Questions P2 (856 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
CAIE P2 2002 June Q5
  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
  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
  1. Prove that \(\tan \theta + \cot \theta \equiv \frac { 2 } { \sin 2 \theta }\).
  2. Hence
    (a) find the exact value of \(\tan \frac { 1 } { 8 } \pi + \cot \frac { 1 } { 8 } \pi\),
    (b) evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 6 } { \tan \theta + \cot \theta } \mathrm { d } \theta\).
CAIE P2 2003 November Q4
  1. Express \(\cos \theta + ( \sqrt { } 3 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact value of \(\alpha\).
  2. Hence show that one solution of the equation $$\cos \theta + ( \sqrt { } 3 ) \sin \theta = \sqrt { } 2$$ is \(\theta = \frac { 7 } { 12 } \pi\), and find the other solution in the interval \(0 < \theta < 2 \pi\).
  3. By sketching a suitable pair of graphs, for \(x < 0\), show that exactly one root of the equation \(x ^ { 2 } = 2 ^ { x }\) is negative.
  4. Verify by calculation that this root lies between - 1.0 and - 0.5 .
  5. Use the iterative formula $$x _ { n + 1 } = - \sqrt { } \left( 2 ^ { x _ { n } } \right)$$ to determine this root correct to 2 significant figures, showing the result of each iteration.
CAIE P2 2007 November Q7
  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
  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.