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Edexcel C12 2015 June Q6
6 marks Moderate -0.8
6. (a) Find the first 3 terms in ascending powers of \(x\) of the binomial expansion of $$( 2 + a x ) ^ { 6 }$$ where \(a\) is a non-zero constant. Give each term in its simplest form. Given that, in the expansion, the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\) (b) find the value of \(a\).
Edexcel C12 2015 June Q7
7 marks Moderate -0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ea81408b-e292-4529-b1e2-e3246503a3ac-09_440_437_285_772} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a circle with centre \(O\) and radius 9 cm . The points \(A\) and \(B\) lie on the circumference of this circle. The minor sector \(O A B\) has perimeter 30 cm and the angle between the radii \(O A\) and \(O B\) of this sector is \(\theta\) radians. Find
  1. the length of the arc \(A B\),
  2. the value of \(\theta\),
  3. the area of the minor sector \(O A B\),
  4. the area of triangle \(O A B\), giving your answer to 3 significant figures.
Edexcel C12 2015 June Q8
7 marks Moderate -0.8
8. A 25-year programme for building new houses began in Core Town in the year 1986 and finished in the year 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in the year 2000 and 108 were built in the year 2010, find
  1. the number of houses built in 1986, the first year of the building programme,
  2. the total number of houses built in the 25 years of the programme.
Edexcel C12 2015 June Q9
7 marks Moderate -0.3
9. The equation \(x ^ { 2 } + ( 6 k + 4 ) x + 3 = 0\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$9 k ^ { 2 } + 12 k + 1 < 0$$
  2. Find the range of possible values for \(k\), giving your boundaries as fully simplified surds.
Edexcel C12 2015 June Q10
8 marks Moderate -0.3
10. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 4 \\ u _ { n + 1 } & = \frac { 2 u _ { n } } { 3 } , \quad n \geqslant 1 \end{aligned}$$
  1. Find the exact values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
  2. Find the value of \(u _ { 20 }\), giving your answer to 3 significant figures.
  3. Evaluate $$12 - \sum _ { i = 1 } ^ { 16 } u _ { i }$$ giving your answer to 3 significant figures.
  4. Explain why \(\sum _ { i = 1 } ^ { N } u _ { i } < 12\) for all positive integer values of \(N\).
Edexcel C12 2015 June Q11
11 marks Moderate -0.3
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where
$$f ^ { \prime } ( x ) = 3 \sqrt { x } - \frac { 9 } { \sqrt { x } } + 2$$ Given that the point \(P ( 9,14 )\) lies on \(C\),
  1. find \(\mathrm { f } ( x )\), simplifying your answer,
  2. find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
Edexcel C12 2015 June Q12
9 marks Moderate -0.3
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ea81408b-e292-4529-b1e2-e3246503a3ac-17_679_1241_274_500} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the \(x\)-axis at the origin and at the point \(( 6,0 )\). The curve has maximum points at \(( 1,6 )\) and \(( 5,6 )\) and has a minimum point at \(( 3,2 )\). On separate diagrams sketch the curve with equation
  1. \(y = - \mathrm { f } ( x )\)
  2. \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\)
  3. \(y = \mathrm { f } ( x + 4 )\) On each diagram show clearly the coordinates of the maximum and minimum points, and the coordinates of the points where the curve crosses the \(x\)-axis.
Edexcel C12 2015 June Q13
9 marks Standard +0.3
  1. (i) Showing each step in your reasoning, prove that
$$( \sin x + \cos x ) ( 1 - \sin x \cos x ) \equiv \sin ^ { 3 } x + \cos ^ { 3 } x$$ (ii) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$3 \sin \theta = \tan \theta$$ giving your answers in degrees to 1 decimal place, as appropriate.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2015 June Q14
10 marks Standard +0.3
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ea81408b-e292-4529-b1e2-e3246503a3ac-21_641_920_260_568} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The finite region \(R\), which is shown shaded in Figure 3, is bounded by the straight line \(l\) with equation \(y = 4 x + 3\) and the curve \(C\) with equation \(y = 2 x ^ { \frac { 3 } { 2 } } - 2 x + 3 , x \geqslant 0\) The line \(l\) meets the curve \(C\) at the point \(A\) on the \(y\)-axis and \(l\) meets \(C\) again at the point \(B\), as shown in Figure 3.
