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 2020 Specimen Q5
5
  1. Sb that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x - \tan ^ { - 1 } x \right) = \frac { x ^ { 2 } } { 1 + x ^ { 2 } }\).
  2. Sth the \(\int _ { 0 } ^ { \sqrt { 3 } } x \tan ^ { - 1 } x \mathrm {~d} x = \frac { 2 } { 3 } \pi - \frac { 1 } { 2 } \sqrt { 3 }\).
CAIE P3 2020 Specimen Q6
6 Th cm plexm b rs \(1 + B\) ad \(4 + \quad 2\) are \(d \mathbf { b }\) ed \(\forall u\) ad \(v\) resp ctie ly.
  1. Fid \(\frac { \mathrm { u } } { \mathrm { V } }\) irt b fo \(\mathrm { m } x + \mathrm { i } y , \mathrm { w } \mathbf { b }\) re \(x\) ad \(y\) are real.
  2. State th argn en \(6 \frac { u } { v }\). In an Arg nd id ag am, with o ign \(O\), th \(\dot { \mathrm { p } } \mathrm { ns } A , B\) ad \(C\) represen th cm p ex m b rs \(u , v\) ad \(u - v\) resp ctie ly.
  3. State fullyt bg m etrical relatio h申 tween \(O C\) ad \(B A\).
  4. Sth the tag e \(A O B = \frac { 1 } { 4 } \pi\) rad as.
CAIE P3 2020 Specimen Q7
7
  1. By first d g co \(\left( x + \Omega ^ { \circ } \right)\), ev ess co \(\left( x + \Omega ^ { \circ } \right) - \sqrt { 2 } \sin x\) in th fo \(\mathrm { m } R \mathrm { co } ( x + \alpha )\), wh re \(R > 0\) ad \(0 ^ { \circ } < \alpha < \theta { } ^ { \circ }\). Gie th le \(6 R\) co rect to 4 sig fican fig res ad th le \(6 \alpha\) co rect tod cimal p aces. [ $\$$
  2. Hen e sb th teq tin $$\text { CB } \left( x + 3 ^ { \circ } \right) - \sqrt { 2 } \sin x = 2$$ fo \(0 ^ { \circ } < x < \boldsymbol { \theta }\)
CAIE P3 2020 Specimen Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{258f9a6f-9339-49c3-8118-6ae9e934f1bb-14_503_727_251_669} In th id ag am, \(O A B C\) is a ply amid in wh ch \(O A = 2\) in ts, \(O B = 4\) in ts ad \(O C = 2\) in ts. Th ed \(O C\) is rtical, th \(\mathbf { b }\) se \(O A B\) is \(\mathbf { b }\) izd al ad ag e \(A O B = \theta ^ { \circ }\). Un t cto s \(\mathbf { i } , \mathbf { j }\) ad \(\mathbf { k }\) are \(\mathbf { p }\) rallel to \(O A\), \(O B\) ad \(O C\) resp ctie ly. Th mij nsg \(A B\) ad \(B C\) are \(M\) ad \(N\) resp ctie ly.
  1. Eq ess th cto s \(\overrightarrow { \mathrm { ON } }\) ad \(\overrightarrow { \mathrm { CM } }\) irt erms \(\boldsymbol { 6 } \mathbf { i } , \mathbf { j }\) ad \(\mathbf { k }\).
  2. Calch ate th ab eb tweert b di rectis \(6 \overrightarrow { \mathrm { ON } }\) ad \(\overrightarrow { \mathrm { CM } }\).
  3. Sth the leg lo th p rp d ich ar from \(M\) to \(O N\) is \(\frac { 3 } { 5 } \sqrt { 5 }\).
CAIE P3 2020 Specimen Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{258f9a6f-9339-49c3-8118-6ae9e934f1bb-16_321_602_260_735} Th d ag am sto \(\mathrm { su } y = \sin ^ { 2 } 2 x \mathrm { co } x\) fo \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), ad ts max mm \(\dot { \mathrm { p } }\) n \(M\).
  1. Fid b \(x\)-co dia te \(6 M\).
    [0pt] [С
  2. Usig th stb tittu in \(u = \sin x\), find th area 6 th sh d d regn \(\mathbf { d } \quad \mathrm { d }\) y th cn e ad th \(x\)-ax s.
CAIE P3 2020 Specimen Q10
10 Ira ch mical reactio a cm \(\mathbf { p } \quad X\) is fo med rm twœ \(\mathbf { m } \mathbf { p } \quad Y\) ad \(Z\).
