Questions M3 - A-Level Maths

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Edexcel M3 2005 January Q3

9 marks Standard +0.8

\includegraphics{figure_2} A uniform lamina occupies the region $R$ bounded by the $x$-axis and the curve $$y = \sin x, \quad 0 \leq x \leq \pi,$$ as shown in Figure 2.

Show, by integration, that the $y$-coordinate of the centre of mass of the lamina is $\frac{\pi}{8}$. [6]

\includegraphics{figure_3} A uniform prism $S$ has cross-section $R$. The prism is placed with its rectangular face on a table which is inclined at an angle $\theta^{\circ}$ to the horizontal. The cross-section $R$ lies in a vertical plane as shown in Figure 3. The table is sufficiently rough to prevent $S$ sliding. Given that $S$ does not topple,

find the largest possible value of $\theta$. [3]

Edexcel M3 2005 January Q4

10 marks Standard +0.8

[diagram]

In a game at a fair, a small target $C$ moves horizontally with simple harmonic motion between the points $A$ and $B$, where $AB = 4L$. The target moves inside a box and takes 3 s to travel from $A$ to $B$. A player has to shoot at $C$, but $C$ is only visible to the player when it passes a window $PQ$, where $PQ = b$. The window is initially placed with $Q$ at the point as shown in Figure 4. The target $C$ takes 0.75 s to pass from $Q$ to $P$.

Show that $b = (2 - \sqrt{2})L$. [5]
Find the speed of $C$ as it passes $P$. [2]

\includegraphics{figure_5} For advanced players, the window $PQ$ is moved to the centre of $AB$ so that $AP = QB$, as shown in Figure 5.

Find the time, in seconds to 2 decimal places, taken for $C$ to pass from $Q$ to $P$ in this new position. [3]

Edexcel M3 2005 January Q5

12 marks Standard +0.8

At time $t = 0$, a particle $P$ is at the origin $O$, moving with speed 18 m s$^{-1}$ along the $x$-axis, in the positive $x$-direction. At time $t$ seconds ($t > 0$) the acceleration of $P$ has magnitude $\frac{3}{\sqrt{(t + 4)}}$ m s$^{-2}$ and is directed towards $O$.

Show that, at time $t$ seconds, the velocity of $P$ is $[30 - 6\sqrt{(t + 4)}]$ m s$^{-1}$. [5]
Find the distance of $P$ from $O$ when $P$ comes to instantaneous rest. [7]

Edexcel M3 2005 January Q6

14 marks Standard +0.3

A light spring of natural length $L$ has one end attached to a fixed point $A$. A particle $P$ of mass $m$ is attached to the other end of the spring. The particle is moving vertically. As it passes through the point $B$ below $A$, where $AB = L$, its speed is $\sqrt{(2gL)}$. The particle comes to instantaneous rest at a point $C$, $4L$ below $A$.

Show that the modulus of elasticity of the spring is $\frac{8mg}{9}$. [4]

At the point $D$ the tension in the spring is $mg$.

Show that $P$ performs simple harmonic motion with centre $D$. [5]
Find, in terms of $L$ and $g$,
1. the period of the simple harmonic motion,
2. the maximum speed of $P$.
[5]

Edexcel M3 2005 January Q7

14 marks Challenging +1.8

\includegraphics{figure_6} A trapeze artiste of mass 60 kg is attached to the end $A$ of a light inextensible rope $OA$ of length 5 m. The artiste must swing in an arc of a vertical circle, centre $O$, from a platform $P$ to another platform $Q$, where $PQ$ is horizontal. The other end of the rope is attached to the fixed point $O$ which lies in the vertical plane containing $PQ$, with $\angle POQ = 120^{\circ}$ and $OP = OQ = 5$ m, as shown in Figure 6. As part of her act, the artiste projects herself from $P$ with speed $\sqrt{15}$ m s$^{-1}$ in a direction perpendicular to the rope $OA$ and in the plane $POQ$. She moves in a circular arc towards $Q$. At the lowest point of her path she catches a ball of mass $m$ kg which is travelling towards her with speed 3 m s$^{-1}$ and parallel to $QP$. After catching the ball, she comes to rest at the point $Q$. By modelling the artiste and the ball as particles and ignoring her air resistance, find

the speed of the artiste immediately before she catches the ball, [4]
the value of $m$, [7]
the tension in the rope immediately after she catches the ball. [3]

Edexcel M3 2011 January Q1

6 marks Standard +0.3

A particle $P$ moves on the positive $x$-axis. When the distance of $P$ from the origin $O$ is $x$ metres, the acceleration of $P$ is $(7 - 2x)$ m s$^{-2}$, measured in the positive $x$-direction. When $t = 0$, $P$ is at $O$ and is moving in the positive $x$-direction with speed 6 m s$^{-1}$. Find the distance of $P$ from $O$ when $P$ first comes to instantaneous rest. [6]

Edexcel M3 2011 January Q2

8 marks Standard +0.3

\includegraphics{figure_1} A toy is formed by joining a uniform solid hemisphere, of radius $r$ and mass $4m$, to a uniform right circular solid cone of mass $km$. The cone has vertex $A$, base radius $r$ and height $2r$. The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the hemisphere is $O$ and $OB$ is a radius of its plane face as shown in Figure 1. The centre of mass of the toy is at $O$.

