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Edexcel C12 2019 January Q11
8 marks Moderate -0.3
11. (i) Given that \(x\) is a positive real number, solve the equation $$\log _ { x } 324 = 4$$ writing your answer as a simplified surd.
(ii) Given that $$\log _ { a } ( 5 y - 4 ) - \log _ { a } ( 2 y ) = 3 \quad y > 0.8,0 < a < 1$$ express \(y\) in terms of \(a\).
Edexcel C12 2019 January Q12
9 marks Moderate -0.5
12. Karen is going to raise money for a charity. She aims to cycle a total distance of 1000 km over a number of days.
On day one she cycles 25 km .
She increases the distance that she cycles each day by \(10 \%\) of the distance cycled on the previous day, until she reaches the total distance of 1000 km . She reaches the total distance of 1000 km on day \(N\), where \(N\) is a positive integer.
  1. Find the value of \(N\). On day one, 50 people donated money to the charity. Each day, 20 more people donated to the charity than did so on the previous day, so that 70 people donated money on day two, 90 people donated money on day three, and so on.
  2. Find the number of people who donated to the charity on day fifteen. Each day, the donation given by each person was \(\pounds 5\)
  3. Find the total amount of money donated by the end of day fifteen.
Edexcel C12 2019 January Q13
10 marks Moderate -0.3
13. \(\mathrm { f } ( x ) = 3 x ^ { 3 } + 3 x ^ { 2 } + c x + 12\), where \(c\) is a constant Given that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\),
  1. show that \(c = - 14\)
  2. Write \(\mathrm { f } ( x )\) in the form $$\mathrm { f } ( x ) = ( x + 3 ) \mathrm { Q } ( x )$$ where \(\mathrm { Q } ( x )\) is a quadratic function.
  3. Use the answer to part (b) to prove that the equation \(\mathrm { f } ( x ) = 0\) has only one real solution. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{75d68987-2314-4c8f-8160-24977c5c4e34-32_595_915_1034_518} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). On separate diagrams sketch the curve with equation
    1. \(y = \mathrm { f } ( 3 x )\)
    2. \(y = - \mathrm { f } ( \mathrm { x } )\) On each diagram show clearly the coordinates of the points where the curve crosses the coordinate axes.
Edexcel C12 2019 January Q14
11 marks Standard +0.3
14. In this question solutions based entirely on graphical or numerical methods are not acceptable.
  1. Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), the equation $$\sin \left( x + 60 ^ { \circ } \right) = - 0.4$$ giving your answers, in degrees, to one decimal place.
  2. (a) Show that the equation $$2 \sin \theta \tan \theta - 3 = \cos \theta$$ can be written in the form $$3 \cos ^ { 2 } \theta + 3 \cos \theta - 2 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$2 \sin \theta \tan \theta - 3 = \cos \theta$$ showing each stage of your working and giving your answers, in degrees, to one decimal place.
Edexcel C12 2019 January Q15
11 marks Standard +0.3
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{75d68987-2314-4c8f-8160-24977c5c4e34-40_545_794_294_584} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The straight line \(l\) with equation \(y = 5 - 3 x\) cuts the curve \(C\), with equation \(y = 20 x - 12 x ^ { 2 }\), at the points \(P\) and \(Q\), as shown in Figure 3.
  1. Use algebra to find the exact coordinates of the points \(P\) and \(Q\). The finite region \(R\), shown shaded in Figure 3, is bounded by the line \(l\), the \(x\)-axis and the curve \(C\).
  2. Use calculus to find the exact area of \(R\).
Edexcel C12 2019 January Q16
16 marks Standard +0.3
16. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{75d68987-2314-4c8f-8160-24977c5c4e34-44_442_822_285_561} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows the design for a container in the shape of a hollow triangular prism. The container is open at the top, which is labelled \(A B C D\). The sides of the container, \(A B F E\) and \(D C F E\), are rectangles. The ends of the container, \(A D E\) and \(B C F\), are congruent right-angled triangles, as shown in Figure 4. The ends of the container are vertical and the edge \(E F\) is horizontal. The edges \(A E , D E\) and \(E F\) have lengths \(4 x\) metres, \(3 x\) metres and \(l\) metres respectively. Given that the container has a capacity of \(0.75 \mathrm {~m} ^ { 3 }\) and is made of material of negligible thickness,
  1. show that the internal surface area of the container, \(S \mathrm {~m} ^ { 2 }\), is given by $$S = 12 x ^ { 2 } + \frac { 7 } { 8 x }$$
  2. Use calculus to find the value of \(x\), for which \(S\) is a minimum. Give your answer to 3 significant figures.
  3. Justify that the value of \(x\) found in part (b) gives a minimum value for \(S\). Using the value of \(x\) found in part (b), find to 2 decimal places,
    1. the length of the edge \(A D\),
    2. the length of the edge \(C D\).
