Questions P3 (1203 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 2024 June Q7
7
  1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta \equiv \cos 2 \theta\).
  2. Hence find the exact value of \(\int _ { - \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 8 } \pi } \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta + 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \right) \mathrm { d } \theta\).
CAIE P3 2024 June Q8
8 The points \(A , B\) and \(C\) have position vectors \(\overrightarrow { \mathrm { OA } } = - 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } , \overrightarrow { \mathrm { OB } } = 5 \mathbf { i } + 2 \mathbf { j }\) and \(\overrightarrow { \mathrm { OC } } = 8 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k }\), where \(O\) is the origin. The line \(l _ { 1 }\) passes through \(B\) and \(C\).
  1. Find a vector equation for \(l _ { 1 }\).
    The line \(l _ { 2 }\) has equation \(\mathbf { r } = - 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\).
  2. Find the coordinates of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
  3. The point \(D\) on \(l _ { 2 }\) is such that \(\mathrm { AB } = \mathrm { BD }\). Find the position vector of \(D\).
    \includegraphics[max width=\textwidth, alt={}, center]{5eb2657c-ed74-4ed2-b8c4-08e9e0f657b5-13_58_1545_388_349}
CAIE P3 2024 June Q9
9 The complex numbers \(z\) and \(\omega\) are defined by \(z = 1 - i\) and \(\omega = - 3 + 3 \sqrt { 3 } i\).
  1. Express \(z \omega\) in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real and in exact surd form.
  2. Express \(z\) and \(\omega\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\) in each case.
  3. On an Argand diagram, the points representing \(\omega\) and \(z \omega\) are \(A\) and \(B\) respectively. Prove that \(O A B\) is an isosceles right-angled triangle, where \(O\) is the origin.
  4. Using your answers to part (b), prove that \(\tan \frac { 5 } { 12 } \pi = \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }\).
CAIE P3 2024 June Q10
10
  1. By writing \(y = \sec ^ { 3 } \theta\) as \(\frac { 1 } { \cos ^ { 3 } \theta }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = 3 \sin \theta \sec ^ { 4 } \theta\).
  2. The variables \(x\) and \(\theta\) satisfy the differential equation $$\left( x ^ { 2 } + 9 \right) \sin \theta \frac { d \theta } { d x } = ( x + 3 ) \cos ^ { 4 } \theta$$ It is given that \(x = 3\) when \(\theta = \frac { 1 } { 3 } \pi\).
    Solve the differential equation to find the value of \(\cos \theta\) when \(x = 0\). Give your answer correct to 3 significant figures.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P3 2024 June Q1
1 Solve the equation \(8 ^ { 3 - 6 x } = 4 \times 5 ^ { - 2 x }\). Give your answer correct to 3 decimal places.
\includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-02_2718_35_141_2012}
CAIE P3 2024 June Q2
2 Find the exact coordinates of the stationary point of the curve \(y = \mathrm { e } ^ { 2 x } \sin 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
CAIE P3 2024 June Q3
3 The square roots of 24-7i can be expressed in the Cartesian form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(24 - 7 \mathrm { i }\) in exact Cartesian form.
\includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-04_2717_35_141_2012}
\includegraphics[max width=\textwidth, alt={}]{b1c4d339-322f-496d-833e-8b2d002d7c48-05_595_896_287_587}
The variables \(x\) and \(y\) satisfy the equation \(k y = \mathrm { e } ^ { c x }\), where \(k\) and \(c\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(2.80,0.372\) ) and ( \(5.10,2.21\) ), as shown in the diagram. Find the values of \(k\) and \(c\). Give each value correct to 2 significant figures.
CAIE P3 2024 June Q5
5 Express \(\frac { 6 x ^ { 2 } - 2 x + 2 } { ( x - 1 ) ( 2 x + 1 ) }\) in partial fractions.
\includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-06_2716_40_141_2010}
CAIE P3 2024 June Q6
6
  1. On an Argand diagram shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z - 4 - 3 i | \leqslant 2\) and \(\arg ( z - 2 - i ) \geqslant \frac { 1 } { 3 } \pi\).
  2. Calculate the greatest value of \(\arg z\) for points in this region.
CAIE P3 2024 June Q7
7 Let \(\mathrm { f } ( x ) = 8 x ^ { 3 } + 54 x ^ { 2 } - 17 x - 21\).
  1. Show that \(x + 7\) is a factor of \(\mathrm { f } ( x )\).
  2. Find the quotient when \(\mathrm { f } ( x )\) is divided by \(x + 7\).
  3. Hence solve the equation $$8 \cos ^ { 3 } \theta + 54 \cos ^ { 2 } \theta - 17 \cos \theta - 21 = 0$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2024 June Q8
8
  1. Express \(3 \cos 2 x - \sqrt { 3 } \sin 2 x\) in the form \(R \cos ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact values of \(R\) and \(\alpha\).
    \includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-10_2715_35_143_2012}
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } \frac { 3 } { ( 3 \cos 2 x - \sqrt { 3 } \sin 2 x ) ^ { 2 } } \mathrm {~d} x\), simplifying your answer.
