Questions — CAIE P2 (699 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 2016 March Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{d53a2d6b-4c5e-4bc6-8aa1-587e97c87920-2_371_839_1409_651} The diagram shows the part of the curve \(y = 3 \mathrm { e } ^ { - x } \sin 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and the stationary point \(M\).
  1. Find the equation of the tangent to the curve at the origin.
  2. Find the coordinates of \(M\), giving each coordinate correct to 3 decimal places.
CAIE P2 2016 March Q7
7 The equation of a curve is \(2 x ^ { 3 } + y ^ { 3 } = 24\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\), and show that the gradient of the curve is never positive.
  2. Find the coordinates of the two points on the curve at which the gradient is - 2 .
CAIE P2 2016 March Q8
8
  1. Show that \(\sin 2 x \cot x \equiv 2 \cos ^ { 2 } x\).
  2. Using the identity in part (i),
    (a) find the least possible value of $$3 \sin 2 x \cot x + 5 \cos 2 x + 8$$ as \(x\) varies,
    (b) find the exact value of \(\int _ { \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 6 } \pi } \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).
CAIE P2 2017 March Q1
1 Solve the equation \(2 \ln ( 2 x ) - \ln ( x + 3 ) = \ln ( 3 x + 5 )\).
CAIE P2 2017 March Q2
2
  1. Given that \(\tan 2 \theta \cot \theta = 8\), show that \(\tan ^ { 2 } \theta = \frac { 3 } { 4 }\).
  2. Hence solve the equation \(\tan 2 \theta \cot \theta = 8\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2017 March Q3
3
  1. Solve the inequality \(| 2 x - 5 | < | x + 3 |\).
  2. Hence find the largest integer \(y\) satisfying the inequality \(| 2 \ln y - 5 | < | \ln y + 3 |\).
CAIE P2 2017 March Q4
4 Find the gradient of the curve $$x ^ { 2 } \sin y + \cos 3 y = 4$$ at the point \(\left( 2 , \frac { 1 } { 2 } \pi \right)\).
CAIE P2 2017 March Q5
5 It is given that \(a\) is a positive constant such that $$\int _ { 0 } ^ { a } \left( 1 + 2 x + 3 \mathrm { e } ^ { 3 x } \right) \mathrm { d } x = 250$$
  1. Show that \(a = \frac { 1 } { 3 } \ln \left( 251 - a - a ^ { 2 } \right)\).
  2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2017 March Q6
6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 17 x - a$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 28 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\).
  1. Find the values of \(a\) and \(b\).
  2. Hence factorise \(\mathrm { p } ( x )\) completely.
  3. State the number of roots of the equation \(\mathrm { p } \left( 2 ^ { y } \right) = 0\), justifying your answer.
    \includegraphics[max width=\textwidth, alt={}, center]{17025451-6f07-4f35-9dfc-869e084b5ed0-10_508_538_310_799} The diagram shows part of the curve $$y = 2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)$$ The shaded region is bounded by the curve and the two axes.
  4. Show that \(2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)\) can be expressed in the form $$k _ { 1 } ( 1 + \cos 4 x ) + k _ { 2 } \sin 4 x ,$$ where the values of the constants \(k _ { 1 }\) and \(k _ { 2 }\) are to be determined.
  5. Find the exact area of the shaded region.
CAIE P2 2019 March Q1
1 Solve the equation \(\sec ^ { 2 } \theta + \tan ^ { 2 } \theta = 5 \tan \theta + 4\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\). Show all necessary working.
CAIE P2 2019 March Q2
2 Given that \(x\) satisfies the equation \(| 2 x + 3 | = | 2 x - 1 |\), find the value of $$| 4 x - 3 | - | 6 x |$$
CAIE P2 2019 March Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{772c14a1-f79a-4147-a293-0ff34f930e20-04_577_569_260_788} The variables \(x\) and \(y\) satisfy the equation \(y = A \mathrm { e } ^ { p x + p }\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 1,2.835 )\) and \(( 6,6.585 )\), as shown in the diagram. Find the values of \(A\) and \(p\).
CAIE P2 2019 March Q4
4
  1. Find the quotient when \(4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 9\) is divided by ( \(2 x + 1\) ), and show that the remainder is 5 .
  2. Show that the equation \(4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 4 = 0\) has exactly one real root.
CAIE P2 2019 March Q5
5 The equation of a curve is \(y = \frac { \mathrm { e } ^ { 2 x } } { 4 x + 1 }\) and the point \(P\) on the curve has \(y\)-coordinate 10 .
