Questions C2 (1550 questions)

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Edexcel C2 2016 June Q2
6 marks Moderate -0.8
2. The curve \(C\) has equation $$y = 8 - 2 ^ { x - 1 } , \quad 0 \leqslant x \leqslant 4$$
  1. Complete the table below with the value of \(y\) corresponding to \(x = 1\)
    \(x\)01234
    \(y\)7.5640
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to find an approximate value for \(\int _ { 0 } ^ { 4 } \left( 8 - 2 ^ { x - 1 } \right) \mathrm { d } x\) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{582cda45-80fc-43a8-90e6-1cae08cb1534-03_650_606_1016_671} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = 8 - 2 ^ { x - 1 } , \quad 0 \leqslant x \leqslant 4\) The curve \(C\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).
    The region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\) and the straight line through \(A\) and \(B\).
  3. Use your answer to part (b) to find an approximate value for the area of \(R\).
Edexcel C2 2016 June Q3
8 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{582cda45-80fc-43a8-90e6-1cae08cb1534-05_791_917_121_484} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale The circle \(C\) has centre \(P ( 7,8 )\) and passes through the point \(Q ( 10,13 )\), as shown in Figure 2.
  1. Find the length \(P Q\), giving your answer as an exact value.
  2. Hence write down an equation for \(C\). The line \(l\) is a tangent to \(C\) at the point \(Q\), as shown in Figure 2.
  3. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C2 2016 June Q4
8 marks Moderate -0.8
4. $$f ( x ) = 6 x ^ { 3 } + 13 x ^ { 2 } - 4$$
  1. Use the remainder theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 3\) ).
  2. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
  3. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2016 June Q5
9 marks Moderate -0.3
  1. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$( 2 - 9 x ) ^ { 4 }$$ giving each term in its simplest form. $$f ( x ) = ( 1 + k x ) ( 2 - 9 x ) ^ { 4 } , \text { where } k \text { is a constant }$$ The expansion, in ascending powers of \(x\), of \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 2 }\) is $$A - 232 x + B x ^ { 2 }$$ where \(A\) and \(B\) are constants.
(b) Write down the value of \(A\).
(c) Find the value of \(k\).
(d) Hence find the value of \(B\).
Edexcel C2 2016 June Q6
9 marks Moderate -0.3
6. (i) Solve, for \(- \pi < \theta \leqslant \pi\), $$1 - 2 \cos \left( \theta - \frac { \pi } { 5 } \right) = 0$$ giving your answers in terms of \(\pi\).
(ii) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \cos ^ { 2 } x + 7 \sin x - 2 = 0$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C2 2016 June Q7
6 marks Moderate -0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{582cda45-80fc-43a8-90e6-1cae08cb1534-12_563_812_244_630} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 3 x - x ^ { \frac { 3 } { 2 } } , \quad x \geqslant 0$$ The finite region \(S\), bounded by the \(x\)-axis and the curve, is shown shaded in Figure 3.
  1. Find $$\int \left( 3 x - x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x$$
  2. Hence find the area of \(S\).
Edexcel C2 2016 June Q8
7 marks Moderate -0.3
8. (i) Given that $$\log _ { 3 } ( 3 b + 1 ) - \log _ { 3 } ( a - 2 ) = - 1 , \quad a > 2$$ express \(b\) in terms of \(a\).
(ii) Solve the equation $$2 ^ { 2 x + 5 } - 7 \left( 2 ^ { x } \right) = 0$$ giving your answer to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C2 2016 June Q9
15 marks Standard +0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{582cda45-80fc-43a8-90e6-1cae08cb1534-15_831_1167_118_513} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a plan view of a sheep enclosure.
The enclosure \(A B C D E A\), as shown in Figure 4, consists of a rectangle \(B C D E\) joined to an equilateral triangle \(B F A\) and a sector \(F E A\) of a circle with radius \(x\) metres and centre \(F\). The points \(B , F\) and \(E\) lie on a straight line with \(F E = x\) metres and \(10 \leqslant x \leqslant 25\)
  1. Find, in \(\mathrm { m } ^ { 2 }\), the exact area of the sector \(F E A\), giving your answer in terms of \(x\), in its simplest form. Given that \(B C = y\) metres, where \(y > 0\), and the area of the enclosure is \(1000 \mathrm {~m} ^ { 2 }\),
  2. show that $$y = \frac { 500 } { x } - \frac { x } { 24 } ( 4 \pi + 3 \sqrt { 3 } )$$
  3. Hence show that the perimeter \(P\) metres of the enclosure is given by $$P = \frac { 1000 } { x } + \frac { x } { 12 } ( 4 \pi + 36 - 3 \sqrt { 3 } )$$
