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Edexcel C2 2011 June Q6
10 marks Moderate -0.8
  1. The second and third terms of a geometric series are 192 and 144 respectively.
For this series, find
  1. the common ratio,
  2. the first term,
  3. the sum to infinity,
  4. the smallest value of \(n\) for which the sum of the first \(n\) terms of the series exceeds 1000.
Edexcel C2 2011 June Q7
10 marks Moderate -0.3
  1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers in degrees to 1 decimal place, $$3 \sin \left( x + 45 ^ { \circ } \right) = 2$$
  2. Find, for \(0 \leqslant x < 2 \pi\), all the solutions of $$2 \sin ^ { 2 } x + 2 = 7 \cos x$$ giving your answers in radians.
    You must show clearly how you obtained your answers.
Edexcel C2 2011 June Q8
11 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c9758792-ca4c-4837-bd7c-e695fe0c0cdf-12_662_719_127_609} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A cuboid has a rectangular cross-section where the length of the rectangle is equal to twice its width, \(x \mathrm {~cm}\), as shown in Figure 2.
The volume of the cuboid is 81 cubic centimetres.
  1. Show that the total length, \(L \mathrm {~cm}\), of the twelve edges of the cuboid is given by $$L = 12 x + \frac { 162 } { x ^ { 2 } }$$
  2. Use calculus to find the minimum value of \(L\).
  3. Justify, by further differentiation, that the value of \(L\) that you have found is a minimum.
Edexcel C2 2011 June Q9
11 marks Moderate -0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c9758792-ca4c-4837-bd7c-e695fe0c0cdf-14_360_956_278_504} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The straight line with equation \(y = x + 4\) cuts the curve with equation \(y = - x ^ { 2 } + 2 x + 24\) at the points \(A\) and \(B\), as shown in Figure 3.
  1. Use algebra to find the coordinates of the points \(A\) and \(B\). The finite region \(R\) is bounded by the straight line and the curve and is shown shaded in Figure 3.
  2. Use calculus to find the exact area of \(R\).
Edexcel C2 2012 June Q1
4 marks Easy -1.2
Find expansion of
Edexcel C2 2012 June Q2
5 marks Standard +0.3
2. Find the values of \(x\) such that $$2 \log _ { 3 } x - \log _ { 3 } ( x - 2 ) = 2$$
Edexcel C2 2012 June Q3
11 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1ef99f0-4ad4-49d8-bee7-d5bb9cc84660-04_734_1262_237_315} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The circle \(C\) with centre \(T\) and radius \(r\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 16 y + 139 = 0$$
  1. Find the coordinates of the centre of \(C\).
  2. Show that \(r = 5\) The line \(L\) has equation \(x = 13\) and crosses \(C\) at the points \(P\) and \(Q\) as shown in Figure 1.
  3. Find the \(y\) coordinate of \(P\) and the \(y\) coordinate of \(Q\). Given that, to 3 decimal places, the angle \(P T Q\) is 1.855 radians,
  4. find the perimeter of the sector \(P T Q\).
Edexcel C2 2012 June Q4
6 marks Moderate -0.8
4. $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 10 x + 24$$
  1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Factorise f(x) completely.
Edexcel C2 2012 June Q5
12 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1ef99f0-4ad4-49d8-bee7-d5bb9cc84660-07_823_1081_267_404} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the line with equation \(y = 10 - x\) and the curve with equation \(y = 10 x - x ^ { 2 } - 8\) The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.
  1. Calculate the coordinates of \(A\) and the coordinates of \(B\). The shaded area \(R\) is bounded by the line and the curve, as shown in Figure 2.
  2. Calculate the exact area of \(R\).
Edexcel C2 2012 June Q6
7 marks Moderate -0.3
  1. Show that the equation $$\tan 2 x = 5 \sin 2 x$$ can be written in the form $$( 1 - 5 \cos 2 x ) \sin 2 x = 0$$
  2. Hence solve, for \(0 \leqslant x \leqslant 180 ^ { \circ }\), $$\tan 2 x = 5 \sin 2 x$$ giving your answers to 1 decimal place where appropriate.
    You must show clearly how you obtained your answers.
