Questions P2 (867 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
Edexcel P2 2018 Specimen Q1
7 marks Moderate -0.8
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
8 marks Moderate -0.3
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
7 marks Moderate -0.8
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
4 marks Moderate -0.8
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
7 marks Moderate -0.8
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
10 marks Moderate -0.5
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
12 marks Moderate -0.3
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 marks Moderate -0.3
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
11 marks Easy -1.2
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
7 marks Moderate -0.3
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
9 marks Moderate -0.3
  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 November Q6
9 marks Moderate -0.3
  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 \(B\).
CAIE P2 2002 June Q5
8 marks Standard +0.3
  1. Find the exact coordinates of \(P\).
  2. Show that the \(x\)-coordinates of \(Q\) and \(R\) satisfy the equation $$x = \frac { 1 } { 4 } e ^ { x } .$$
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 4 } e ^ { x _ { n } }$$ with initial value \(x _ { 1 } = 0\), to find the \(x\)-coordinate of \(Q\) correct to 2 decimal places, showing the value of each approximation that you calculate.
CAIE P2 2010 June Q6
8 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 2 - x ^ { 2 }$$ has only one root.
  2. Verify by calculation that this root lies between \(x = 1.3\) and \(x = 1.4\).
  3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)$$ converges, then it converges to the root of the equation in part (i).
  4. Use the iterative formula \(x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2014 June Q5
8 marks Standard +0.3
  1. Prove that \(\tan \theta + \cot \theta \equiv \frac { 2 } { \sin 2 \theta }\).
  2. Hence
    1. find the exact value of \(\tan \frac { 1 } { 8 } \pi + \cot \frac { 1 } { 8 } \pi\),
    2. evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 6 } { \tan \theta + \cot \theta } \mathrm { d } \theta\).
CAIE P2 2007 November Q7
8 marks Standard +0.3
  1. Prove the identity $$( \cos x + 3 \sin x ) ^ { 2 } \equiv 5 - 4 \cos 2 x + 3 \sin 2 x$$
  2. Using the identity, or otherwise, find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \cos x + 3 \sin x ) ^ { 2 } d x$$
CAIE P2 2017 November Q5
9 marks Standard +0.3
  1. Show that the \(x\)-coordinate of \(Q\) satisfies the equation \(x = \frac { 9 } { 8 } - \frac { 1 } { 2 } \mathrm { e } ^ { - 2 x }\).
  2. Use an iterative formula based on the equation in part (i) to find the \(x\)-coordinate of \(Q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2024 June Q1
4 marks Standard +0.3
Solve the inequality \(|5x + 7| > |2x - 3|\). [4]
CAIE P2 2024 June Q2
4 marks Standard +0.3
Use logarithms to solve the equation \(6^{2x-1} = 5e^{3x+2}\). Give your answer correct to 4 significant figures. [4]
CAIE P2 2024 June Q3
8 marks Moderate -0.3
\includegraphics{figure_3} The diagram shows the curve with equation \(y = 8e^{-x} - e^{2x}\). The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.
  1. Find the gradient of the curve at A. [3]
  2. Show that the x-coordinate of B is \(\ln 2\) and hence find the area of the shaded region. [5]
CAIE P2 2024 June Q4
7 marks Standard +0.3
A curve is defined by the parametric equations $$x = 4\cos^2 t, \quad y = \sqrt{3}\sin 2t,$$ for values of \(t\) such that \(0 < t < \frac{1}{2}\pi\). Find the equation of the normal to the curve at the point for which \(t = \frac{1}{6}\pi\). Give your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [7]
CAIE P2 2024 June Q5
8 marks Standard +0.3
The polynomial \(p(x)\) is defined by \(p(x) = 9x^3 + 18x^2 + 5x + 4\).
  1. Find the quotient when \(p(x)\) is divided by \((3x + 2)\), and show that the remainder is 6. [3]
  2. Find the value of \(\int_0^2 \frac{p(x)}{3x + 2} \, dx\), giving your answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers. [5]
CAIE P2 2024 June Q6
9 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{\ln(2x + 1)}{x + 3}\). The curve has a maximum point M.
  1. Find an expression for \(\frac{dy}{dx}\). [2]
  2. Show that the x-coordinate of M satisfies the equation \(x = \frac{x + 3}{\ln(2x + 1)} - 0.5\). [2]
  3. Show by calculation that the x-coordinate of M lies between 2.5 and 3.0. [2]
  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. [3]
CAIE P2 2024 June Q7
10 marks Standard +0.3
  1. Prove that \(2\sin\theta\cosec 2\theta \equiv \sec\theta\). [2]
  2. Solve the equation \(\tan^2\theta + 7\sin\theta\cosec 2\theta = 8\) for \(-\pi < \theta < \pi\). [5]
  3. Find \(\int 8\sin^2\frac{1}{2}x\cosec^2 x \, dx\). [3]
CAIE P2 2023 March Q1
4 marks Standard +0.3
Find the exact value of \(\int_0^{\frac{\pi}{4}} 2 \tan^2(\frac{1}{2}x) \, dx\). [4]