Questions P2 (856 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 P2 2022 October Q9
  1. In this question you must show detailed reasoning.
Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f926d53-c6de-4eb7-9d18-596f61ec26e1-26_723_455_413_804} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows
  • the curve \(C _ { 1 }\) with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 3 x + 14\)
  • the circle \(C _ { 2 }\) with centre \(T\)
The point \(T\) is the minimum turning point of \(C _ { 1 }\)
Using Figure 3 and calculus,
  1. find the coordinates of \(T\) The curve \(C _ { 1 }\) intersects the circle \(C _ { 2 }\) at the point \(A\) with \(x\) coordinate 2
  2. Find an equation of the circle \(C _ { 2 }\) The line \(l\) shown in Figure 3, is the tangent to circle \(C _ { 2 }\) at \(A\)
  3. Show that an equation of \(l\) is $$y = \frac { 1 } { 3 } x + \frac { 22 } { 3 }$$ The region \(R\), shown shaded in Figure 3, is bounded by \(C _ { 1 } , l\) and the \(y\)-axis.
  4. Find the exact area of \(R\).
Edexcel P2 2022 October Q10
  1. Given \(a = \log _ { 2 } 3\)
    1. write, in simplest form, in terms of \(a\),
      (a) \(\log _ { 2 } 9\)
      (b) \(\log _ { 2 } \left( \frac { \sqrt { 3 } } { 16 } \right)\)
    2. Solve
    $$3 ^ { x } \times 2 ^ { x + 4 } = 6$$ giving your answer, in simplest form, in terms of \(a\).
Edexcel P2 2023 October Q1
  1. Given that \(a , b\) and \(c\) are integers greater than 0 such that
  • \(c = 3 a + 1\)
  • \(a + b + c = 15\)
    prove, by exhaustion, that the product \(a b c\) is always a multiple of 4
    You may use the table below to illustrate your answer.
You may not need to use all rows of this table.
\(a\)\(b\)\(c\)\(a b c\)
Edexcel P2 2023 October Q2
  1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
$$\begin{gathered} u _ { 1 } = 3
u _ { n + 1 } = 2 - \frac { 4 } { u _ { n } } \end{gathered}$$
  1. Find the value of \(u _ { 2 }\), the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\)
  2. Find the value of $$\sum _ { r = 1 } ^ { 100 } u _ { r }$$
Edexcel P2 2023 October Q3
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\) the equation $$2 \tan \theta + 3 \sin \theta = 0$$ giving your answers, as appropriate, to one decimal place.
  2. Hence, or otherwise, find the smallest positive solution of $$2 \tan \left( 2 x + 40 ^ { \circ } \right) + 3 \sin \left( 2 x + 40 ^ { \circ } \right) = 0$$ giving your answer to one decimal place.
Edexcel P2 2023 October Q4
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } + a x ^ { 2 } - 29 x + b$$ where \(a\) and \(b\) are constants.
Given that \(( 2 x + 1 )\) is a factor of \(\mathrm { f } ( x )\),
  1. show that $$a + 4 b = - 56$$ Given also that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is - 25
  2. find a second simplified equation linking \(a\) and \(b\).
  3. Hence, using algebra and showing your working,
    1. find the value of \(a\) and the value of \(b\),
    2. fully factorise \(\mathrm { f } ( x )\).
Edexcel P2 2023 October Q5
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Solve $$3 ^ { a } = 70$$ giving the answer to 3 decimal places.
  2. Find the exact value of \(b\) such that $$4 + 3 \log _ { 3 } b = \log _ { 3 } 5 b$$
Edexcel P2 2023 October Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{66abdef1-072e-41eb-a933-dd51a96330ff-14_488_1511_246_278} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A river is being studied.
At one particular place, the river is 15 m wide.
The depth, \(y\) metres, of the river is measured at a point \(x\) metres from one side of the river. Figure 1 shows a plot of the cross-section of the river and the coordinate values \(( x , y )\)
  1. Use the trapezium rule with all the \(y\) values given in Figure 1 to estimate the cross-sectional area of the river. The water in the river is modelled as flowing at a constant speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) across the whole of the cross-section.
  2. Use the model and the answer to part (a) to estimate the volume of water flowing through this section of the river each minute, giving your answer in \(\mathrm { m } ^ { 3 }\) to 2 significant figures. Assuming the model,
  3. state, giving a reason for your answer, whether your answer for part (b) is an overestimate or an underestimate of the true volume of water flowing through this section of the river each minute.