  1. Use algebra to find the coordinates of \(A\) and \(B\).
  2. Use integration to find the area of the shaded region \(R\).
Edexcel C12 2015 June Q15
14 marks Standard +0.3
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ea81408b-e292-4529-b1e2-e3246503a3ac-23_830_938_269_520} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Diagram not drawn to scale The circle shown in Figure 4 has centre \(P ( 5,6 )\) and passes through the point \(A ( 12,7 )\). Find
  1. the exact radius of the circle,
  2. an equation of the circle,
  3. an equation of the tangent to the circle at the point \(A\). The circle also passes through the points \(B ( 0,1 )\) and \(C ( 4,13 )\).
  4. Use the cosine rule on triangle \(A B C\) to find the size of the angle \(B C A\), giving your answer in degrees to 3 significant figures.
Edexcel C12 2015 June Q16
13 marks Standard +0.3
  1. \hspace{0pt} [In this question you may assume the formula for the area of a circle and the following formulae:
    a sphere of radius \(r\) has volume \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) and surface area \(S = 4 \pi r ^ { 2 }\) a cylinder of radius \(r\) and height \(h\) has volume \(V = \pi r ^ { 2 } h\) and curved surface area \(S = 2 \pi r h ]\)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ea81408b-e292-4529-b1e2-e3246503a3ac-25_414_478_566_726} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows the model for a building. The model is made up of three parts. The roof is modelled by the curved surface of a hemisphere of radius \(R \mathrm {~cm}\). The walls are modelled by the curved surface of a circular cylinder of radius \(R \mathrm {~cm}\) and height \(H \mathrm {~cm}\). The floor is modelled by a circular disc of radius \(R \mathrm {~cm}\). The model is made of material of negligible thickness, and the walls are perpendicular to the base. It is given that the volume of the model is \(800 \pi \mathrm {~cm} ^ { 3 }\) and that \(0 < R < 10.6\)
  1. Show that $$H = \frac { 800 } { R ^ { 2 } } - \frac { 2 } { 3 } R$$
  2. Show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of the model is given by $$A = \frac { 5 \pi R ^ { 2 } } { 3 } + \frac { 1600 \pi } { R }$$
  3. Use calculus to find the value of \(R\), to 3 significant figures, for which \(A\) is a minimum.
  4. Prove that this value of \(R\) gives a minimum value for \(A\).
  5. Find, to 3 significant figures, the value of \(H\) which corresponds to this value for \(R\).
Edexcel C12 2016 June Q1
5 marks Moderate -0.8
  1. The first three terms in ascending powers of \(x\) in the binomial expansion of \(( 1 + p x ) ^ { 8 }\) are given by
$$1 + 12 x + q x ^ { 2 }$$ where \(p\) and \(q\) are constants.
Find the value of \(p\) and the value of \(q\).
Edexcel C12 2016 June Q3
7 marks Moderate -0.8
3. Answer this question without a calculator, showing all your working and giving your answers in their simplest form.
  1. Solve the equation $$4 ^ { 2 x + 1 } = 8 ^ { 4 x }$$
  2. (a) Express $$3 \sqrt { 18 } - \sqrt { 32 }$$ in the form \(k \sqrt { 2 }\), where \(k\) is an integer.
