Questions — CAIE P3 (1070 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
CAIE P3 2018 November Q7
4 marks
7
\includegraphics[max width=\textwidth, alt={}, center]{2a3df76c-2323-470c-8586-009753a4c1e3-12_357_565_260_790} The diagram shows the curve \(y = 5 \sin ^ { 2 } x \cos ^ { 3 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.
  2. Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\). [4]
CAIE P3 2018 November Q9
9 Let \(f ( x ) = \frac { 6 x ^ { 2 } + 8 x + 9 } { ( 2 - x ) ( 3 + 2 x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence, showing all necessary working, show that \(\int _ { - 1 } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x = 1 + \frac { 1 } { 2 } \ln \left( \frac { 3 } { 4 } \right)\).
CAIE P3 2019 November Q1
1 Given that \(\ln \left( 1 + \mathrm { e } ^ { 2 y } \right) = x\), express \(y\) in terms of \(x\).
CAIE P3 2019 November Q2
2 Solve the inequality \(| 2 x - 3 | > 4 | x + 1 |\).
CAIE P3 2019 November Q3
3 The parametric equations of a curve are $$x = 2 t + \sin 2 t , \quad y = \ln ( 1 - \cos 2 t )$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosec } 2 t\).
CAIE P3 2019 November Q4
4 The number of insects in a population \(t\) weeks after the start of observations is denoted by \(N\). The population is decreasing at a rate proportional to \(N \mathrm { e } ^ { - 0.02 t }\). The variables \(N\) and \(t\) are treated as continuous, and it is given that when \(t = 0 , N = 1000\) and \(\frac { \mathrm { d } N } { \mathrm {~d} t } = - 10\).
  1. Show that \(N\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = - 0.01 \mathrm { e } ^ { - 0.02 t } N .$$
  2. Solve the differential equation and find the value of \(t\) when \(N = 800\).
  3. State what happens to the value of \(N\) as \(t\) becomes large.
CAIE P3 2019 November Q5
5 The curve with equation \(y = \mathrm { e } ^ { - 2 x } \ln ( x - 1 )\) has a stationary point when \(x = p\).
  1. Show that \(p\) satisfies the equation \(x = 1 + \exp \left( \frac { 1 } { 2 ( x - 1 ) } \right)\), where \(\exp ( x )\) denotes \(\mathrm { e } ^ { x }\).
  2. Verify by calculation that \(p\) lies between 2.2 and 2.6.
  3. Use an iterative formula based on the equation in part (i) to determine \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2019 November Q6
6
  1. By differentiating \(\frac { \cos x } { \sin x }\), show that if \(y = \cot x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
  2. Show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } x \operatorname { cosec } ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 4 } ( \pi + \ln 4 )\).
    \(7 \quad\) Two lines \(l\) and \(m\) have equations \(\mathbf { r } = a \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\) respectively, where \(a\) is a constant. It is given that the lines intersect.
CAIE P3 2019 November Q8
8 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + x + 6 } { x ^ { 2 } ( x + 2 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence, showing full working, show that the exact value of \(\int _ { 1 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x\) is \(\frac { 9 } { 4 }\).
CAIE P3 2019 November Q9
9
  1. By first expanding \(\cos ( 2 x + x )\), show that \(\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x\).
  2. Hence solve the equation \(\cos 3 x + 3 \cos x + 1 = 0\), for \(0 \leqslant x \leqslant \pi\).
  3. Find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 3 } x \mathrm {~d} x\).
CAIE P3 2019 November Q10
5 marks
10
  1. The complex number \(u\) is given by \(u = - 3 - ( 2 \sqrt { } 10 )\) i. Showing all necessary working and without using a calculator, find the square roots of \(u\). Give your answers in the form \(a + \mathrm { i } b\), where the numbers \(a\) and \(b\) are real and exact.
  2. On a sketch of an Argand diagram shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 3 - \mathrm { i } | \leqslant 3 , \arg z \geqslant \frac { 1 } { 4 } \pi\) and \(\operatorname { Im } z \geqslant 2\), where \(\operatorname { Im } z\) denotes the imaginary part of the complex number \(z\).
    [0pt] [5] 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 P3 2019 November Q1
1 Solve the equation \(5 \ln \left( 4 - 3 ^ { x } \right) = 6\). Show all necessary working and give the answer correct to 3 decimal places.
