Questions - CAIE - A-Level Maths

CAIE FP1 2013 November Q7

10 marks Standard +0.3

7 The square matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that $\mathbf { e }$ is an eigenvector of $\mathbf { A } ^ { 2 }$ and state the corresponding eigenvalue. Find the eigenvalues of the matrix $\mathbf { B }$, where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0 \\ 2 & 0 & 2 \\ 1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of $\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix.

CAIE FP1 2013 November Q8

10 marks Standard +0.3

8 The plane $\Pi _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)$. Find a cartesian equation of $\Pi _ { 1 }$. The plane $\Pi _ { 2 }$ has equation $2 x - y + z = 10$. Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$. Find an equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$.

CAIE FP1 2013 November Q10

12 marks Standard +0.3

10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where $\lambda$ and $\mu$ are real numbers and $\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}$ is a basis for $K$. 7 The square matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that $\mathbf { e }$ is an eigenvector of $\mathbf { A } ^ { 2 }$ and state the corresponding eigenvalue. Find the eigenvalues of the matrix $\mathbf { B }$, where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of $\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix. 8 The plane $\Pi _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)$. Find a cartesian equation of $\Pi _ { 1 }$. The plane $\Pi _ { 2 }$ has equation $2 x - y + z = 10$. Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$. Find an equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$. 9 The curve $C$ has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis. Find the coordinates of the centroid of the region bounded by $C$, the $x$-axis and the line $x = 1$. 10 The curve $C$ has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 } ,$$ where $p$ is a positive constant and $p \neq 3$.

Obtain the equations of the asymptotes of $C$.
Find the value of $p$ for which the $x$-axis is a tangent to $C$, and sketch $C$ in this case.
For the case $p = 1$, show that $C$ has no turning points, and sketch $C$, giving the exact coordinates of the points of intersection of $C$ with the $x$-axis.

CAIE FP1 2013 November Q16

Standard +0.3

16
10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where $\lambda$ and $\mu$ are real numbers and $\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}$ is a basis for $K$. 7 The square matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that $\mathbf { e }$ is an eigenvector of $\mathbf { A } ^ { 2 }$ and state the corresponding eigenvalue. Find the eigenvalues of the matrix $\mathbf { B }$, where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of $\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix. 8 The plane $\Pi _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)$. Find a cartesian equation of $\Pi _ { 1 }$. The plane $\Pi _ { 2 }$ has equation $2 x - y + z = 10$. Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$. Find an equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$. 9 The curve $C$ has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis. Find the coordinates of the centroid of the region bounded by $C$, the $x$-axis and the line $x = 1$. 10 The curve $C$ has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 } ,$$ where $p$ is a positive constant and $p \neq 3$.\\

Obtain the equations of the asymptotes of $C$.
Find the value of $p$ for which the $x$-axis is a tangent to $C$, and sketch $C$ in this case.
For the case $p = 1$, show that $C$ has no turning points, and sketch $C$, giving the exact coordinates of the points of intersection of $C$ with the $x$-axis. 11 Answer only one of the following two alternatives. \section*{EITHER} State the fifth roots of unity in the form $\cos \theta + \mathrm { i } \sin \theta$, where $- \pi < \theta \leqslant \pi$. Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right)$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form $\cos \theta \pm \mathrm { i } \sin \theta$. Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$ OR Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that $v = y ^ { 3 }$, show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for $y$ in terms of $x$, given that when $x = 0 , y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1$.

CAIE FP1 2015 November Q5

8 marks Standard +0.8

5 The cubic equation $x ^ { 3 } + p x ^ { 2 } + q x + r = 0$, where $p , q$ and $r$ are integers, has roots $\alpha , \beta$ and $\gamma$, such that $$\begin{aligned} \alpha + \beta + \gamma & = 15 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83 \end{aligned}$$ Write down the value of $p$ and find the value of $q$. Given that $\alpha , \beta$ and $\gamma$ are all real and that $\alpha \beta + \alpha \gamma = 36$, find $\alpha$ and hence find the value of $r$.

CAIE FP1 2015 November Q6

10 marks Standard +0.8

6 The matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that $\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)$ is an eigenvector of $\mathbf { A }$. Find the corresponding eigenvalue. Find a diagonal matrix $\mathbf { D }$ and matrices $\mathbf { P }$ and $\mathbf { P } ^ { - 1 }$ such that $\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }$.

