Questions — Edexcel (9670 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 C2 2009 January Q8
8 marks Moderate -0.3
8. (a) Show that the equation $$4 \sin ^ { 2 } x + 9 \cos x - 6 = 0$$ can be written as $$4 \cos ^ { 2 } x - 9 \cos x + 2 = 0$$ (b) Hence solve, for \(0 \leqslant x < 720 ^ { \circ }\), $$4 \sin ^ { 2 } x + 9 \cos x - 6 = 0$$ giving your answers to 1 decimal place.
Edexcel C2 2009 January Q9
10 marks Moderate -0.3
  1. The first three terms of a geometric series are ( \(k + 4\) ), \(k\) and ( \(2 k - 15\) ) respectively, where \(k\) is a positive constant.
    1. Show that \(k ^ { 2 } - 7 k - 60 = 0\).
    2. Hence show that \(k = 12\).
    3. Find the common ratio of this series.
    4. Find the sum to infinity of this series.
Edexcel C2 2009 January Q10
12 marks Standard +0.3
10. A solid right circular cylinder has radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The total surface area of the cylinder is \(800 \mathrm {~cm} ^ { 2 }\).
  1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = 400 r - \pi r ^ { 3 }$$ Given that \(r\) varies,
  2. use calculus to find the maximum value of \(V\), to the nearest \(\mathrm { cm } ^ { 3 }\).
  3. Justify that the value of \(V\) you have found is a maximum.
    \includegraphics[max width=\textwidth, alt={}, center]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-16_103_63_2477_1873}
Edexcel C2 2010 January Q1
4 marks Easy -1.2
  1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$( 3 - x ) ^ { 6 }$$ and simplify each term.
Edexcel C2 2010 January Q2
6 marks Moderate -0.3
2. (a) Show that the equation $$5 \sin x = 1 + 2 \cos ^ { 2 } x$$ can be written in the form $$2 \sin ^ { 2 } x + 5 \sin x - 3 = 0$$ (b) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$2 \sin ^ { 2 } x + 5 \sin x - 3 = 0$$
Edexcel C2 2010 January Q3
9 marks Moderate -0.3
3. $$f ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x - 6$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is - 5 .
When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) there is no remainder.
  1. Find the value of \(a\) and the value of \(b\).
  2. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2010 January Q4
7 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e3faf018-37a8-48ef-b100-81402a8ec87f-05_556_1189_237_413} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} An emblem, as shown in Figure 1, consists of a triangle \(A B C\) joined to a sector \(C B D\) of a circle with radius 4 cm and centre \(B\). The points \(A , B\) and \(D\) lie on a straight line with \(A B = 5 \mathrm {~cm}\) and \(B D = 4 \mathrm {~cm}\). Angle \(B A C = 0.6\) radians and \(A C\) is the longest side of the triangle \(A B C\).
  1. Show that angle \(A B C = 1.76\) radians, correct to 3 significant figures.
  2. Find the area of the emblem.
Edexcel C2 2010 January Q5
8 marks Moderate -0.3
5. (a) Find the positive value of \(x\) such that $$\log _ { x } 64 = 2$$ (b) Solve for \(x\) $$\log _ { 2 } ( 11 - 6 x ) = 2 \log _ { 2 } ( x - 1 ) + 3$$
Edexcel C2 2010 January Q6
9 marks Moderate -0.8
6. A car was purchased for \(\pounds 18000\) on 1 st January. On 1st January each following year, the value of the car is \(80 \%\) of its value on 1st January in the previous year.
  1. Show that the value of the car exactly 3 years after it was purchased is \(\pounds 9216\). The value of the car falls below \(\pounds 1000\) for the first time \(n\) years after it was purchased.