CAIE P3 2019 November Q2
2 The curve with equation \(y = \frac { \mathrm { e } ^ { - 2 x } } { 1 - x ^ { 2 } }\) has a stationary point in the interval \(- 1 < x < 1\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the \(x\)-coordinate of this stationary point, giving the answer correct to 3 decimal places.
CAIE P3 2019 November Q3
3 The polynomial \(x ^ { 4 } + 3 x ^ { 3 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + x - 1\) the remainder is \(2 x + 3\). Find the values of \(a\) and \(b\).
CAIE P3 2019 November Q4
4
  1. Express \(( \sqrt { } 6 ) \sin x + \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 3 decimal places.
  2. Hence solve the equation \(( \sqrt { } 6 ) \sin 2 \theta + \cos 2 \theta = 2\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2019 November Q5
5 The equation of a curve is \(2 x ^ { 2 } y - x y ^ { 2 } = a ^ { 3 }\), where \(a\) is a positive constant. Show that there is only one point on the curve at which the tangent is parallel to the \(x\)-axis and find the \(y\)-coordinate of this point.
CAIE P3 2019 November Q6
6 The variables \(x\) and \(\theta\) satisfy the differential equation $$\sin \frac { 1 } { 2 } \theta \frac { d x } { d \theta } = ( x + 2 ) \cos \frac { 1 } { 2 } \theta$$ for \(0 < \theta < \pi\). It is given that \(x = 1\) when \(\theta = \frac { 1 } { 3 } \pi\). Solve the differential equation and obtain an expression for \(x\) in terms of \(\cos \theta\).
CAIE P3 2019 November Q7
7
  1. Find the complex number \(z\) satisfying the equation $$z + \frac { \mathrm { i } z } { z ^ { * } } - 2 = 0$$ where \(z ^ { * }\) denotes the complex conjugate of \(z\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On a single Argand diagram sketch the loci given by the equations \(| z - 2 \mathrm { i } | = 2\) and \(\operatorname { Im } z = 3\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\).
    2. In the first quadrant the two loci intersect at the point \(P\). Find the exact argument of the complex number represented by \(P\).
CAIE P3 2019 November Q8
8 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + x + 8 } { ( 2 x - 1 ) \left( x ^ { 2 } + 2 \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence, showing full working, find \(\int _ { 1 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x\), giving the answer in the form \(\ln c\), where \(c\) is an integer.
CAIE P3 2019 November Q9
9 It is given that \(\int _ { 0 } ^ { a } x \cos \frac { 1 } { 3 } x \mathrm {~d} x = 3\), where the constant \(a\) is such that \(0 < a < \frac { 3 } { 2 } \pi\).
  1. Show that \(a\) satisfies the equation $$a = \frac { 4 - 3 \cos \frac { 1 } { 3 } a } { \sin \frac { 1 } { 3 } a }$$
  2. Verify by calculation that \(a\) lies between 2.5 and 3 .
  3. Use an iterative formula based on the equation in part (i) to calculate \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2019 November Q10
10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )\). The plane \(p\) has equation \(2 x + y - 3 z = 5\).
  1. Find the position vector of the point of intersection of \(l\) and \(p\).
  2. Calculate the acute angle between \(l\) and \(p\).
  3. A second plane \(q\) is perpendicular to the plane \(p\) and contains the line \(l\). Find the equation of \(q\), giving your answer in the form \(a x + b y + c z = d\).
    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 P3 2019 November Q1
1 Solve the inequality \(2 | x + 2 | > | 3 x - 1 |\).
CAIE P3 2019 November Q2
2 The polynomial \(6 x ^ { 3 } + a x ^ { 2 } + b x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is - 24 . Find the values of \(a\) and \(b\).
CAIE P3 2019 November Q3
3 Showing all necessary working, solve the equation \(\frac { 3 ^ { 2 x } + 3 ^ { - x } } { 3 ^ { 2 x } - 3 ^ { - x } } = 4\). Give your answer correct to 3 decimal places.
CAIE P3 2019 November Q4
4
  1. By first expanding \(\tan ( 2 x + x )\), show that the equation \(\tan 3 x = 3 \cot x\) can be written in the form \(\tan ^ { 4 } x - 12 \tan ^ { 2 } x + 3 = 0\).
  2. Hence solve the equation \(\tan 3 x = 3 \cot x\) for \(0 ^ { \circ } < x < 90 ^ { \circ }\).