Questions — Edexcel C2 (579 questions)

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Edexcel C2 2014 January Q2
6 marks Moderate -0.3
2. \(\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants. Given that when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) the remainder is 25 ,
  1. show that \(2 a + b = 5\) Given also that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\),
  2. find the value of \(a\) and the value of \(b\). \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-05_90_97_2613_1784} \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-05_52_169_2709_1765}
Edexcel C2 2014 January Q3
11 marks Moderate -0.8
3. The curve \(C\) has equation $$y = 2 \sqrt { } x + \frac { 18 } { \sqrt { } x } - 1 , \quad x > 0$$
  1. Find
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
  2. Use calculus to find the coordinates of the stationary point of \(C\).
  3. Determine whether the stationary point is a maximum or minimum, giving a reason for your answer. \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-09_138_154_2597_1804}
Edexcel C2 2014 January Q4
8 marks Moderate -0.3
4. The first term of a geometric series is 5 and the common ratio is 1.2 For this series find, to 1 decimal place,
    1. the \(20 ^ { \text {th } }\) term,
    2. the sum of the first 20 terms. The sum of the first \(n\) terms of the series is greater than 3000
  2. Calculate the smallest possible value of \(n\).
Edexcel C2 2014 January Q5
7 marks Moderate -0.8
5. The height of water, \(H\) metres, in a harbour on a particular day is given by the equation $$H = 10 + 5 \sin \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight.
  1. Show that the height of the water 1 hour after midnight is 12.5 metres.
  2. Find, to the nearest minute, the times before midday when the height of the water is 9 metres.
Edexcel C2 2014 January Q6
5 marks Standard +0.3
6. Given that $$\log _ { x } ( 7 y + 1 ) - \log _ { x } ( 2 y ) = 1 , \quad x > 4 , \quad 0 < y < 1$$ express \(y\) in terms of \(x\).
Edexcel C2 2014 January Q7
13 marks Standard +0.8
7. \begin{figure}[h]
[diagram]
\captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 5$$ The point \(P ( 4,9 )\) lies on \(C\).
  1. Show that the normal to \(C\) at the point \(P\) has equation $$x + 9 y = 85$$ The region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the \(y\)-axis and the normal to \(C\) at \(P\).
  2. Showing all your working, calculate the exact area of \(R\).
Edexcel C2 2014 January Q8
11 marks Moderate -0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e7043e7a-2c8f-425a-8471-f647828cc297-22_1015_1542_267_185} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a circle \(C\) with centre \(O\) and radius 5
  1. Write down the cartesian equation of \(C\). The points \(P ( - 3 , - 4 )\) and \(Q ( 3 , - 4 )\) lie on \(C\).
  2. Show that the tangent to \(C\) at the point \(Q\) has equation $$3 x - 4 y = 25$$
  3. Show that, to 3 decimal places, angle \(P O Q\) is 1.287 radians. The tangent to \(C\) at \(P\) and the tangent to \(C\) at \(Q\) intersect on the \(y\)-axis at the point \(R\).
  4. Find the area of the shaded region \(P Q R\) shown in Figure 2. \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-25_177_154_2576_1804}
Edexcel C2 2014 January Q9
9 marks Standard +0.3
9. (a) Show that the equation $$5 \sin x - \cos ^ { 2 } x + 2 \sin ^ { 2 } x = 1$$ can be written in the form $$3 \sin ^ { 2 } x + 5 \sin x - 2 = 0$$ (b) Hence solve, for \(- 180 ^ { \circ } \leqslant \theta < 180 ^ { \circ }\), the equation $$5 \sin 2 \theta - \cos ^ { 2 } 2 \theta + 2 \sin ^ { 2 } 2 \theta = 1$$ giving your answers to 2 decimal places.
Edexcel C2 2005 June Q1
4 marks Easy -1.2
Find the coordinates of the stationary point on the curve with equation \(y = 2 x ^ { 2 } - 12 x\).
Edexcel C2 2005 June Q5
8 marks Moderate -0.8
5. Solve, for \(0 \leqslant x \leqslant 180 ^ { \circ }\), the equation
  1. \(\quad \sin \left( x + 10 ^ { \circ } \right) = \frac { \sqrt { } 3 } { 2 }\),
  2. \(\cos 2 x = - 0.9\), giving your answers to 1 decimal place.