CAIE M1 2024 June Q1

3 marks Moderate -0.3

1 Two particles $P$ and $Q$ of masses 0.2 kg and 0.5 kg respectively are at rest on a smooth horizontal plane. Particle $P$ is projected with a speed $6 \mathrm {~ms} ^ { - 1 }$ directly towards $Q$. After $P$ and $Q$ collide, $P$ moves with a speed of $1 \mathrm {~ms} ^ { - 1 }$. Find the two possible speeds of $Q$ after the collision. \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-02_2716_35_143_2012}

CAIE M1 2024 June Q2

3 marks Moderate -0.8

2 \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-03_721_622_296_724} A particle of mass 0.2 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a vertical wall. The particle is held in equilibrium by a force of magnitude $X \mathrm {~N}$, perpendicular to the string, with the string taut and making an angle of $30 ^ { \circ }$ with the wall (see diagram). Find the tension in the string and the value of $X$.

CAIE M1 2024 June Q3

6 marks Standard +0.3

3 A car travels along a straight road with constant acceleration $a \mathrm {~ms} ^ { - 2 }$, where $a > 0$. The car passes through points $A , B$ and $C$ in that order. The speed of the car at $A$ is $u \mathrm {~ms} ^ { - 1 }$ in the direction $A B$. The distance $B C$ is twice the distance $A B$. The car takes 8 seconds to travel from $A$ to $B$ and 10 seconds to travel from $B$ to $C$.

Find $u$ in terms of $a$.
Find the speed of the car at $C$ in terms of $a$.

CAIE M1 2024 June Q4

7 marks Standard +0.3

4 A particle travels in a straight line. The velocity of the particle at time $t \mathrm {~s}$ after leaving a point $O$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $$v = k t ^ { 2 } - 4 t + 3$$ The distance travelled by the particle in the first 2 s of its motion is 6 m . You may assume that $v > 0$ in the first 2s of its motion.

Find the value of $k$.
Find the value of the minimum velocity of the particle. You do not need to show that this velocity is a minimum.

CAIE M1 2024 June Q5

11 marks Standard +0.3

5 A van of mass 4500 kg is towing a trailer of mass 750 kg down a straight hill inclined at an angle of $\theta$ to the horizontal where $\sin \theta = 0.05$. The van and the trailer are connected by a light rigid tow-bar which is parallel to the road. There are constant resistance forces of 2500 N on the van and 300 N on the trailer.

It is given that the tension in the tow-bar is 450 N . Find the acceleration of the trailer and the driving force of the van's engine.
On another occasion, the van and trailer ascend a straight hill inclined at an angle of $\alpha$ to the horizontal where $\sin \alpha = 0.09$. The driving force of the van's engine is now 9100 N , and the speed of the van at the bottom of the hill is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resistances to motion are unchanged.
1. Find the acceleration of the van and the tension in the tow-bar.
2. Find the speed of the van when it has travelled a distance of 375 m up the hill.

CAIE M1 2024 June Q6

11 marks Standard +0.3

6 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg . There is a constant resistance force of magnitude 32 N to the cyclist's motion. At an instant when she is travelling at $7 \mathrm {~ms} ^ { - 1 }$, her acceleration is $0.1 \mathrm {~ms} ^ { - 2 }$.

Find the power output of the cyclist.
Find the steady speed that the cyclist can maintain if her power output and the resistance force are both unchanged. \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-08_2718_35_141_2012} \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-09_2724_35_136_20} The cyclist later descends a straight hill of length 32.2 m , inclined at an angle of $\sin ^ { - 1 } \left( \frac { 1 } { 20 } \right)$ to the horizontal. Her power output is now 120 W , and the resistance force now has variable magnitude such that the work done against this force in descending the hill is 1128 J . The time taken to descend the hill is 4 s .
Given that the speed of the cyclist at the top of the hill is $7.5 \mathrm {~ms} ^ { - 1 }$, find her speed at the bottom of the hill.

CAIE M1 2024 June Q7

9 marks Standard +0.8

7 \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-10_323_1308_292_376} The diagram shows a track $A B C D$ which lies in a vertical plane. The section $A B$ is a straight line inclined at an angle of $30 ^ { \circ }$ to the horizontal and is smooth. The section $B C$ is a horizontal straight line and is rough. The section CD is a straight line inclined at an angle of $30 ^ { \circ }$ to the horizontal and is rough. The lengths $A B , B C$ and $C D$ are each 2 m . A particle is released from rest at $A$. The coefficient of friction between the particle and both $B C$ and $C D$ is $\mu$. There is no change in the speed of the particle when it passes through either of the points $B$ or $C$.

