Questions - CAIE P3 - A-Level Maths

CAIE P3 2021 June Q10

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 $x = 0.9 ( 2 - \cos x ) \sin x$.
Verify by calculation that $x$ lies between 0.5 and 0.7 .
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).
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

Find the exact coordinates of $M$.
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

Find the $x$-coordinate of the stationary point of the curve with equation $y = \mathrm { f } ( x )$.
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

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}
Find the exact area of the region $R$.

CAIE P3 2012 June Q7

Express $u$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
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.
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

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.
Hence solve the equation $$\cos \left( x + 45 ^ { \circ } \right) - ( \sqrt { } 2 ) \sin x = 2$$ for $0 ^ { \circ } < x < 360 ^ { \circ }$.
Express $\frac { 1 } { x ^ { 2 } ( 2 x + 1 ) }$ in the form $\frac { A } { x ^ { 2 } } + \frac { B } { x } + \frac { C } { 2 x + 1 }$.
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

Express $\mathrm { f } ( x )$ in partial fractions.
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

Sketch this diagram and state fully the geometrical relationship between $O B$ and $A C$.
Find, in the form $x + \mathrm { i } y$, where $x$ and $y$ are real, the complex number $\frac { u } { v }$.
Prove that angle $A O B = \frac { 3 } { 4 } \pi$.

CAIE P3 2017 June Q5

Show that $x$ satisfies the equation $x = \frac { 1 } { 3 } ( \pi + \sin x )$.
Verify by calculation that $x$ lies between 1 and 1.5.
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

Showing all your working, verify that $u$ is a root of the equation $\mathrm { p } ( z ) = 0$.
Find the other three roots of the equation $\mathrm { p } ( z ) = 0$.

CAIE P3 2013 November Q7

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.
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

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.
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

Find the modulus and argument of $z$.
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) $\frac { z ^ { * } } { \mathrm { i } z }$.
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

Express $\mathrm { f } ( x )$ in partial fractions.
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

Find the value of $a$.
When $a$ has this value, find the equation of the plane containing $l$ and $m$.

CAIE P3 2019 November Q6

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 $\arg v = \arg u + \frac { 1 } { 3 } \pi$.
Sketch an Argand diagram showing the points representing $u$ and $v$.
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.

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 $u$.

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 $b$.
On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying both the inequalities $| z - u | < 1$ and $| z | < | z + \mathrm { i } |$.

CAIE P3 2019 June Q5

5 Throughout this question the use of a calculator is not permitted. It is given that the complex number $- 1 + ( \sqrt { } 3 ) \mathrm { i }$ is a root of the equation $$k x ^ { 3 } + 5 x ^ { 2 } + 10 x + 4 = 0$$ where $k$ is a real constant.

Write down another root of the equation.
Find the value of $k$ and the third root of the equation.

CAIE P3 2013 November Q8

8 Throughout this question the use of a calculator is not permitted.

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.
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 2014 November Q5

5 Throughout this question the use of a calculator is not permitted. The complex numbers $w$ and $z$ satisfy the relation $$w = \frac { z + \mathrm { i } } { \mathrm { i } z + 2 }$$

Given that $z = 1 + \mathrm { i }$, find $w$, giving your answer in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.
Given instead that $w = z$ and the real part of $z$ is negative, find $z$, giving your answer in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.

Questions — CAIE P3 (1070 questions)

Browse by module

Browse by board