  2. Find the value of \(n\). An insurance company has a scheme to cover the maintenance of the car. The cost is \(\pounds 200\) for the first year, and for every following year the cost increases by \(12 \%\) so that for the 3rd year the cost of the scheme is \(\pounds 250.88\)
  3. Find the cost of the scheme for the 5th year, giving your answer to the nearest penny.
  4. Find the total cost of the insurance scheme for the first 15 years.
    \section*{LU}
Edexcel C2 2010 January Q7
10 marks Moderate -0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e3faf018-37a8-48ef-b100-81402a8ec87f-09_696_821_205_516} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The curve \(C\) has equation \(y = x ^ { 2 } - 5 x + 4\). It cuts the \(x\)-axis at the points \(L\) and \(M\) as shown in Figure 2.
  1. Find the coordinates of the point \(L\) and the point \(M\).
  2. Show that the point \(N ( 5,4 )\) lies on \(C\).
  3. Find \(\int \left( x ^ { 2 } - 5 x + 4 \right) \mathrm { d } x\). The finite region \(R\) is bounded by \(L N , L M\) and the curve \(C\) as shown in Figure 2.
  4. Use your answer to part (c) to find the exact value of the area of \(R\).
    \section*{LU}
Edexcel C2 2010 January Q8
12 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e3faf018-37a8-48ef-b100-81402a8ec87f-11_1262_1178_203_386} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the circle \(C\) with centre \(N\) and equation $$( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = \frac { 169 } { 4 }$$
  1. Write down the coordinates of \(N\).
  2. Find the radius of \(C\). The chord \(A B\) of \(C\) is parallel to the \(x\)-axis, lies below the \(x\)-axis and is of length 12 units as shown in Figure 3.
  3. Find the coordinates of \(A\) and the coordinates of \(B\).
  4. Show that angle \(A N B = 134.8 ^ { \circ }\), to the nearest 0.1 of a degree. The tangents to \(C\) at the points \(A\) and \(B\) meet at the point \(P\).
  5. Find the length \(A P\), giving your answer to 3 significant figures.
Edexcel C2 2010 January Q9
10 marks Moderate -0.8
9. The curve \(C\) has equation \(y = 12 \sqrt { } ( x ) - x ^ { \frac { 3 } { 2 } } - 10 , \quad x > 0\)
  1. Use calculus to find the coordinates of the turning point on \(C\).
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  3. State the nature of the turning point.
Edexcel C2 2011 January Q1
7 marks Moderate -0.3
1. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is 7 .
  1. Show that \(a + b = 3\). When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\), the remainder is - 8 .
  2. Find the value of \(a\) and the value of \(b\).
Edexcel C2 2011 January Q2
6 marks Moderate -0.8
2. In the triangle \(A B C , A B = 11 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and \(C A = 8 \mathrm {~cm}\).
  1. Find the size of angle \(C\), giving your answer in radians to 3 significant figures.
  2. Find the area of triangle \(A B C\), giving your answer in \(\mathrm { cm } ^ { 2 }\) to 3 significant figures.
Edexcel C2 2011 January Q3
7 marks Moderate -0.3
3. The second and fifth terms of a geometric series are 750 and - 6 respectively. Find
  1. the common ratio of the series,
  2. the first term of the series,
  3. the sum to infinity of the series.
Edexcel C2 2011 January Q4
7 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be8f9187-055a-476f-974d-22e8e16e9996-05_547_798_251_575} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = ( x + 1 ) ( x - 5 )$$ The curve crosses the \(x\)-axis at the points \(A\) and \(B\).
  1. Write down the \(x\)-coordinates of \(A\) and \(B\). The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis.
  2. Use integration to find the area of \(R\).
Edexcel C2 2011 January Q5
4 marks Moderate -0.8
  1. Given that \(\binom { 40 } { 4 } = \frac { 40 ! } { 4 ! b ! }\),
    1. write down the value of \(b\).
    In the binomial expansion of \(( 1 + x ) ^ { 40 }\), the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) are \(p\) and \(q\) respectively.