    (b) Hence, or otherwise, solve $$3 \sqrt { 18 } - \sqrt { 32 } = \sqrt { n }$$
Edexcel C12 2016 June Q4
8 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-05_476_1338_251_360} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt { x + 2 } , x \geqslant - 2\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line \(x = 6\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { x + 2 }\)
\(x\)- 20246
\(y\)01.414222.8284
  1. Complete the table above, giving the missing value of \(y\) to 4 decimal places.
  2. Use the trapezium rule, with all of the values of \(y\) in the completed table, to find an approximate value for the area of \(R\), giving your answer to 3 decimal places. Use your answer to part (b) to find approximate values of
    1. \(\int _ { - 2 } ^ { 6 } \frac { \sqrt { x + 2 } } { 2 } \mathrm {~d} x\)
    2. \(\int _ { - 2 } ^ { 6 } ( 2 + \sqrt { x + 2 } ) \mathrm { d } x\)
Edexcel C12 2016 June Q5
6 marks Standard +0.3
5. (i) $$U _ { n + 1 } = \frac { U _ { n } } { U _ { n } - 3 } , \quad n \geqslant 1$$ Given \(U _ { 1 } = 4\), find
  1. \(U _ { 2 }\)
  2. \(\sum _ { n = 1 } ^ { 100 } U _ { n }\) (ii) Given $$\sum _ { r = 1 } ^ { n } ( 100 - 3 r ) < 0$$ find the least value of the positive integer \(n\).
Edexcel C12 2016 June Q6
7 marks Moderate -0.8
6. (a) Show that \(\frac { x ^ { 2 } - 4 } { 2 \sqrt { } x }\) can be written in the form \(A x ^ { p } + B x ^ { q }\), where \(A , B , p\) and \(q\) are constants to be determined.
(b) Hence find $$\int \frac { x ^ { 2 } - 4 } { 2 \sqrt { x } } \mathrm {~d} x , \quad x > 0$$ giving your answer in its simplest form.
Edexcel C12 2016 June Q7
10 marks Moderate -0.3
7. $$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + b x - 10 \text {, where } a \text { and } b \text { are constants. }$$ Given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\),
  1. use the factor theorem to show that \(2 a + b = - 7\) Given also that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 36
  2. find the value of \(a\) and the value of \(b\). \(\mathrm { f } ( x )\) can be written in the form $$\mathrm { f } ( x ) = ( x - 2 ) \mathrm { Q } ( x ) \text {, where } \mathrm { Q } ( x ) \text { is a quadratic function. }$$
    1. Find \(\mathrm { Q } ( x )\).
    2. Prove that the equation \(\mathrm { f } ( x ) = 0\) has only one real root. You must justify your answer and show all your working.
Edexcel C12 2016 June Q8
7 marks Standard +0.3
8. In this question the angle \(\theta\) is measured in degrees throughout.
  1. Show that the equation $$\frac { 5 + \sin \theta } { 3 \cos \theta } = 2 \cos \theta , \quad \theta \neq ( 2 n + 1 ) 90 ^ { \circ } , \quad n \in \mathbb { Z }$$ may be rewritten as $$6 \sin ^ { 2 } \theta + \sin \theta - 1 = 0$$
  2. Hence solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), the equation $$\frac { 5 + \sin \theta } { 3 \cos \theta } = 2 \cos \theta$$ Give your answers to one decimal place, where appropriate.
Edexcel C12 2016 June Q9
8 marks Moderate -0.3
  1. The first term of a geometric series is 6 and the common ratio is 0.92
For this series, find
    1. the \(25 ^ { \text {th } }\) term, giving your answer to 2 significant figures,
    2. the sum to infinity. The sum to \(n\) terms of this series is greater than 72
  2. Calculate the smallest possible value of \(n\).
    VJYV SIHI NITIIIUMION, OC
    VILV SIHI NI JAHM ION OO
    VILV SIHI NI JIIIM ION OO
Edexcel C12 2016 June Q10
9 marks Moderate -0.8
10. The curve \(C\) has equation \(y = \sin \left( x + \frac { \pi } { 4 } \right) , \quad 0 \leqslant x \leqslant 2 \pi\)
  1. On the axes below, sketch the curve \(C\).
  2. Write down the exact coordinates of all the points at which the curve \(C\) meets or intersects the \(x\)-axis and the \(y\)-axis.
