Questions — Edexcel C12 (247 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel C12 2014 June Q11
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
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
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
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
  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
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. 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
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
  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.)
Edexcel C12 2015 June Q6
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. \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
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
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
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
  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
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
  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
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
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
  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
  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
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
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
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
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.