CAIE P3 2024 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-12_431_547_280_758} A container in the shape of a cuboid has a square base of side \(x\) and a height of ( \(10 - x\) ). It is given that \(x\) varies with time, \(t\), where \(t > 0\). The container decreases in volume at a rate which is inversely proportional to \(t\). When \(t = \frac { 1 } { 10 } , x = \frac { 1 } { 2 }\) and the rate of decrease of \(x\) is \(\frac { 20 } { 37 }\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { - 1 } { 2 t \left( 20 x - 3 x ^ { 2 } \right) }$$ \includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-12_2739_47_123_2001}
  2. Solve the differential equation, obtaining an expression for \(t\) in terms of \(x\).
CAIE P3 2024 June Q10
10 The equations of two straight lines are $$\mathbf { r } = \mathbf { i } + \mathbf { j } + 2 a \mathbf { k } + \lambda ( 3 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) ,$$ where \(a\) is a constant.
  1. Given that the acute angle between the directions of these lines is \(\frac { 1 } { 4 } \pi\), find the possible values of \(a\).
    \includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-14_2715_35_144_2012}
  2. Given instead that the lines intersect, find the value of \(a\) and the position vector of the point of intersection.
CAIE P3 2024 June Q11
11 Use the substitution \(2 x = \tan \theta\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 12 } { \left( 1 + 4 x ^ { 2 } \right) ^ { 2 } } d x$$ Give your answer in the form \(a + b \pi\), where \(a\) and \(b\) are rational numbers.
\includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-16_2716_38_143_2009}
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-18_2715_35_143_2012}
CAIE P3 2020 March Q1
1
  1. Sketch the graph of \(y = | x - 2 |\).
  2. Solve the inequality \(| x - 2 | < 3 x - 4\).
CAIE P3 2020 March Q2
2 Solve the equation \(\ln 3 + \ln ( 2 x + 5 ) = 2 \ln ( x + 2 )\). Give your answer in a simplified exact form.
CAIE P3 2020 March Q3
3
  1. By sketching a suitable pair of graphs, show that the equation \(\sec x = 2 - \frac { 1 } { 2 } x\) has exactly one root in the interval \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 0.8 and 1 .
  3. Use the iterative formula \(x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 2 } { 4 - x _ { n } } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2020 March Q4
4 Find \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } x \sec ^ { 2 } x \mathrm {~d} x\). Give your answer in a simplified exact form.
CAIE P3 2020 March Q5
5
  1. Show that \(\frac { \cos 3 x } { \sin x } + \frac { \sin 3 x } { \cos x } = 2 \cot 2 x\).
  2. Hence solve the equation \(\frac { \cos 3 x } { \sin x } + \frac { \sin 3 x } { \cos x } = 4\), for \(0 < x < \pi\).
CAIE P3 2020 March Q6
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + 4 y ^ { 2 } } { \mathrm { e } ^ { x } }$$ It is given that \(y = 0\) when \(x = 1\).
  1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
  2. State what happens to the value of \(y\) as \(x\) tends to infinity.
CAIE P3 2020 March Q7
7 The equation of a curve is \(x ^ { 3 } + 3 x y ^ { 2 } - y ^ { 3 } = 5\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } + y ^ { 2 } } { y ^ { 2 } - 2 x y }\).
  2. Find the coordinates of the points on the curve where the tangent is parallel to the \(y\)-axis.
CAIE P3 2020 March Q8
4 marks
8
\includegraphics[max width=\textwidth, alt={}, center]{8f81a526-783c-4321-b540-c9deccfee17b-12_639_713_262_715} In the diagram, \(O A B C D E F G\) is a cuboid in which \(O A = 2\) units, \(O C = 3\) units and \(O D = 2\) units. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively. The point \(M\) on \(A B\) is such that \(M B = 2 A M\). The midpoint of \(F G\) is \(N\).
  1. Express the vectors \(\overrightarrow { O M }\) and \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Find a vector equation for the line through \(M\) and \(N\).
  3. Find the position vector of \(P\), the foot of the perpendicular from \(D\) to the line through \(M\) and \(N\). [4]
CAIE P3 2020 March Q9
9 Let \(\mathrm { f } ( x ) = \frac { 2 + 11 x - 10 x ^ { 2 } } { ( 1 + 2 x ) ( 1 - 2 x ) ( 2 + x ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2020 March Q10
10
  1. The complex numbers \(v\) and \(w\) satisfy the equations $$v + \mathrm { i } w = 5 \quad \text { and } \quad ( 1 + 2 \mathrm { i } ) v - w = 3 \mathrm { i } .$$ Solve the equations for \(v\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying \(| z - 2 - 3 \mathrm { i } | = 1\).
    2. Calculate the least value of \(\arg z\) for points on this locus.
      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 2021 March Q1
1 Solve the equation \(\ln \left( x ^ { 3 } - 3 \right) = 3 \ln x - \ln 3\). Give your answer correct to 3 significant figures.