  1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 2 } \ln ( 40 x + 10 )\).
  2. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 40 x _ { n } + 10 \right)\) with \(x _ { 1 } = 2.3\) to find the \(x\)-coordinate of \(P\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
  3. Find the gradient of the curve at \(P\), giving the answer correct to 3 significant figures.
CAIE P2 2019 March Q6
6
  1. Show that \(\int _ { 1 } ^ { 4 } \left( \frac { 2 } { x } + \frac { 2 } { 2 x + 1 } \right) \mathrm { d } x = \ln 48\).
  2. Find \(\int \sin 2 x ( \cot x + 2 \operatorname { cosec } x ) \mathrm { d } x\).
CAIE P2 2019 March Q7
7 The parametric equations of a curve are $$x = 2 t - \sin 2 t , \quad y = 5 t + \cos 2 t$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). At the point \(P\) on the curve, the gradient of the curve is 2 .
  1. Show that the value of the parameter at \(P\) satisfies the equation \(2 \sin 2 t - 4 \cos 2 t = 1\).
  2. By first expressing \(2 \sin 2 t - 4 \cos 2 t\) in the form \(R \sin ( 2 t - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), find the coordinates of \(P\). Give each coordinate correct to 3 significant figures.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2002 November Q1
1 Solve the inequality \(| 2 x - 1 | < | 3 x |\).
CAIE P2 2002 November Q2
2 The cubic polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\), and that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is - 5 . Find the values of \(a\) and \(b\).
CAIE P2 2002 November Q3
3
  1. Express \(9 ^ { x }\) in terms of \(y\), where \(y = 3 ^ { x }\).
  2. Hence solve the equation $$2 \left( 9 ^ { x } \right) - 7 \left( 3 ^ { x } \right) + 3 = 0 ,$$ expressing your answers for \(x\) in terms of logarithms where appropriate.
CAIE P2 2002 November Q4
4
  1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) in the interval \(0 < x < \frac { 1 } { 2 } \pi\) that is a root of the equation $$\sin x = \frac { 1 } { x ^ { 2 } }$$
  2. Verify by calculation that this root lies between 1 and 1.5.
  3. Show that this value of \(x\) is also a root of the equation $$x = \sqrt { } ( \operatorname { cosec } x )$$
  4. Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \operatorname { cosec } x _ { n } \right)$$ to determine this root correct to 3 significant figures, showing the value of each approximation that you calculate.
CAIE P2 2002 November Q5
5 The angle \(x\), measured in degrees, satisfies the equation $$\cos \left( x - 30 ^ { \circ } \right) = 3 \sin \left( x - 60 ^ { \circ } \right)$$
  1. By expanding each side, show that the equation may be simplified to $$( 2 \sqrt { } 3 ) \cos x = \sin x$$
  2. Find the two possible values of \(x\) lying between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
  3. Find the exact value of \(\cos 2 x\), giving your answer as a fraction.
CAIE P2 2002 November Q6
6
  1. Find the value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } ( \sin 2 x + \cos x ) \mathrm { d } x\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{9894d97f-3b7b-4dbe-b94a-2c8415442038-3_517_880_422_669} The diagram shows part of the curve \(y = \frac { 1 } { x + 1 }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 1 , y = 0\) and \(x = p\).
    1. Find, in terms of \(p\), the area of \(R\).
    2. Hence find, correct to 1 decimal place, the value of \(p\) for which the area of \(R\) is equal to 2 .
CAIE P2 2002 November Q7
7 The equation of a curve is $$2 x ^ { 2 } + 3 y ^ { 2 } - 2 x y = 10 .$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - 2 x } { 3 y - x }\).
  2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.
CAIE P2 2003 November Q1
1 Find the set of values of \(x\) satisfying the inequality \(| 8 - 3 x | < 2\).
CAIE P2 2003 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{f5be66c3-7234-4039-9b66-199b35430c7d-2_750_1102_404_520} Two variable quantities \(x\) and \(y\) are related by the equation $$y = k \left( a ^ { - x } \right)$$ where \(a\) and \(k\) are constants. Four pairs of values of \(x\) and \(y\) are measured experimentally. The result of plotting \(\ln y\) against \(x\) is shown in the diagram. Use the diagram to estimate the values of \(a\) and \(k\).