Questions — Edexcel C2 (579 questions)

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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
  1. 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. (i) Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\)
$$\sin \left( 2 \theta - 30 ^ { \circ } \right) + 1 = 0.4$$ giving your answers to 1 decimal place.
(ii) 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. (a) Use the binomial theorem to find all the terms of the expansion of $$( 2 + 3 x ) ^ { 4 }$$ Give each term in its simplest form.
(b) 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.
Edexcel C2 2013 June Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c51b071-5cb1-4841-b031-80bde9027433-08_598_1297_118_319} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a plan view of a garden.
The plan of the garden \(A B C D E A\) consists of a triangle \(A B E\) joined to a sector \(B C D E\) of a circle with radius 12 m and centre \(B\).
The points \(A , B\) and \(C\) lie on a straight line with \(A B = 23 \mathrm {~m}\) and \(B C = 12 \mathrm {~m}\).
Given that the size of angle \(A B E\) is exactly 0.64 radians, find
  1. the area of the garden, giving your answer in \(\mathrm { m } ^ { 2 }\), to 1 decimal place,
  2. the perimeter of the garden, giving your answer in metres, to 1 decimal place.
Edexcel C2 2013 June Q6
8 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c51b071-5cb1-4841-b031-80bde9027433-10_697_1182_210_386} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = x ( x + 4 ) ( x - 2 )$$ The curve \(C\) crosses the \(x\)-axis at the origin \(O\) and at the points \(A\) and \(B\).
  1. Write down the \(x\)-coordinates of the points \(A\) and \(B\). The finite region, shown shaded in Figure 3, is bounded by the curve \(C\) and the \(x\)-axis.
  2. Use integration to find the total area of the finite region shown shaded in Figure 3.
Edexcel C2 2013 June Q7
7 marks Moderate -0.3
7. (i) Find the exact value of \(x\) for which $$\log _ { 2 } ( 2 x ) = \log _ { 2 } ( 5 x + 4 ) - 3$$ (ii) Given that $$\log _ { a } y + 3 \log _ { a } 2 = 5$$ express \(y\) in terms of \(a\).
Give your answer in its simplest form.
Edexcel C2 2013 June Q8
11 marks Standard +0.3
8. (i) Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), $$\tan \left( x - 40 ^ { \circ } \right) = 1.5$$ giving your answers to 1 decimal place.
(ii) (a) Show that the equation $$\sin \theta \tan \theta = 3 \cos \theta + 2$$ can be written in the form $$4 \cos ^ { 2 } \theta + 2 \cos \theta - 1 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$\sin \theta \tan \theta = 3 \cos \theta + 2$$ showing each stage of your working.
Edexcel C2 2013 June Q9
9 marks Moderate -0.3
  1. The curve with equation
$$y = x ^ { 2 } - 32 \sqrt { } ( x ) + 20 , \quad x > 0$$ has a stationary point \(P\). Use calculus
  1. to find the coordinates of \(P\),
  2. to determine the nature of the stationary point \(P\).
Edexcel C2 2013 June Q10
6 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c51b071-5cb1-4841-b031-80bde9027433-16_723_979_207_495} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The circle \(C\) has radius 5 and touches the \(y\)-axis at the point \(( 0,9 )\), as shown in Figure 4.
  1. Write down an equation for the circle \(C\), that is shown in Figure 4. A line through the point \(P ( 8 , - 7 )\) is a tangent to the circle \(C\) at the point \(T\).
  2. Find the length of \(P T\).
Edexcel C2 2014 June Q1
4 marks Moderate -0.8
  1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 1 + \frac { 3 x } { 2 } \right) ^ { 8 }$$ giving each term in its simplest form.
Edexcel C2 2014 June Q2
8 marks Standard +0.3
2. A geometric series has first term \(a\), where \(a \neq 0\), and common ratio \(r\). The sum to infinity of this series is 6 times the first term of the series.
  1. Show that \(r = \frac { 5 } { 6 }\) Given that the fourth term of this series is 62.5
  2. find the value of \(a\),
  3. find the difference between the sum to infinity and the sum of the first 30 terms, giving your answer to 3 significant figures.
Edexcel C2 2014 June Q3
5 marks Easy -1.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07cc9ed-a820-46c8-a3a3-3c780cf20fa7-05_821_1273_118_338} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt { } ( 2 x - 1 ) , x \geqslant 0.5\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines with equations \(x = 2\) and \(x = 10\). The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { } ( 2 x - 1 )\).
\(x\)246810
\(y\)\(\sqrt { } 3\)\(\sqrt { } 11\)\(\sqrt { } 19\)
  1. Complete the table with the values of \(y\) corresponding to \(x = 4\) and \(x = 8\).
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to find an approximate value for the area of \(R\), giving your answer to 2 decimal places.
  3. State whether your approximate value in part (b) is an overestimate or an underestimate for the area of \(R\).
Edexcel C2 2014 June Q4
7 marks Moderate -0.3
4. \(\mathrm { f } ( x ) = - 4 x ^ { 3 } + a x ^ { 2 } + 9 x - 18\), where \(a\) is a constant. Given that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\),
  1. find the value of \(a\),
  2. factorise \(\mathrm { f } ( x )\) completely,
  3. find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ).
Edexcel C2 2014 June Q5
6 marks Moderate -0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07cc9ed-a820-46c8-a3a3-3c780cf20fa7-08_566_725_127_614} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the shape \(A B C D E A\) which consists of a right-angled triangle \(B C D\) joined to a sector \(A B D E A\) of a circle with radius 7 cm and centre \(B\). \(A , B\) and \(C\) lie on a straight line with \(A B = 7 \mathrm {~cm}\).
Given that the size of angle \(A B D\) is exactly 2.1 radians,
  1. find, in cm, the length of the arc \(D E A\),
  2. find, in cm, the perimeter of the shape \(A B C D E A\), giving your answer to 1 decimal place.