Questions C2 (1550 questions)

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Edexcel C2 2010 June Q2
7 marks Moderate -0.5
2. $$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 58 x + 40$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ). Given that \(( x - 5 )\) is a factor of \(\mathrm { f } ( x )\),
  2. find all the solutions of \(\mathrm { f } ( x ) = 0\).
Edexcel C2 2010 June Q3
4 marks Moderate -0.8
3. $$y = x ^ { 2 } - k \sqrt { } x , \text { where } k \text { is a constant. }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Given that \(y\) is decreasing at \(x = 4\), find the set of possible values of \(k\).
Edexcel C2 2010 June Q4
6 marks Moderate -0.8
4. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + a x ) ^ { 7 }\), where \(a\) is a constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 2 }\) in this expansion is 525 ,
(b) find the possible values of \(a\).
Edexcel C2 2010 June Q5
6 marks Moderate -0.3
5. (a) Given that \(5 \sin \theta = 2 \cos \theta\), find the value of \(\tan \theta\).
(b) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$5 \sin 2 x = 2 \cos 2 x$$ giving your answers to 1 decimal place.
Edexcel C2 2010 June Q6
9 marks Standard +0.3
6. \includegraphics[max width=\textwidth, alt={}, center]{571780c2-945b-4636-b7c3-0bd558d28710-07_458_809_258_569} \section*{Figure 1} Figure 1 shows the sector \(O A B\) of a circle with centre \(O\), radius 9 cm and angle 0.7 radians.
  1. Find the length of the arc \(A B\).
  2. Find the area of the sector \(O A B\). The line \(A C\) shown in Figure 1 is perpendicular to \(O A\), and \(O B C\) is a straight line.
  3. Find the length of \(A C\), giving your answer to 2 decimal places. The region \(H\) is bounded by the arc \(A B\) and the lines \(A C\) and \(C B\).
  4. Find the area of \(H\), giving your answer to 2 decimal places.
    \section*{LU}
Edexcel C2 2010 June Q7
7 marks Standard +0.3
7. (a) Given that $$2 \log _ { 3 } ( x - 5 ) - \log _ { 3 } ( 2 x - 13 ) = 1$$ show that \(x ^ { 2 } - 16 x + 64 = 0\).
(b) Hence, or otherwise, solve \(2 \log _ { 3 } ( x - 5 ) - \log _ { 3 } ( 2 x - 13 ) = 1\).
Edexcel C2 2010 June Q8
9 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{571780c2-945b-4636-b7c3-0bd558d28710-10_611_831_210_575} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 10 x ^ { 2 } + k x$$ where \(k\) is a constant. The point \(P\) on \(C\) is the maximum turning point.
Given that the \(x\)-coordinate of \(P\) is 2 ,
  1. show that \(k = 28\). The line through \(P\) parallel to the \(x\)-axis cuts the \(y\)-axis at the point \(N\). The region \(R\) is bounded by \(C\), the \(y\)-axis and \(P N\), as shown shaded in Figure 2.
  2. Use calculus to find the exact area of \(R\).
Edexcel C2 2010 June Q9
10 marks Standard +0.3
9. The adult population of a town is 25000 at the end of Year 1. A model predicts that the adult population of the town will increase by \(3 \%\) each year, forming a geometric sequence.
  1. Show that the predicted adult population at the end of Year 2 is 25750.
  2. Write down the common ratio of the geometric sequence. The model predicts that Year \(N\) will be the first year in which the adult population of the town exceeds 40000.
  3. Show that $$( N - 1 ) \log 1.03 > \log 1.6$$
  4. Find the value of \(N\). At the end of each year, each member of the adult population of the town will give \(\pounds 1\) to a charity fund. Assuming the population model,
  5. find the total amount that will be given to the charity fund for the 10 years from the end of Year 1 to the end of Year 10, giving your answer to the nearest \(\pounds 1000\).
Edexcel C2 2010 June Q10
11 marks Standard +0.3
10. The circle \(C\) has centre \(A ( 2,1 )\) and passes through the point \(B ( 10,7 )\).
  1. Find an equation for \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(B\).
  2. Find an equation for \(l _ { 1 }\). The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the mid-point of \(A B\).
    Given that \(l _ { 2 }\) intersects \(C\) at the points \(P\) and \(Q\),
  3. find the length of \(P Q\), giving your answer in its simplest surd form. \includegraphics[max width=\textwidth, alt={}, center]{571780c2-945b-4636-b7c3-0bd558d28710-15_115_127_2461_1814}
Edexcel C2 2011 June Q1
8 marks Moderate -0.8
1. $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 5 x + 4$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\).
  2. Use the factor theorem to show that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\).
  3. Factorise f(x) completely.
Edexcel C2 2011 June Q2
6 marks Moderate -0.8
2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 3 + b x ) ^ { 5 }$$ where \(b\) is a non-zero constant. Give each term in its simplest form. Given that, in this expansion, the coefficient of \(x ^ { 2 }\) is twice the coefficient of \(x\),
(b) find the value of \(b\).
Edexcel C2 2011 June Q3
4 marks Easy -1.2
3. Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which
  1. \(5 ^ { x } = 10\),
  2. \(\log _ { 3 } ( x - 2 ) = - 1\).
Edexcel C2 2011 June Q4
8 marks Moderate -0.8
4. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - 11 = 0$$ Find
  1. the coordinates of the centre of \(C\),
  2. the radius of \(C\),
  3. the coordinates of the points where \(C\) crosses the \(y\)-axis, giving your answers as simplified surds.
Edexcel C2 2011 June Q5
7 marks Standard +0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c9758792-ca4c-4837-bd7c-e695fe0c0cdf-06_426_417_260_760} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The shape shown in Figure 1 is a pattern for a pendant. It consists of a sector \(O A B\) of a circle centre \(O\), of radius 6 cm , and angle \(A O B = \frac { \pi } { 3 }\). The circle \(C\), inside the sector, touches the two straight edges, \(O A\) and \(O B\), and the \(\operatorname { arc } A B\) as shown. Find
  1. the area of the sector \(O A B\),
  2. the radius of the circle \(C\). The region outside the circle \(C\) and inside the sector \(O A B\) is shown shaded in Figure 1.
  3. Find the area of the shaded region.
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. (a) Solve for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers in degrees to 1 decimal place,
$$3 \sin \left( x + 45 ^ { \circ } \right) = 2$$ (b) 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. (a) Show that the equation
$$\tan 2 x = 5 \sin 2 x$$ can be written in the form $$( 1 - 5 \cos 2 x ) \sin 2 x = 0$$ (b) 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.