      END
Edexcel C12 2014 June Q1
6 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-02_856_700_214_630} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the position of three stationary fishing boats \(A , B\) and \(C\), which are assumed to be in the same horizontal plane. Boat \(A\) is 10 km due north of boat \(B\). Boat \(C\) is 8 km on a bearing of \(065 ^ { \circ }\) from boat \(B\).
  1. Find the distance of boat \(C\) from boat \(A\), giving your answer to the nearest 10 metres.
  2. Find the bearing of boat \(C\) from boat \(A\), giving your answer to one decimal place.
Edexcel C12 2014 June Q2
4 marks Moderate -0.8
2. Without using your calculator, solve $$x \sqrt { 27 } + 21 = \frac { 6 x } { \sqrt { 3 } }$$ Write your answer in the form \(a \sqrt { b }\) where \(a\) and \(b\) are integers. You must show all stages of your working.
Edexcel C12 2014 June Q3
7 marks Moderate -0.3
3. Solve, giving each answer to 3 significant figures, the equations
  1. \(4 ^ { a } = 20\)
  2. \(3 + 2 \log _ { 2 } b = \log _ { 2 } ( 30 b )\) (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2014 June Q4
8 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-05_716_725_219_603} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } + \frac { 16 } { x } , \quad x > 0$$ The curve has a minimum turning point at \(A\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\).
  2. Hence find the coordinates of \(A\).
  3. Use your answer to part (b) to write down the turning point of the curve with equation
    1. \(y = \mathrm { f } ( x + 1 )\),
    2. \(y = \frac { 1 } { 2 } \mathrm { f } ( x )\).
Edexcel C12 2014 June Q5
7 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-07_953_929_219_422} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Diagram not drawn to scale Figure 3 shows the points \(P , Q\) and \(R\). Points \(P\) and \(Q\) have coordinates ( \(- 1,4\) ) and ( 4,7 ) respectively.
  1. Find an equation for the straight line passing through points \(P\) and \(Q\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The point \(R\) has coordinates ( \(p , - 3\) ), where \(p\) is a positive constant. Given that angle \(Q P R = 90 ^ { \circ }\),
  2. find the value of \(p\).
Edexcel C12 2014 June Q6
7 marks Moderate -0.3
6. (a) Show that $$\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } \equiv 1 - \tan ^ { 2 } x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant x < 2 \pi\), $$\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } + 2 = 0$$ Give your answers in terms of \(\pi\).
Edexcel C12 2014 June Q7
10 marks Moderate -0.8
7. (i) A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 2,3 )\). Given that $$f ^ { \prime } ( x ) = \frac { 4 } { x ^ { 3 } } + 2 x - 1$$ find the value of \(\mathrm { f } ( 1 )\).
(ii) Given that $$\int _ { 1 } ^ { 4 } ( 3 \sqrt { x } + A ) \mathrm { d } x = 21$$ find the exact value of the constant \(A\).
Edexcel C12 2014 June Q8
7 marks Moderate -0.3
8. Given that $$1 + 12 x + 70 x ^ { 2 } + \ldots$$ is the binomial expansion, in ascending powers of \(x\) of \(( 1 + b x ) ^ { n }\), where \(n \in \mathbb { N }\) and \(b\) is a constant,
  1. show that \(n b = 12\)
  2. find the values of the constants \(b\) and \(n\).
Edexcel C12 2014 June Q9
7 marks Moderate -0.8
9. (i) Find the value of \(\sum _ { r = 1 } ^ { 20 } ( 3 + 5 r )\) (ii) Given that \(\sum _ { r = 0 } ^ { \infty } \frac { a } { 4 ^ { r } } = 16\), find the value of the constant \(a\).
Edexcel C12 2014 June Q10
7 marks Moderate -0.3
10. The equation $$k x ^ { 2 } + 4 x + k = 2 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 4 < 0$$
  2. Hence find the set of all possible values of \(k\).
Edexcel C12 2014 June Q11
15 marks Challenging +1.2
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-17_1000_956_264_500} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the circle \(C\) with centre \(Q\) and equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 2 y + 5 = 0$$
  1. Find
    1. the coordinates of \(Q\),
    2. the exact value of the radius of \(C\). The tangents to \(C\) from the point \(T ( 8,4 )\) meet \(C\) at the points \(M\) and \(N\), as shown in Figure 4.