Tb masses in g ams \(\varnothing \quad X , Y\) ad \(Z \mathrm { p }\) esen at time \(t\) secd s after th start \(\varnothing\) th reactin are \(x , \mathbb { Q } - x\) ad \(0 - x\) resp ctie ly. At ay time th rate 6 fo matin \(6 X\) is p p tio l to to pd t \(\mathbf { 6 }\) th masses \(6 Y\) ad \(Z \mathrm { p }\) esen at th time. Wh \(\mathrm { n } t = \rho x = 0 \mathrm {~d} \frac { \mathrm { dx } } { \mathrm { dt } } = 2\).
  1. Sh the t \(x\) ad \(t\) satisfyt b d fferen ial eq tin $$\frac { \mathrm { dx } } { \mathrm { dt } } = \left( \frac { 1 } { 1 } \quad x \quad x \right) \left( \begin{array} { l l } 1 & x \end{array} \right) .$$
  2. Sb this d fferen ial eq tin \(\mathbf { C }\) aira ressift \(\mathbf { D }\) irt erms \(\boldsymbol { 6 } t\).
  3. State wh th p в to b \& le \(6 x\) wd \(n t\) b cm es larg If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin \(\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )\) ms tb clearlys n n
CAIE P3 2020 June Q7
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Using your answer to part (a), show that $$( f ( x ) ) ^ { 2 } = \frac { 1 } { ( 2 x - 1 ) ^ { 2 } } - \frac { 1 } { 2 x - 1 } + \frac { 1 } { 2 x + 1 } + \frac { 1 } { ( 2 x + 1 ) ^ { 2 } }$$
  3. Hence show that \(\int _ { 1 } ^ { 2 } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x = \frac { 2 } { 5 } + \frac { 1 } { 2 } \ln \left( \frac { 5 } { 9 } \right)\).
CAIE P3 2021 June Q10
  1. Given that the sum of the areas of the shaded sectors is \(90 \%\) of the area of the trapezium, show that \(x\) satisfies the equation \(x = 0.9 ( 2 - \cos x ) \sin x\).
  2. Verify by calculation that \(x\) lies between 0.5 and 0.7 .
  3. Show that if a sequence of values in the interval \(0 < x < \frac { 1 } { 2 } \pi\) given by the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( 2 - \frac { x _ { n } } { 0.9 \sin x _ { n } } \right)$$ converges, then it converges to the root of the equation in part (a).
  4. Use this iterative formula to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2023 June Q10
  1. Find the exact coordinates of \(M\).
  2. Using the substitution \(u = 3 - 2 x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis. Give your answer in the form \(a \sqrt { 13 }\), where \(a\) is a rational number. [5]
CAIE P3 2021 November Q9
  1. Find the \(x\)-coordinate of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\).
  2. Using the substitution \(u = \sqrt { x }\), show that \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 3 } \ln 5\).
CAIE P3 2024 November Q6
  1. Given that the \(x\)-coordinate of \(M\) lies in the interval \(\frac { 1 } { 2 } \pi < x < \frac { 3 } { 4 } \pi\), find the exact coordinates of \(M\).
    \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-10_2718_35_107_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-11_2725_35_99_20}
  2. Find the exact area of the region \(R\).
CAIE P3 2012 June Q7
  1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Show on a sketch of an Argand diagram the points \(A , B\) and \(C\) representing the complex numbers \(u , 1 + 2 \mathrm { i }\) and \(1 - 3 \mathrm { i }\) respectively.
  3. By considering the arguments of \(1 + 2 \mathrm { i }\) and \(1 - 3 \mathrm { i }\), show that $$\tan ^ { - 1 } 2 + \tan ^ { - 1 } 3 = \frac { 3 } { 4 } \pi$$
CAIE P3 2013 June Q6
  1. By first expanding \(\cos \left( x + 45 ^ { \circ } \right)\), express \(\cos \left( x + 45 ^ { \circ } \right) - ( \sqrt { } 2 ) \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$\cos \left( x + 45 ^ { \circ } \right) - ( \sqrt { } 2 ) \sin x = 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
  3. Express \(\frac { 1 } { x ^ { 2 } ( 2 x + 1 ) }\) in the form \(\frac { A } { x ^ { 2 } } + \frac { B } { x } + \frac { C } { 2 x + 1 }\).
  4. The variables \(x\) and \(y\) satisfy the differential equation $$y = x ^ { 2 } ( 2 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$ and \(y = 1\) when \(x = 1\). Solve the differential equation and find the exact value of \(y\) when \(x = 2\). Give your value of \(y\) in a form not involving logarithms.
