Questions — Edexcel C2 (476 questions)

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Edexcel C2 2013 June Q9
  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
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
  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
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
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
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
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.
Edexcel C2 2014 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07cc9ed-a820-46c8-a3a3-3c780cf20fa7-09_796_1132_121_397} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 1 } { 8 } x ^ { 3 } + \frac { 3 } { 4 } x ^ { 2 } , \quad x \in \mathbb { R }$$ The curve \(C\) has a maximum turning point at the point \(A\) and a minimum turning point at the origin \(O\). The line \(l\) touches the curve \(C\) at the point \(A\) and cuts the curve \(C\) at the point \(B\). The \(x\) coordinate of \(A\) is - 4 and the \(x\) coordinate of \(B\) is 2 . The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\) and the line \(l\).
Use integration to find the area of the finite region \(R\).
Edexcel C2 2014 June Q7
7. (i) Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$\frac { \sin 2 \theta } { ( 4 \sin 2 \theta - 1 ) } = 1$$ giving your answers to 1 decimal place.
(ii) Solve, for \(0 \leqslant x < 2 \pi\), the equation $$5 \sin ^ { 2 } x - 2 \cos x - 5 = 0$$ giving your answers to 2 decimal places. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C2 2014 June Q8
8. (i) Solve $$5 ^ { y } = 8$$ giving your answer to 3 significant figures.
(ii) Use algebra to find the values of \(x\) for which $$\log _ { 2 } ( x + 15 ) - 4 = \frac { 1 } { 2 } \log _ { 2 } x$$
Edexcel C2 2014 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07cc9ed-a820-46c8-a3a3-3c780cf20fa7-14_899_686_212_639} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows the plan of a pool. The shape of the pool \(A B C D E F A\) consists of a rectangle \(B C E F\) joined to an equilateral triangle \(B F A\) and a semi-circle \(C D E\), as shown in Figure 4. Given that \(A B = x\) metres, \(E F = y\) metres, and the area of the pool is \(50 \mathrm {~m} ^ { 2 }\),
  1. show that $$y = \frac { 50 } { x } - \frac { x } { 8 } ( \pi + 2 \sqrt { } 3 )$$
  2. Hence show that the perimeter, \(P\) metres, of the pool is given by $$P = \frac { 100 } { x } + \frac { x } { 4 } ( \pi + 8 - 2 \sqrt { } 3 )$$
  3. Use calculus to find the minimum value of \(P\), giving your answer to 3 significant figures.
  4. Justify, by further differentiation, that the value of \(P\) that you have found is a minimum.
Edexcel C2 2014 June Q10
  1. The circle \(C\), with centre \(A\), passes through the point \(P\) with coordinates ( \(- 9,8\) ) and the point \(Q\) with coordinates \(( 15 , - 10 )\).
Given that \(P Q\) is a diameter of the circle \(C\),
  1. find the coordinates of \(A\),
  2. find an equation for \(C\). A point \(R\) also lies on the circle \(C\).
    Given that the length of the chord \(P R\) is 20 units,
  3. find the length of the shortest distance from \(A\) to the chord \(P R\). Give your answer as a surd in its simplest form.
  4. Find the size of the angle \(A R Q\), giving your answer to the nearest 0.1 of a degree.
Edexcel C2 2014 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6b490c0-80c4-4e15-b587-ac052ee27db7-02_738_1257_274_340} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt { } \left( x ^ { 2 } + 1 \right) , x \geqslant 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) The table below shows corresponding values for \(x\) and \(y\) for \(y = \sqrt { } \left( x ^ { 2 } + 1 \right)\).
\(x\)11.251.51.752
\(y\)1.4141.8032.0162.236
  1. Complete the table above, giving the missing value of \(y\) to 3 decimal places.
  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.
Edexcel C2 2014 June Q2
2. $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 4 x + 4$$
  1. Use the factor theorem to show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2014 June Q3
3. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 - 3 x ) ^ { 6 }$$ giving each term in its simplest form.
(b) Hence, or otherwise, find the first 3 terms, in ascending powers of \(x\), of the expansion of $$\left( 1 + \frac { x } { 2 } \right) ( 2 - 3 x ) ^ { 6 }$$
Edexcel C2 2014 June Q4
  1. Use integration to find
$$\int _ { 1 } ^ { \sqrt { 3 } } \left( \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 3 x ^ { 2 } } \right) \mathrm { d } x$$ giving your answer in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are constants to be determined.
Edexcel C2 2014 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6b490c0-80c4-4e15-b587-ac052ee27db7-07_531_1127_264_411} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The shape \(A B C D E A\), as shown in Figure 2, consists of a right-angled triangle \(E A B\) and a triangle \(D B C\) joined to a sector \(B D E\) of a circle with radius 5 cm and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(B C = 7.5 \mathrm {~cm}\).
