212_744} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height \(h \mathrm {~cm}\). The cross section is a sector of a circle. The sector has radius \(r \mathrm {~cm}\) and angle 1 radian. The volume of the box is \(300 \mathrm {~cm} ^ { 3 }\).
  1. Show that the surface area of the box, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = r ^ { 2 } + \frac { 1800 } { r }$$
  2. Use calculus to find the value of \(r\) for which \(S\) is stationary.
  3. Prove that this value of \(r\) gives a minimum value of \(S\).
  4. Find, to the nearest \(\mathrm { cm } ^ { 2 }\), this minimum value of \(S\).
Edexcel C2 2010 June Q1
6 marks Easy -1.2
1. $$y = 3 ^ { x } + 2 x$$
  1. Complete the table below, giving the values of \(y\) to 2 decimal places.
    \(x\)00.20.40.60.81
    \(y\)11.655
  2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximate value for \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 2 x \right) d x\).