    (a) The complex number \(w\) is such that \(\operatorname { Re } w > 0\) and \(w + 3 w ^ { * } = \mathrm { i } w ^ { 2 }\), where \(w ^ { * }\) denotes the complex conjugate of \(w\). Find \(w\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    (b) On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z - 2 i | \leqslant 2\) and \(0 \leqslant \arg ( z + 2 ) \leqslant \frac { 1 } { 4 } \pi\). Calculate the greatest value of \(| z |\) for points in this region, giving your answer correct to 2 decimal places.
CAIE P3 2015 June Q8
  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 \(x ^ { 2 }\). [5]
CAIE P3 2016 June Q9
  1. Sketch this diagram and state fully the geometrical relationship between \(O B\) and \(A C\).
  2. Find, in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex number \(\frac { u } { v }\).
  3. Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).
CAIE P3 2017 June Q5
  1. Show that \(x\) satisfies the equation \(x = \frac { 1 } { 3 } ( \pi + \sin x )\).
  2. Verify by calculation that \(x\) lies between 1 and 1.5.
  3. Use an iterative formula based on the equation in part (i) to determine \(x\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2017 March Q8
  1. Showing all your working, verify that \(u\) is a root of the equation \(\mathrm { p } ( z ) = 0\).
  2. Find the other three roots of the equation \(\mathrm { p } ( z ) = 0\).
CAIE P3 2013 November Q7
  1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2 v = 2 \mathrm { i } \quad \text { and } \quad \mathrm { i } u + v = 3$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(| z + \mathrm { i } | = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg ( w - 2 ) = \frac { 3 } { 4 } \pi\). Find the least value of \(| z - w |\) for points on these loci.
CAIE P3 2016 November Q9
  1. Solve the equation \(( 1 + 2 \mathrm { i } ) w ^ { 2 } + 4 w - ( 1 - 2 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z - 1 - \mathrm { i } | \leqslant 2\) and \(- \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).
CAIE P3 2016 November Q7
  1. Find the modulus and argument of \(z\).
  2. Express each of the following in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact:
    (a) \(z + 2 z ^ { * }\);
    (b) \(\frac { z ^ { * } } { \mathrm { i } z }\).
  3. On a sketch of an Argand diagram with origin \(O\), show the points \(A\) and \(B\) representing the complex numbers \(z ^ { * }\) and \(\mathrm { i } z\) respectively. Prove that angle \(A O B\) is equal to \(\frac { 1 } { 6 } \pi\).
CAIE P3 2017 November Q8
  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 2019 November Q7
  1. Find the value of \(a\).
  2. When \(a\) has this value, find the equation of the plane containing \(l\) and \(m\).
CAIE P3 2019 November Q6
  1. Express \(w\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The complex number \(v\) is such that \(| v | = 2 | u |\) and \(\arg v = \arg u + \frac { 1 } { 3 } \pi\).
  2. Sketch an Argand diagram showing the points representing \(u\) and \(v\).
  3. Explain why \(v\) can be expressed as \(2 u w\). Hence find \(v\), giving your answer in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and exact.
Edexcel P3 2022 October Q8
  1. Express \(8 \sin x - 15 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\), and give the value of \(\alpha\), in radians, to 4 significant figures. $$\mathrm { f } ( x ) = \frac { 15 } { 41 + 16 \sin x - 30 \cos x } \quad x > 0$$
  2. Find
    1. the minimum value of \(\mathrm { f } ( x )\)
    2. the smallest value of \(x\) at which this minimum value occurs.
  3. State the \(y\) coordinate of the minimum points on the curve with equation $$y = 2 \mathrm { f } ( x ) - 5 \quad x > 0$$
  4. State the smallest value of \(x\) at which a maximum point occurs for the curve with equation $$y = - \mathrm { f } ( 2 x ) \quad x > 0$$ \section*{8. In this question you must show all stages of your working.
    In this question you must show all stages of your working.}
CAIE P3 2017 June Q6
6 Throughout this question the use of a calculator is not permitted. The complex number \(2 - \mathrm { i }\) is denoted by \(u\).
  1. It is given that \(u\) is a root of the equation \(x ^ { 3 } + a x ^ { 2 } - 3 x + b = 0\), where the constants \(a\) and \(b\) are real. Find the values of \(a\) and \(b\).
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - u | < 1\) and \(| z | < | z + \mathrm { i } |\).