Edexcel C2 2005 June Q6
8 marks Moderate -0.8
6. A river, running between parallel banks, is 20 m wide. The depth, \(y\) metres, of the river measured at a point \(x\) metres from one bank is given by the formula $$y = \frac { 1 } { 10 } x \sqrt { } ( 20 - x ) , \quad 0 \leqslant x \leqslant 20$$
  1. Complete the table below, giving values of \(y\) to 3 decimal places.
    \(x\)048121620
    \(y\)02.7710
  2. Use the trapezium rule with all the values in the table to estimate the cross-sectional area of the river. Given that the cross-sectional area is constant and that the river is flowing uniformly at \(2 \mathrm {~ms} ^ { - 1 }\),
  3. estimate, in \(\mathrm { m } ^ { 3 }\), the volume of water flowing per minute, giving your answer to 3 significant figures.
Edexcel C2 2005 June Q7
6 marks Standard +0.3
7. In the triangle \(A B C , A B = 8 \mathrm {~cm} , A C = 7 \mathrm {~cm} , \angle A B C = 0.5\) radians and \(\angle A C B = x\) radians.
  1. Use the sine rule to find the value of \(\sin x\), giving your answer to 3 decimal places. Given that there are two possible values of \(x\),
  2. find these values of \(x\), giving your answers to 2 decimal places.
Edexcel C2 2005 June Q8
9 marks Standard +0.3
8. The circle \(C\), with centre at the point \(A\), has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 9 = 0\). Find
  1. the coordinates of \(A\),
  2. the radius of \(C\),
  3. the coordinates of the points at which \(C\) crosses the \(x\)-axis. Given that the line \(l\) with gradient \(\frac { 7 } { 2 }\) is a tangent to \(C\), and that \(l\) touches \(C\) at the point \(T\),
  4. find an equation of the line which passes through \(A\) and \(T\).
Edexcel C2 2005 June Q9
10 marks Easy -1.2
9. (a) A geometric series has first term \(a\) and common ratio \(r\). Prove that the sum of the first \(n\) terms of the series is $$\frac { a \left( 1 - r ^ { n } \right) } { 1 - r } .$$ Mr. King will be paid a salary of \(\pounds 35000\) in the year 2005 . Mr. King's contract promises a \(4 \%\) increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence.
(b) Find, to the nearest \(\pounds 100\), Mr. King's salary in the year 2008. Mr. King will receive a salary each year from 2005 until he retires at the end of 2024.
(c) Find, to the nearest \(\pounds 1000\), the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024.
Edexcel C2 2005 June Q10
12 marks Standard +0.3
10. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{135bc546-9274-4862-b2e7-c11e9c8e2c4f-13_1018_1029_287_445}
\end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = 2 x + \frac { 8 } { x ^ { 2 } } - 5 , x > 0\).
The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 4 respectively. The region \(R\), shaded in Figure 1, is bounded by \(C\) and the straight line joining \(P\) and \(Q\).
  1. Find the exact area of \(R\).
  2. Use calculus to show that \(y\) is increasing for \(x > 2\).
Edexcel C2 2006 June Q1
4 marks Easy -1.2
Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 + x ) ^ { 6 }\), giving each term in its simplest form.
Edexcel C2 2006 June Q5
8 marks Easy -1.2
5. (a) In the space provided, sketch the graph of \(y = 3 ^ { x } , x \in \mathbb { R }\), showing the coordinates of the point at which the graph meets the \(y\)-axis.
(b) Complete the table, giving the values of \(3 ^ { x }\) to 3 decimal places.
\(x\)00.20.40.60.81
\(3 ^ { x }\)1.2461.5523
(c) Use the trapezium rule, with all the values from your table, to find an approximation for the value of \(\int _ { 0 } ^ { 1 } 3 ^ { x } \mathrm {~d} x\).
Edexcel C2 2006 June Q6
4 marks Moderate -0.8
6. (a) Given that \(\sin \theta = 5 \cos \theta\), find the value of \(\tan \theta\).