  3. Solve, for \(0 \leqslant x \leqslant 2 \pi\), the equation $$\sin \left( x + \frac { \pi } { 4 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answers in the form \(k \pi\), where \(k\) is a rational number. \includegraphics[max width=\textwidth, alt={}, center]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-14_677_1031_1446_445}
Edexcel C12 2016 June Q11
8 marks Standard +0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-16_892_825_228_548} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale Figure 2 shows the design for a sail \(A P B C A\). The curved edge \(A P B\) of the sail is an arc of a circle centre \(O\) and radius \(r \mathrm {~m}\). The straight edge \(A C B\) is a chord of the circle. The height \(A B\) of the sail is 2.4 m . The maximum width \(C P\) of the sail is 0.4 m .
  1. Show that \(r = 2\)
  2. Show, to 4 decimal places, that angle \(A O B = 1.2870\) radians.
  3. Hence calculate the area of the sail, giving your answer, in \(\mathrm { m } ^ { 2 }\), to 3 decimal places.
Edexcel C12 2016 June Q12
5 marks Moderate -0.5
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-18_636_887_274_534} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a circle \(C\) \(C\) touches the \(y\)-axis and has centre at the point ( \(a , 0\) ) where \(a\) is a positive constant.
  1. Write down an equation for \(C\) in terms of \(a\) Given that the point \(P ( 4 , - 3 )\) lies on \(C\),
  2. find the value of \(a\)
Edexcel C12 2016 June Q13
8 marks Standard +0.3
  1. (a) Show that the equation
$$2 \log _ { 2 } y = 5 - \log _ { 2 } x \quad x > 0 , y > 0$$ may be written in the form \(y ^ { 2 } = \frac { k } { x }\) where \(k\) is a constant to be found.
(b) Hence, or otherwise, solve the simultaneous equations $$\begin{gathered} 2 \log _ { 2 } y = 5 - \log _ { 2 } x \\ \log _ { x } y = - 3 \end{gathered}$$ for \(x > 0 , y > 0\)
Edexcel C12 2016 June Q14
8 marks Moderate -0.8
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-21_831_919_127_509} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the graph of \(y = g ( x ) , - 3 \leqslant x \leqslant 4\) and part of the line \(l\) with equation \(y = \frac { 1 } { 2 } x\) The graph of \(y = \mathrm { g } ( x )\) consists of three line segments, from \(P ( - 3,4 )\) to \(Q ( 0,4 )\), from \(Q ( 0,4 )\) to \(R ( 2,0 )\) and from \(R ( 2,0 )\) to \(S ( 4,10 )\). The line \(l\) intersects \(y = \mathrm { g } ( x )\) at the points \(A\) and \(B\) as shown in Figure 4.
  1. Use algebra to find the \(x\) coordinate of the point \(A\) and the \(x\) coordinate of the point \(B\). Show each step of your working and give your answers as exact fractions.
  2. Sketch the graph with equation $$y = \frac { 3 } { 2 } g ( x ) , \quad - 3 \leqslant x \leqslant 4$$ On your sketch show the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.
Edexcel C12 2016 June Q15
11 marks Standard +0.3
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-23_609_493_223_762} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a design for a water barrel.
It is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and radius \(r \mathrm {~cm}\). The barrel has a base but has no lid, is open at the top and is made of material of negligible thickness. The barrel is designed to hold \(60000 \mathrm {~cm} ^ { 3 }\) of water when full.
  1. Show that the total external surface area, \(S \mathrm {~cm} ^ { 2 }\), of the barrel is given by the formula $$S = \pi r ^ { 2 } + \frac { 120000 } { r }$$
  2. Use calculus to find the minimum value of \(S\), giving your answer to 3 significant figures.
  3. Justify that the value of \(S\) you found in part (b) is a minimum.