Edexcel P2 2023 October Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{66abdef1-072e-41eb-a933-dd51a96330ff-16_949_940_246_566} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of
  • the circle \(C\) with centre \(X ( 4 , - 3 )\)
  • the line \(l\) with equation \(y = \frac { 5 } { 2 } x - \frac { 55 } { 2 }\)
Given that \(l\) is the tangent to \(C\) at the point \(N\),
  1. show that an equation for the straight line passing through \(X\) and \(N\) is $$2 x + 5 y + 7 = 0$$
  2. Hence find
    1. the coordinates of \(N\),
    2. an equation for \(C\).
Edexcel P2 2023 October Q8
  1. In a large theatre there are \(n\) rows of seats, where \(n\) is a constant.
The number of seats in the first row is \(a\), where \(a\) is a constant.
In each subsequent row there are 4 more seats than in the previous row so that
  • in the 2 nd row there are \(( a + 4 )\) seats
  • in the 3rd row there are ( \(a + 8\) ) seats
  • the number of seats in each row form an arithmetic sequence
Given that the total number of seats in the first 10 rows is 360
  1. find the value of \(a\). Given also that the total number of seats in the \(n\) rows is 2146
  2. show that $$n ^ { 2 } + 8 n - 1073 = 0$$
  3. Hence
    1. state the number of rows of seats in the theatre,
    2. find the maximum number of seats in any one row.
Edexcel P2 2023 October Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{66abdef1-072e-41eb-a933-dd51a96330ff-24_803_1050_251_511} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 2 } { 3 } x ^ { 2 } - 9 \sqrt { x } + 13 \quad x \geqslant 0$$
  1. Find, using calculus, the range of values of \(x\) for which \(y\) is increasing. The point \(P\) lies on \(C\) and has coordinates (9, 40).
    The line \(l\) is the tangent to \(C\) at the point \(P\).
    The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the line \(l\), the \(x\)-axis and the \(y\)-axis.
  2. Find, using calculus, the exact area of \(R\).
Edexcel P2 2023 October Q10
  1. (i) (a) Find, in ascending powers of \(x\), the 2nd, 3rd and 5th terms of the binomial expansion of
$$( 3 + 2 x ) ^ { 6 }$$ For a particular value of \(x\), these three terms form consecutive terms in a geometric series.
(b) Find this value of \(x\).
(ii) In a different geometric series,
  • the first term is \(\sin ^ { 2 } \theta\)
  • the common ratio is \(2 \cos \theta\)
  • the sum to infinity is \(\frac { 8 } { 5 }\)
    (a) Show that
$$5 \cos ^ { 2 } \theta - 16 \cos \theta + 3 = 0$$ (b) Hence find the exact value of the 2nd term in the series.
Edexcel P2 2018 Specimen Q1
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 \(b\)
    VIIIV SIHI NI JIIIM ION OCVIIV SIHI NI JINAM ION OCVEYV SIHI NI JULIM ION OO
Edexcel P2 2018 Specimen Q2
2. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\). The sum to infinity of the series is \(S _ { \infty }\)
  1. Find the value of \(S _ { \infty }\) The sum to \(N\) terms of the series is \(S _ { N }\)
  2. Find, to 1 decimal place, the value of \(S _ { 12 }\)
  3. Find the smallest value of \(N\), for which \(S _ { \infty } - S _ { N } < 0.5\)
    2. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\). The sum to infinity
    of the series is \(S _ { \infty }\)
Edexcel P2 2018 Specimen Q3
3. $$y = \sqrt { \left( 3 ^ { x } + x \right) }$$
  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.
  3. Explain how the trapezium rule could be used to obtain a more accurate estimate for the value of $$\int _ { 0 } ^ { 1 } \sqrt { \left( 3 ^ { x } + x \right) } d x$$
    \includegraphics[max width=\textwidth, alt={}]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-10_2673_1948_107_118}
Edexcel P2 2018 Specimen Q4
Given \(n \in \mathbb { N }\), prove, by exhaustion, that \(n ^ { 2 } + 2\) is not divisible by 4 .
\includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-12_2658_1943_111_118}
Edexcel P2 2018 Specimen Q6
6. (i) Find the exact value of \(x\) for which $$\log _ { 2 } ( 2 x ) = \log _ { 2 } ( 5 x + 4 ) - 3$$ (ii) Given that $$\log _ { a } y + 3 \log _ { a } 2 = 5$$ express \(y\) in terms of \(a\). Give your answer in its simplest form.
\includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-18_2674_1948_107_118}
Edexcel P2 2018 Specimen Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-19_739_871_260_532} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The circle with equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 16 y + 139 = 0$$ had centre \(C\) and radius \(r\).