(b) Hence, or otherwise, find the values of \(\theta\) in the interval \(0 \leqslant \theta < 360 ^ { \circ }\) for which $$\sin \theta = 5 \cos \theta ,$$ giving your answers to 1 decimal place.
Edexcel C2 2006 June Q7
8 marks Moderate -0.8
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{29c7baa1-6929-448a-a756-319ea75dffa7-08_611_682_296_641}
\end{figure} The line \(y = 3 x - 4\) is a tangent to the circle \(C\), touching \(C\) at the point \(P ( 2,2 )\), as shown in Figure 1. The point \(Q\) is the centre of \(C\).
  1. Find an equation of the straight line through \(P\) and \(Q\). Given that \(Q\) lies on the line \(y = 1\),
  2. show that the \(x\)-coordinate of \(Q\) is 5,
  3. find an equation for \(C\).
Edexcel C2 2006 June Q8
9 marks Moderate -0.8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{29c7baa1-6929-448a-a756-319ea75dffa7-10_620_636_301_660}
\end{figure} Figure 2 shows the cross section \(A B C D\) of a small shed. The straight line \(A B\) is vertical and has length 2.12 m . The straight line \(A D\) is horizontal and has length 1.86 m . The curve \(B C\) is an arc of a circle with centre \(A\), and \(C D\) is a straight line. Given that the size of \(\angle B A C\) is 0.65 radians, find
  1. the length of the arc \(B C\), in m , to 2 decimal places,
  2. the area of the sector \(B A C\), in \(\mathrm { m } ^ { 2 }\), to 2 decimal places,
  3. the size of \(\angle C A D\), in radians, to 2 decimal places,
  4. the area of the cross section \(A B C D\) of the shed, in \(\mathrm { m } ^ { 2 }\), to 2 decimal places.
Edexcel C2 2006 June Q9
11 marks Moderate -0.3
  1. A geometric series has first term \(a\) and common ratio \(r\). The second term of the series is 4 and the sum to infinity of the series is 25.
    1. Show that \(25 r ^ { 2 } - 25 r + 4 = 0\).
    2. Find the two possible values of \(r\).
    3. Find the corresponding two possible values of \(a\).
    4. Show that the sum, \(S _ { n }\), of the first \(n\) terms of the series is given by
    $$S _ { n } = 25 \left( 1 - r ^ { n } \right) .$$ Given that \(r\) takes the larger of its two possible values,
  2. find the smallest value of \(n\) for which \(S _ { n }\) exceeds 24 .
Edexcel C2 2006 June Q10
14 marks Moderate -0.3
10. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{29c7baa1-6929-448a-a756-319ea75dffa7-14_636_956_285_513}
\end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = x ^ { 3 } - 8 x ^ { 2 } + 20 x\). The curve has stationary points \(A\) and \(B\).
  1. Use calculus to find the \(x\)-coordinates of \(A\) and \(B\).
  2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\), and hence verify that \(A\) is a maximum. The line through \(B\) parallel to the \(y\)-axis meets the \(x\)-axis at the point \(N\).
    The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line from \(A\) to \(N\).
  3. Find \(\int \left( x ^ { 3 } - 8 x ^ { 2 } + 20 x \right) \mathrm { d } x\).
  4. Hence calculate the exact area of \(R\).
Edexcel C2 2007 June Q1
4 marks Moderate -0.8
Evaluate \(\int _ { 1 } ^ { 8 } \frac { 1 } { \sqrt { } x } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.
Edexcel C2 2007 June Q3
6 marks Moderate -0.3
3. (a) Find the first four terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + k x ) ^ { 6 }\), where \(k\) is a non-zero constant. Given that, in this expansion, the coefficients of \(x\) and \(x ^ { 2 }\) are equal, find
(b) the value of \(k\),
(c) the coefficient of \(x ^ { 3 }\).
Edexcel C2 2007 June Q4
5 marks Moderate -0.8
4. Figure 1 Figure 1 shows the triangle \(A B C\), with \(A B = 6 \mathrm {~cm} , B C = 4 \mathrm {~cm}\) and \(C A = 5 \mathrm {~cm}\).
  1. Show that \(\cos A = \frac { 3 } { 4 }\).
  2. Hence, or otherwise, find the exact value of \(\sin A\).