  2. Show that the obtuse angle \(M Q N\) is 2.498 radians to 3 decimal places. The region \(R\), shown shaded in Figure 4, is bounded by the tangent \(T N\), the minor arc \(N M\), and the tangent \(M T\).
  3. Find the area of region \(R\).
Edexcel C12 2014 June Q12
15 marks Standard +0.3
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-19_1011_1349_237_310} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 } C\) touches the \(x\)-axis at the origin and cuts the \(x\)-axis at the point \(A\).
  1. Show that the coordinates of \(A\) are \(( 3,0 )\).
  2. Show that the equation of the tangent to \(C\) at the point \(A\) is \(y = - 3 x + 9\) The tangent to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 5.
  3. Use algebra to find the \(x\) coordinate of \(B\). The region \(R\), shown shaded in Figure 5, is bounded by the curve \(C\) and the tangent to \(C\) at \(A\).
  4. Find, by using calculus, the area of region \(R\).
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2014 June Q13
10 marks Standard +0.3
13. The height of sea water, \(h\) metres, on a harbour wall at time \(t\) hours after midnight is given by $$h = 3.7 + 2.5 \cos ( 30 t - 40 ) ^ { \circ } , \quad 0 \leqslant t < 24$$
  1. Calculate the maximum value of \(h\) and the exact time of day when this maximum first occurs. Fishing boats cannot enter the harbour if \(h\) is less than 3
  2. Find the times during the morning between which fishing boats cannot enter the harbour.
    Give these times to the nearest minute.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2014 June Q14
15 marks Standard +0.3
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-23_650_1182_212_383} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} Figure 6 shows a solid triangular prism \(A B C D E F\) in which \(A B = 2 x \mathrm {~cm}\) and \(C D = l \mathrm {~cm}\). The cross section \(A B C\) is an equilateral triangle. The rectangle \(B C D F\) is horizontal and the triangles \(A B C\) and \(D E F\) are vertical.
The total surface area of the prism is \(S \mathrm {~cm} ^ { 2 }\) and the volume of the prism is \(V \mathrm {~cm} ^ { 3 }\).
  1. Show that \(S = 2 x ^ { 2 } \sqrt { 3 } + 6 x l\) Given that \(S = 960\),
  2. show that \(V = 160 x \sqrt { 3 } - x ^ { 3 }\)
  3. Use calculus to find the maximum value of \(V\), giving your answer to the nearest integer.
  4. Justify that the value of \(V\) found in part (c) is a maximum. \includegraphics[max width=\textwidth, alt={}, center]{b85872d4-00b2-499b-9765-f7559d3de66a-24_63_52_2690_1886}
Edexcel C12 2015 June Q1
4 marks Easy -1.2
  1. The line \(l _ { 1 }\) has equation
$$10 x - 2 y + 7 = 0$$
  1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(\left( - \frac { 1 } { 3 } , \frac { 4 } { 3 } \right)\).
  2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C12 2015 June Q2
5 marks Moderate -0.5
2. $$f ( x ) = x ^ { 4 } - x ^ { 3 } + 3 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is 4
When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is 22
Find the value of \(a\) and the value of \(b\).
Edexcel C12 2015 June Q3
3 marks Easy -1.2
3. Given that $$y = \frac { 1 } { 27 } x ^ { 3 }$$ express each of the following in the form \(k x ^ { n }\) where \(k\) and \(n\) are constants.
  1. \(y ^ { \frac { 1 } { 3 } }\)
  2. \(3 y ^ { - 1 }\)
  3. \(\sqrt { ( 27 y ) }\)
Edexcel C12 2015 June Q4
6 marks Easy -1.2
4. (a) Sketch the graph of \(y = \frac { 1 } { x } , x > 0\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 1 } { x }\), with the values for \(y\) rounded to 3 decimal places where necessary.
\(x\)11.522.53
\(y\)10.6670.50.40.333
(b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value, to 2 decimal places, for \(\int _ { 1 } ^ { 3 } \frac { 1 } { x } \mathrm {~d} x\)
Edexcel C12 2015 June Q5
6 marks Moderate -0.8
  1. (i) Find, giving your answer to 3 significant figures, the value of \(y\) for which
$$3 ^ { y } = 12$$ (ii) Solve, giving an exact answer, the equation $$\log _ { 2 } ( x + 3 ) - \log _ { 2 } ( 2 x + 4 ) = 4$$ (You should show each step in your working.)