  1. Find the coordinates of \(C\).
  2. Show that \(r = 5\) The line with equation \(x = 13\) crosses the circle at the points \(P\) and \(Q\) as shown in Figure 1 .
  3. Find the \(y\) coordinate of \(P\) and the \(y\) coordinate of \(Q\). A tangent to the circle from \(O\) touches the circle at point \(X\).
  4. Find, in surd form, the length \(O X\).
    \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-22_2673_1948_107_118}
Edexcel P2 2018 Specimen Q8
8. Figure 2 Figure 2 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations $$\begin{array} { l l } C _ { 1 } : y = 10 x - x ^ { 2 } - 8 & x > 0
C _ { 2 } : y = x ^ { 3 } & x > 0 \end{array}$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(A\) and \(B\).
  1. Verify that the point \(A\) has coordinates (1, 1)
  2. Use algebra to find the coordinates of the point \(B\) The finite region \(R\) is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
  3. Use calculus to find the exact area of \(R\)
    \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-23_936_759_118_582} \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-26_2674_1948_107_118}
Edexcel P2 2018 Specimen Q9
9. (i) Solve, for \(0 \leqslant \theta < \pi\), the equation $$\sin 3 \theta - \sqrt { 3 } \cos 3 \theta = 0$$ giving your answers in terms of \(\pi\)
(ii) Given that $$4 \sin ^ { 2 } x + \cos x = 4 - k , \quad 0 \leqslant k \leqslant 3$$
  1. find \(\cos x\) in terms of \(k\)
  2. When \(k = 3\), find the values of \(x\) in the range \(0 \leqslant x < 360 ^ { \circ }\)
    \includegraphics[max width=\textwidth, alt={}]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-30_2671_1942_107_121}
Edexcel P2 2018 Specimen Q5
An arithmetic series has first term \(a\) and common difference \(d\).
  1. Prove that the sum of the first \(n\) terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]$$ A company, which is making 200 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to 220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week \(N\).
  2. Find the value of \(N\) The company then plans to continue to make 600 mobile phones each week.
  3. Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.
    \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-16_2673_1948_107_118}
CAIE P2 2024 November Q4
4
  1. Sketch the graphs of \(y = 1 + \mathrm { e } ^ { 2 x }\) and \(y = | x - 4 |\) on the same diagram.
  2. The two graphs meet at the point \(P\) .
    Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 2 } \ln ( 3 - x )\) .
    \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-06_2716_38_109_2012}
  3. Use an iterative formula, based on the equation in part (b), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Use an initial value of 0.45 and give the result of each iteration to 5 significant figures.
CAIE P2 2024 June Q6
  1. Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 significant figures.
  2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.
    \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_65_1548_379_349}
    \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_67_1566_466_328} ........................................................................................................................................
    \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_70_1570_646_324}
    \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1570_735_324}
    \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1570_826_324}
    \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_74_1570_916_324} ........................................................................................................................................ .
    \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1572_1096_322}
    \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_70_1570_1187_324}
    \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_67_1570_1279_324}
    \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_70_1570_1367_324}
    \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_62_1570_1462_324} .......................................................................................................................................... ......................................................................................................................................... .
    \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1570_1724_324} .......................................................................................................................................... .
    \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_71_1570_1905_324}
    \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_74_1570_1994_324}
    \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_76_1570_2083_324} ......................................................................................................................................... . ........................................................................................................................................ ......................................................................................................................................... ........................................................................................................................................ . ......................................................................................................................................... . ........................................................................................................................................
CAIE P2 2024 June Q6
  1. Find an expression for \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
  2. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { x + 3 } { \ln ( 2 x + 1 ) } - 0.5\).
  3. Show by calculation that the \(x\)-coordinate of \(M\) lies between 2.5 and 3.0 .
  4. Use an iterative formula based on the equation in part (b) to find the \(x\)-coordinate of \(M\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2024 November Q6
  1. Use the trapezium rule with two intervals to find an approximation to the area of region \(A\). Give your answer correct to 3 significant figures.
    \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_2720_38_105_2010}
    \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-11_2716_29_107_22}
  2. Find the exact total area of regions \(A\) and \(B\). Give your answer in the form \(k \ln m\), where \(k\) and \(m\) are constants.
  3. Deduce an approximation to the area of region \(B\). Give your answer correct to 3 significant figures.
  4. State, with a reason, whether your answer to part (c) is an over-estimate or an under-estimate of the area of region \(B\).