Edexcel C2 2012 June Q7
6 marks Moderate -0.8
7. $$y = \sqrt { } \left( 3 ^ { x } + x \right)$$
  1. Complete the table below, giving the values of \(y\) to 3 decimal places.
    \(x\)00.250.50.751
    \(y\)11.2512
  2. Use the trapezium rule with all the values of \(y\) from your table to find an approximation for the value of \(\int _ { 0 } ^ { 1 } \sqrt { } \left( 3 ^ { x } + x \right) \mathrm { d } x\) You must show clearly how you obtained your answer.
Edexcel C2 2012 June Q8
13 marks Moderate -0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1ef99f0-4ad4-49d8-bee7-d5bb9cc84660-11_305_446_223_749} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A manufacturer produces pain relieving tablets. Each tablet is in the shape of a solid circular cylinder with base radius \(x \mathrm {~mm}\) and height \(h \mathrm {~mm}\), as shown in Figure 3. Given that the volume of each tablet has to be \(60 \mathrm {~mm} ^ { 3 }\),
  1. express \(h\) in terms of \(x\),
  2. show that the surface area, \(A \mathrm {~mm} ^ { 2 }\), of a tablet is given by \(A = 2 \pi x ^ { 2 } + \frac { 120 } { x }\) The manufacturer needs to minimise the surface area \(A \mathrm {~mm} ^ { 2 }\), of a tablet.
  3. Use calculus to find the value of \(x\) for which \(A\) is a minimum.
  4. Calculate the minimum value of \(A\), giving your answer to the nearest integer.
  5. Show that this value of \(A\) is a minimum.
Edexcel C2 2012 June Q9
11 marks Moderate -0.5
A geometric series is \(a + a r + a r ^ { 2 } + \ldots\)
  1. Prove that the sum of the first \(n\) terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ The third and fifth terms of a geometric series are 5.4 and 1.944 respectively and all the terms in the series are positive. For this series find,
  2. the common ratio,
  3. the first term,
  4. the sum to infinity.
Edexcel C2 2013 June Q1
6 marks Moderate -0.8
  1. Using calculus, find the coordinates of the stationary point on the curve with equation
$$y = 2 x + 3 + \frac { 8 } { x ^ { 2 } } , \quad x > 0$$
Edexcel C2 2013 June Q2
5 marks Moderate -0.8
2. $$y = \frac { x } { \sqrt { ( 1 + x ) } }$$
  1. Complete the table below with the value of \(y\) corresponding to \(x = 1.3\), giving your answer to 4 decimal places.
    \(x\)11.11.21.31.41.5
    \(y\)0.70710.75910.80900.90370.9487
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an approximate value for $$\int _ { 1 } ^ { 1.5 } \frac { x } { \sqrt { } ( 1 + x ) } \mathrm { d } x$$ giving your answer to 3 decimal places.
    You must show clearly each stage of your working.
Edexcel C2 2013 June Q3
4 marks Easy -1.2
3. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 1 } { 2 } x \right) ^ { 8 }$$ giving each term in its simplest form.
Edexcel C2 2013 June Q4
9 marks Moderate -0.3
4. \(\mathrm { f } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } + b x + 4\), where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ) the remainder is 55
When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 9
  1. Find the value of \(a\) and the value of \(b\). Given that \(( 3 x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
  2. factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2013 June Q5
11 marks Standard +0.3
5. The first three terms of a geometric series are \(4 p , ( 3 p + 15 )\) and ( \(5 p + 20\) ) respectively, where \(p\) is a positive constant.
  1. Show that \(11 p ^ { 2 } - 10 p - 225 = 0\)
  2. Hence show that \(p = 5\)
  3. Find the common ratio of this series.
  4. Find the sum of the first ten terms of the series, giving your answer to the nearest integer.
Edexcel C2 2013 June Q6
9 marks Moderate -0.3
6. Given that \(\log _ { 3 } x = a\), find in terms of \(a\),
  1. \(\log _ { 3 } ( 9 x )\)
  2. \(\log _ { 3 } \left( \frac { x ^ { 5 } } { 81 } \right)\) giving each answer in its simplest form.