  2. Find the value of \(\frac { q } { p }\).
Edexcel C2 2011 January Q6
9 marks Moderate -0.8
6. $$y = \frac { 5 } { 3 x ^ { 2 } - 2 }$$
  1. Complete the table below, giving the values of \(y\) to 2 decimal places.
    \(x\)22.252.52.753
    \(y\)0.50.380.2
  2. Use the trapezium rule, with all the values of \(y\) from your table, to find an approximate value for \(\int _ { 2 } ^ { 3 } \frac { 5 } { 3 x ^ { 2 } - 2 } \mathrm {~d} x\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{be8f9187-055a-476f-974d-22e8e16e9996-08_537_743_941_603} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \frac { 5 } { 3 x ^ { 2 } - 2 } , x > 1\).
    At the points \(A\) and \(B\) on the curve, \(x = 2\) and \(x = 3\) respectively.
    The region \(S\) is bounded by the curve, the straight line through \(B\) and ( 2,0 ), and the line through \(A\) parallel to the \(y\)-axis. The region \(S\) is shown shaded in Figure 2.
  3. Use your answer to part (b) to find an approximate value for the area of \(S\).
Edexcel C2 2011 January Q7
7 marks Moderate -0.3
  1. (a) Show that the equation
$$3 \sin ^ { 2 } x + 7 \sin x = \cos ^ { 2 } x - 4$$ can be written in the form $$4 \sin ^ { 2 } x + 7 \sin x + 3 = 0$$ (b) Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$3 \sin ^ { 2 } x + 7 \sin x = \cos ^ { 2 } x - 4$$ giving your answers to 1 decimal place where appropriate.
Edexcel C2 2011 January Q8
8 marks Moderate -0.8
  1. (a) Sketch the graph of \(y = 7 ^ { x } , x \in \mathbb { R }\), showing the coordinates of any points at which the graph crosses the axes.
    (b) Solve the equation
$$7 ^ { 2 x } - 4 \left( 7 ^ { x } \right) + 3 = 0$$ giving your answers to 2 decimal places where appropriate.
Edexcel C2 2011 January Q9
10 marks Moderate -0.8
9. The points \(A\) and \(B\) have coordinates \(( - 2,11 )\) and \(( 8,1 )\) respectively. Given that \(A B\) is a diameter of the circle \(C\),
  1. show that the centre of \(C\) has coordinates \(( 3,6 )\),
  2. find an equation for \(C\).
  3. Verify that the point \(( 10,7 )\) lies on \(C\).
  4. Find an equation of the tangent to \(C\) at the point (10, 7), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C2 2011 January Q10
10 marks Standard +0.3
  1. The volume \(V \mathrm {~cm} ^ { 3 }\) of a box, of height \(x \mathrm {~cm}\), is given by
$$V = 4 x ( 5 - x ) ^ { 2 } , \quad 0 < x < 5$$
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
  2. Hence find the maximum volume of the box.
  3. Use calculus to justify that the volume that you found in part (b) is a maximum.
Edexcel C2 2012 January Q1
6 marks Easy -1.2
  1. A geometric series has first term \(a = 360\) and common ratio \(r = \frac { 7 } { 8 }\)
Giving your answers to 3 significant figures where appropriate, find
  1. the 20 th term of the series,
  2. the sum of the first 20 terms of the series,
  3. the sum to infinity of the series.
Edexcel C2 2012 January Q2
4 marks Easy -1.8
2. A circle \(C\) has centre \(( - 1,7 )\) and passes through the point \(( 0,0 )\). Find an equation for \(C\).
(4)
Edexcel C2 2012 January Q3
7 marks Moderate -0.8
3. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$\left( 1 + \frac { x } { 4 } \right) ^ { 8 }$$ giving each term in its simplest form.
(b) Use your expansion to estimate the value of \(( 1.025 ) ^ { 8 }\), giving your answer to 4 decimal places.