It is given that $\mu = 0.1$. Find the distance which the particle has moved up the section $C D$ when its speed is $1 \mathrm {~ms} ^ { - 1 }$. \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-10_2716_33_143_2014}
It is given instead that with a different value of $\mu$ the particle travels 1 m up the track from $C$ before it comes instantaneously to rest. Find the value of $\mu$ and the speed of the particle at the instant that it passes $C$ for the second time.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE P1 2021 March Q10

8 marks Challenging +1.2

For the case where angle $B A C = \frac { 1 } { 6 } \pi$ radians, find $k$ correct to 4 significant figures.
For the general case in which angle $B A C = \theta$ radians, where $0 < \theta < \frac { 1 } { 2 } \pi$, it is given that $\frac { \theta } { \sin \theta } > 1$. Find the set of possible values of $k$.

CAIE P1 2022 March Q6

8 marks Standard +0.3

Find, by calculation, the coordinates of $A$ and $B$.
Find an equation of the circle which has its centre at $C$ and for which the line with equation $y = 3 x - 20$ is a tangent to the circle.

CAIE P1 2022 March Q10

8 marks Standard +0.3

Find the perimeter of the shaded region.
Find the area of the shaded region.

CAIE P1 2024 March Q10

12 marks Standard +0.3

Find the equation of the tangent to the circle at the point $( - 6,9 )$.
Find the equation of the circle in the form $x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$.
Find the value of $\theta$ correct to 4 significant figures.
Find the perimeter and area of the segment shaded in the diagram.

CAIE P1 2020 November Q10

10 marks Moderate -0.3

[diagram]

The diagram shows a sector $CAB$ which is part of a circle with centre $C$. A circle with centre $O$ and radius $r$ lies within the sector and touches it at $D$, $E$, and $F$, where $COD$ is a straight line and angle $ACD$ is $\theta$ radians.

Find $C D$ in terms of $r$ and $\sin \theta$.
It is now given that $r = 4$ and $\theta = \frac { 1 } { 6 } \pi$.
Find the perimeter of sector $C A B$ in terms of $\pi$.
Find the area of the shaded region in terms of $\pi$ and $\sqrt { 3 }$.

CAIE P1 2021 November Q6

7 marks Standard +0.3

Find the perimeter of the plate, giving your answer in terms of $\pi$.
Find the area of the plate, giving your answer in terms of $\pi$ and $\sqrt { 3 }$.

CAIE P1 2022 November Q10

8 marks Standard +0.3

Find the perimeter of the cross-section RASB, giving your answer correct to 2 decimal places.
Find the difference in area of the two triangles $A O B$ and $A P B$, giving your answer correct to 2 decimal places.
Find the area of the cross-section RASB, giving your answer correct to 1 decimal place.

CAIE P1 2022 November Q10

10 marks Standard +0.3

Find the coordinates of $A$.
Find the volume of revolution when the shaded region is rotated through $360 ^ { \circ }$ about the $x$-axis. Give your answer in the form $\frac { \pi } { a } ( b \sqrt { c } - d )$, where $a , b , c$ and $d$ are integers.
Find an exact expression for the perimeter of the shaded region.

CAIE P1 2017 June Q4

7 marks Moderate -0.8

Express the perimeter of the shaded region in terms of $r$ and $\theta$.
In the case where $r = 5$ and $\theta = \frac { 1 } { 6 } \pi$, find the area of the shaded region.

CAIE P1 2018 June Q5

6 marks Moderate -0.5

Express each of the vectors $\overrightarrow { D A }$ and $\overrightarrow { C A }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.
Use a scalar product to find angle $C A D$.

CAIE P1 2017 March Q4

6 marks Standard +0.3

Show that angle $C B D = \frac { 9 } { 14 } \pi$ radians.
Find the perimeter of the shaded region.

CAIE P1 2005 November Q5

7 marks Standard +0.3

Express $h$ in terms of $r$ and hence show that the volume, $V \mathrm {~cm} ^ { 3 }$, of the cylinder is given by $$V = 12 \pi r ^ { 2 } - 2 \pi r ^ { 3 }$$
Given that $r$ varies, find the stationary value of $V$.

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