  3. Solve, for \(x\), $$\log _ { 3 } ( 9 x ) + \log _ { 3 } \left( \frac { x ^ { 5 } } { 81 } \right) = 3$$ giving your answer to 4 significant figures. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4f4eac7b-8908-480f-bb39-049944203fff-10_775_1605_221_159} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The line with equation \(y = 10\) cuts the curve with equation \(y = x ^ { 2 } + 2 x + 2\) at the points \(A\) and \(B\) as shown in Figure 1. The figure is not drawn to scale.
Edexcel C2 2013 June Q8
10 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4f4eac7b-8908-480f-bb39-049944203fff-12_556_1392_210_283} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the design for a triangular garden \(A B C\) where \(A B = 7 \mathrm {~m} , A C = 13 \mathrm {~m}\) and \(B C = 10 \mathrm {~m}\). Given that angle \(B A C = \theta\) radians,
  1. show that, to 3 decimal places, \(\theta = 0.865\) The point \(D\) lies on \(A C\) such that \(B D\) is an arc of the circle centre \(A\), radius 7 m .
    The shaded region \(S\) is bounded by the arc \(B D\) and the lines \(B C\) and \(D C\). The shaded region \(S\) will be sown with grass seed, to make a lawned area. Given that 50 g of grass seed are needed for each square metre of lawn,
  2. find the amount of grass seed needed, giving your answer to the nearest 10 g .
Edexcel C2 2013 June Q9
12 marks Standard +0.3
  1. Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$\sin \left( 2 \theta - 30 ^ { \circ } \right) + 1 = 0.4$$ giving your answers to 1 decimal place.
  2. Find all the values of \(x\), in the interval \(0 \leqslant x < 360 ^ { \circ }\), for which $$9 \cos ^ { 2 } x - 11 \cos x + 3 \sin ^ { 2 } x = 0$$ giving your answers to 1 decimal place. You must show clearly how you obtained your answers.
Edexcel C2 2013 June Q1
4 marks Moderate -0.8
  1. The first three terms of a geometric series are
$$18,12 \text { and } p$$ respectively, where \(p\) is a constant. Find
  1. the value of the common ratio of the series,
  2. the value of \(p\),
  3. the sum of the first 15 terms of the series, giving your answer to 3 decimal places.
Edexcel C2 2013 June Q2
5 marks Easy -1.2
2.
  1. Use the binomial theorem to find all the terms of the expansion of $$( 2 + 3 x ) ^ { 4 }$$ Give each term in its simplest form.
  2. Write down the expansion of $$( 2 - 3 x ) ^ { 4 }$$ in ascending powers of \(x\), giving each term in its simplest form.
Edexcel C2 2013 June Q3
9 marks Standard +0.3
3. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + a x + 18$$ where \(a\) is a constant. Given that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\),
  1. show that \(a = - 9\)
  2. factorise \(\mathrm { f } ( x )\) completely. Given that $$\mathrm { g } ( y ) = 2 \left( 3 ^ { 3 y } \right) - 5 \left( 3 ^ { 2 y } \right) - 9 \left( 3 ^ { y } \right) + 18$$
  3. find the values of \(y\) that satisfy \(\mathrm { g } ( y ) = 0\), giving your answers to 2 decimal places where appropriate.
Edexcel C2 2013 June Q4
7 marks Moderate -0.8
4. $$y = \frac { 5 } { \left( x ^ { 2 } + 1 \right) }$$
  1. Complete the table below, giving the missing value of \(y\) to 3 decimal places.
    \(x\)00.511.522.53
    \(y\)542.510.6900.5
    (1) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1c51b071-5cb1-4841-b031-80bde9027433-06_732_1118_826_411} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the region \(R\) which is bounded by the curve with equation \(y = \frac { 5 } { \left( x ^ { 2 } + 1 \right) }\),
    the \(x\)-axis and the lines \(x = 0\) and \(x = 3\) the \(x\)-axis and the lines \(x = 0\) and \(x = 3\)
  2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximate value for the area of \(R\).
  3. Use your answer to part (b) to find an approximate value for $$\int _ { 0 } ^ { 3 } \left( 4 + \frac { 5 } { \left( x ^ { 2 } + 1 \right) } \right) d x$$ giving your answer to 2 decimal places.