Angle \(E A B = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.4\) radians and \(C D = 6.1 \mathrm {~cm}\).
  1. Find, in \(\mathrm { cm } ^ { 2 }\), the area of the sector \(B D E\).
  2. Find the size of the angle \(D B C\), giving your answer in radians to 3 decimal places.
  3. Find, in \(\mathrm { cm } ^ { 2 }\), the area of the shape \(A B C D E A\), giving your answer to 3 significant figures.
Edexcel C2 2014 June Q6
6. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\) The sum to infinity of the series is \(S _ { \infty }\)
  1. Find the value of \(S _ { \infty }\) The sum to \(N\) terms of the series is \(S _ { N }\)
  2. Find, to 1 decimal place, the value of \(S _ { 12 }\)
  3. Find the smallest value of \(N\), for which $$S _ { \infty } - S _ { N } < 0.5$$
Edexcel C2 2014 June Q7
7. (i) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$9 \sin \left( \theta + 60 ^ { \circ } \right) = 4$$ giving your answers to 1 decimal place.
You must show each step of your working.
(ii) Solve, for \(- \pi \leqslant x < \pi\), the equation $$2 \tan x - 3 \sin x = 0$$ giving your answers to 2 decimal places where appropriate. [Solutions based entirely on graphical or numerical methods are not acceptable.]
Edexcel C2 2014 June Q8
8. (a) Sketch the graph of $$y = 3 ^ { x } , \quad x \in \mathbb { R }$$ showing the coordinates of any points at which the graph crosses the axes.
(b) Use algebra to solve the equation $$3 ^ { 2 x } - 9 \left( 3 ^ { x } \right) + 18 = 0$$ giving your answers to 2 decimal places where appropriate.
Edexcel C2 2014 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6b490c0-80c4-4e15-b587-ac052ee27db7-15_761_1082_210_424} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a circle \(C\) with centre \(Q\) and radius 4 and the point \(T\) which lies on \(C\). The tangent to \(C\) at the point \(T\) passes through the origin \(O\) and \(O T = 6 \sqrt { } 5\) Given that the coordinates of \(Q\) are \(( 11 , k )\), where \(k\) is a positive constant, (a) find the exact value of \(k\),
(b) find an equation for \(C\).
Edexcel C2 2014 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6b490c0-80c4-4e15-b587-ac052ee27db7-17_929_584_237_287} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6b490c0-80c4-4e15-b587-ac052ee27db7-17_716_544_452_1069} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 4 shows a closed letter box \(A B F E H G C D\), which is made to be attached to a wall of a house. The letter box is a right prism of length \(y \mathrm {~cm}\) as shown in Figure 4. The base \(A B F E\) of the prism is a rectangle. The total surface area of the six faces of the prism is \(S \mathrm {~cm} ^ { 2 }\). The cross section \(A B C D\) of the letter box is a trapezium with edges of lengths \(D A = 9 x \mathrm {~cm}\), \(A B = 4 x \mathrm {~cm} , B C = 6 x \mathrm {~cm}\) and \(C D = 5 x \mathrm {~cm}\) as shown in Figure 5.
The angle \(D A B = 90 ^ { \circ }\) and the angle \(A B C = 90 ^ { \circ }\). The volume of the letter box is \(9600 \mathrm {~cm} ^ { 3 }\).
  1. Show that $$y = \frac { 320 } { x ^ { 2 } }$$
  2. Hence show that the surface area of the letter box, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 60 x ^ { 2 } + \frac { 7680 } { x }$$
  3. Use calculus to find the minimum value of \(S\).
  4. Justify, by further differentiation, that the value of \(S\) you have found is a minimum.
Edexcel C2 2015 June Q1
  1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$ giving each term in its simplest form.
Edexcel C2 2015 June Q2
2. A circle \(C\) with centre at the point \(( 2 , - 1 )\) passes through the point \(A\) at \(( 4 , - 5 )\).
  1. Find an equation for the circle \(C\).
  2. Find an equation of the tangent to the circle \(C\) at the point \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C2 2015 June Q3
3. \(\mathrm { f } ( x ) = 6 x ^ { 3 } + 3 x ^ { 2 } + A x + B\), where \(A\) and \(B\) are constants. Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is 45 ,
  1. show that \(B - A = 48\) Given also that ( \(2 x + 1\) ) is a factor of \(\mathrm { f } ( x )\),
  2. find the value of \(A\) and the value of \(B\).
  3. Factorise f(x) fully.