  4. Use calculus to find the minimum value of \(P\), giving your answer to the nearest metre.
  5. Justify, by further differentiation, that the value of \(P\) you have found is a minimum.
Edexcel C2 2017 June Q1
4 marks Easy -1.2
  1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 3 - \frac { 1 } { 3 } x \right) ^ { 5 }$$ giving each term in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{752efc6c-8d0e-46a6-b75d-5125956969d8-03_104_107_2631_1774}
Edexcel C2 2017 June Q2
4 marks Standard +0.3
2. In the triangle \(A B C , A B = 16 \mathrm {~cm} , A C = 13 \mathrm {~cm}\), angle \(A B C = 50 ^ { \circ }\) and angle \(B C A = x ^ { \circ }\) Find the two possible values for \(x\), giving your answers to one decimal place. \includegraphics[max width=\textwidth, alt={}, center]{752efc6c-8d0e-46a6-b75d-5125956969d8-05_104_107_2631_1774}
Edexcel C2 2017 June Q3
6 marks Moderate -0.8
3. (a) \(\quad y = 5 ^ { x } + \log _ { 2 } ( x + 1 ) , \quad 0 \leqslant x \leqslant 2\) Complete the table below, by giving the value of \(y\) when \(x = 1\)
\(x\)00.511.52
\(y\)12.82112.50226.585
(b) Use the trapezium rule, with all the values of \(y\) from the completed table, to find an approximate value for $$\int _ { 0 } ^ { 2 } \left( 5 ^ { x } + \log _ { 2 } ( x + 1 ) \right) \mathrm { d } x$$ giving your answer to 2 decimal places.
(c) Use your answer to part (b) to find an approximate value for $$\int _ { 0 } ^ { 2 } \left( 5 + 5 ^ { x } + \log _ { 2 } ( x + 1 ) \right) d x$$ giving your answer to 2 decimal places.
Edexcel C2 2017 June Q4
8 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{752efc6c-8d0e-46a6-b75d-5125956969d8-10_508_960_212_477} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Diagram not drawn to scale Figure 1 is a sketch representing the cross-section of a large tent \(A B C D E F\). \(A B\) and \(D E\) are line segments of equal length.
Angle \(F A B\) and angle \(D E F\) are equal. \(F\) is the midpoint of the straight line \(A E\) and \(F C\) is perpendicular to \(A E\). \(B C D\) is an arc of a circle of radius 3.5 m with centre at \(F\).
It is given that $$\begin{aligned} A F & = F E = 3.7 \mathrm {~m} \\ B F & = F D = 3.5 \mathrm {~m} \\ \text { angle } B F D & = 1.77 \text { radians } \end{aligned}$$ Find
  1. the length of the arc \(B C D\) in metres to 2 decimal places,
  2. the area of the sector \(F B C D\) in \(\mathrm { m } ^ { 2 }\) to 2 decimal places,
  3. the total area of the cross-section of the tent in \(\mathrm { m } ^ { 2 }\) to 2 decimal places.
Edexcel C2 2017 June Q5
7 marks Moderate -0.8
5. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 10 x + 6 y + 30 = 0$$ Find
  1. the coordinates of the centre of \(C\),
  2. the radius of \(C\),
  3. the \(y\) coordinates of the points where the circle \(C\) crosses the line with equation \(x = 4\), giving your answers as simplified surds.
Edexcel C2 2017 June Q6
9 marks Standard +0.3
6. $$f ( x ) = - 6 x ^ { 3 } - 7 x ^ { 2 } + 40 x + 21$$
  1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\)
  2. Factorise f(x) completely.
  3. Hence solve the equation $$6 \left( 2 ^ { 3 y } \right) + 7 \left( 2 ^ { 2 y } \right) = 40 \left( 2 ^ { y } \right) + 21$$ giving your answer to 2 decimal places.
Edexcel C2 2017 June Q7
7 marks Moderate -0.3
7. (i) \(2 \log ( x + a ) = \log \left( 16 a ^ { 6 } \right)\), where \(a\) is a positive constant Find \(x\) in terms of \(a\), giving your answer in its simplest form.
(ii) \(\quad \log _ { 3 } ( 9 y + b ) - \log _ { 3 } ( 2 y - b ) = 2\), where \(b\) is a positive constant Find \(y\) in terms of \(b\), giving your answer in its simplest form.
Edexcel C2 2017 June Q8
8 marks Moderate -0.3
8. (a) Show that the equation $$\cos ^ { 2 } x = 8 \sin ^ { 2 } x - 6 \sin x$$ can be written in the form $$( 3 \sin x - 1 ) ^ { 2 } = 2$$ (b) Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$\cos ^ { 2 } x = 8 \sin ^ { 2 } x - 6 \sin x$$ giving your answers to 2 decimal places.
Edexcel C2 2017 June Q9
12 marks Standard +0.3
9. The first three terms of a geometric sequence are $$7 k - 5,5 k - 7,2 k + 10$$ where \(k\) is a constant.
  1. Show that \(11 k ^ { 2 } - 130 k + 99 = 0\) Given that \(k\) is not an integer,
  2. show that \(k = \frac { 9 } { 11 }\) For this value of \(k\),
    1. evaluate the fourth term of the sequence, giving your answer as an exact fraction,
    2. evaluate the sum of the first ten terms of the sequence.
Edexcel C2 2017 June Q10
10 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{752efc6c-8d0e-46a6-b75d-5125956969d8-28_761_1120_258_411} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = 4 x ^ { 3 } + 9 x ^ { 2 } - 30 x - 8 , \quad - 0.5 \leqslant x \leqslant 2.2$$ The curve has a turning point at the point \(A\).
  1. Using calculus, show that the \(x\) coordinate of \(A\) is 1 The curve crosses the \(x\)-axis at the points \(B ( 2,0 )\) and \(C \left( - \frac { 1 } { 4 } , 0 \right)\) The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the line \(A B\), and the \(x\)-axis.
  2. Use integration to find the area of the finite region \(R\), giving your answer to 2 decimal places.
Edexcel C2 2018 June Q1
5 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8daf56fa-bfce-454e-bbb8-fecd8170d77e-02_575_812_214_566} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = \frac { ( x + 2 ) ^ { \frac { 3 } { 2 } } } { 4 } , \quad x \geqslant - 2$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line with equation \(x = 10\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { ( x + 2 ) ^ { \frac { 3 } { 2 } } } { 4 }\)
  1. Complete the table, giving values of \(y\) corresponding to \(x = 2\) and \(x = 6\)
    \(x\)- 22610
    \(y\)0\(6 \sqrt { } 3\)
  2. Use the trapezium rule, with all the values of \(y\) from the completed table, to find an approximate value for the area of \(R\), giving your answer to 3 decimal places.
Edexcel C2 2018 June Q2
7 marks Moderate -0.8
  1. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of
$$( 2 + k x ) ^ { 7 }$$ where \(k\) is a non-zero constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 3 }\) in this expansion is 1890
(b) find the value of \(k\).
Edexcel C2 2018 June Q3
8 marks Moderate -0.3
3. $$f ( x ) = 24 x ^ { 3 } + A x ^ { 2 } - 3 x + B$$ where \(A\) and \(B\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is 30
  1. Show that \(A + 4 B = 114\) Given also that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\),
  2. find another equation in \(A\) and \(B\).
  3. Find the value of \(A\) and the value of \(B\).
  4. Hence find a quadratic factor of \(\mathrm { f } ( x )\).
Edexcel C2 2018 June Q4
9 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8daf56fa-bfce-454e-bbb8-fecd8170d77e-10_310_716_214_621} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Not to scale Figure 2 shows a flag \(X Y W Z X\). The flag consists of a triangle \(X Y Z\) joined to a sector \(Z Y W\) of a circle with radius 5 cm and centre \(Y\). The angle of the sector, angle \(Z Y W\), is 0.7 radians. The points \(X , Y\) and \(W\) lie on a straight line with \(X Y = 7 \mathrm {~cm}\) and \(Y W = 5 \mathrm {~cm}\). Find
  1. the area of the sector \(Z Y W\) in \(\mathrm { cm } ^ { 2 }\),
  2. the area of the flag, in \(\mathrm { cm } ^ { 2 }\), to 2 decimal places,
  3. the length of the perimeter, \(X Y W Z X\), of the flag, in cm to 2 decimal places.
Edexcel C2 2018 June Q5
10 marks Moderate -0.8
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } - 2 x + 14 y = 0$$ Find
  1. the coordinates of the centre of \(C\),
  2. the exact value of the radius of \(C\),
  3. the \(y\) coordinates of the points where the circle \(C\) crosses the \(y\)-axis.
  4. Find an equation of the tangent to \(C\) at the point ( 2,0 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C2 2018 June Q6
7 marks Moderate -0.8
  1. A geometric series with common ratio \(r = - 0.9\) has sum to infinity 10000 For this series,
    1. find the first term,
    2. find the fifth term,
    3. find the sum of the first twelve terms, giving this answer to the nearest integer.
Edexcel C2 2018 June Q7
8 marks Moderate -0.3
7. (i) Find the value of \(y\) for which $$1.01 ^ { y - 1 } = 500$$ Give your answer to 2 decimal places.
(ii) Given that $$2 \log _ { 4 } ( 3 x + 5 ) = \log _ { 4 } ( 3 x + 8 ) + 1 , \quad x > - \frac { 5 } { 3 }$$
  1. show that $$9 x ^ { 2 } + 18 x - 7 = 0$$
  2. Hence solve the equation $$2 \log _ { 4 } ( 3 x + 5 ) = \log _ { 4 } ( 3 x + 8 ) + 1 , \quad x > - \frac { 5 } { 3 }$$ DO NOTI WRITE IN THIS AREA