Find the value of $k$. [4]

A metal stud of mass $2m$ is attached to the toy at $A$. The toy is now suspended by a light string attached to $B$ and hangs freely at rest. The angle between $OB$ and the vertical is $30°$.

Find the value of $\lambda$. [4]

Edexcel M3 2011 January Q3

10 marks Standard +0.8

\includegraphics{figure_2} The region $R$ is bounded by the curve with equation $y = e^x$, the line $x = 1$, the line $x = 2$ and the $x$-axis as shown in Figure 2. A uniform solid $S$ is formed by rotating $R$ through $2\pi$ about the $x$-axis.

Show that the volume of $S$ is $\frac{1}{2}\pi (e^4 - e^2)$. [4]
Find, to 3 significant figures, the $x$-coordinate of the centre of mass of $S$. [6]

Edexcel M3 2011 January Q4

11 marks Standard +0.3

A particle $P$ moves along the $x$-axis. At time $t$ seconds its displacement, $x$ metres, from the origin $O$ is given by $x = 5 \sin (\frac{1}{4}\pi t)$.

Prove that $P$ is moving with simple harmonic motion. [3]
Find the period and the amplitude of the motion. [2]
Find the maximum speed of $P$. [2]

The points $A$ and $B$ on the positive $x$-axis are such that $OA = 2$ m and $OB = 3$ m.

Find the time taken by $P$ to travel directly from $A$ to $B$. [4]

Edexcel M3 2011 January Q5

10 marks Standard +0.3

\includegraphics{figure_3} A small ball $P$ of mass $m$ is attached to the ends of two light inextensible strings of length $l$. The other ends of the strings are attached to fixed points $A$ and $B$, where $A$ is vertically above $B$. Both strings are taut and $AP$ is perpendicular to $BP$ as shown in Figure 3. The system rotates about the line $AB$ with constant angular speed $\omega$. The ball moves in a horizontal circle.

Find, in terms of $m$, $g$, $l$ and $\omega$, the tension in $AP$ and the tension in $BP$. [8]
Show that $\omega^2 \geq \frac{g\sqrt{2}}{l}$. [2]

Edexcel M3 2011 January Q6

13 marks Standard +0.8

\includegraphics{figure_4} A small ball of mass $3m$ is attached to the ends of two light elastic strings $AP$ and $BP$, each of natural length $l$ and modulus of elasticity $kmg$. The ends $A$ and $B$ of the strings are attached to fixed points on the same horizontal level, with $AB = 2l$. The mid-point of $AB$ is $C$. The ball hangs in equilibrium at a distance $\frac{3}{4}l$ vertically below $C$ as shown in Figure 4.

Show that $k = 10$ [7]

The ball is now pulled vertically downwards until it is at a distance $\frac{15}{8}l$ below $C$. The ball is released from rest.

Find the speed of the ball as it reaches $C$. [6]

Edexcel M3 2011 January Q7

17 marks Challenging +1.2

\includegraphics{figure_5} A particle $P$ of mass $m$ is attached to one end of a light rod of length $l$. The other end of the rod is attached to a fixed point $O$. The rod can turn freely in a vertical plane about $O$. The particle is projected with speed $u$ from a point $A$, where $OA$ makes an angle $\alpha$ with the upward vertical through $O$ and $0 < \alpha < \frac{\pi}{2}$. When $OP$ makes an angle $\theta$ with the upward vertical through $O$ the speed of $P$ is $v$ as shown in Figure 5.

Show that $v^2 = u^2 + 2gl (\cos \alpha - \cos \theta)$. [4]

It is given that $\cos \alpha = \frac{3}{5}$ and that $P$ moves in a complete vertical circle.

Show that $u > 2\sqrt{\frac{gl}{5}}$. [4]

As the rod rotates the least tension in the rod is $T$ and the greatest tension is $5T$.

Show that $u^2 = \frac{33}{10}gl$. [9]

Edexcel M3 2001 June Q1

7 marks Moderate -0.3

A particle $P$ moves along the x-axis in the positive direction. At time $t$ seconds, the velocity of $P$ is $v$ m s$^{-1}$ and its acceleration is $\frac{1}{5}e^{-2t}$ m s$^{-2}$. When $t = 0$ the speed of $P$ is 10 m s$^{-1}$.

Express $v$ in terms of $t$. [4]
Find, to 3 significant figures, the speed of $P$ when $t = 3$. [2]
Find the limiting value of $v$. [1]

Edexcel M3 2001 June Q2

7 marks Challenging +1.2

\includegraphics{figure_1} A smooth solid hemisphere, of radius 0.8 m and centre $O$, is fixed with its plane face on a horizontal table. A particle of mass 0.5 kg is projected horizontally with speed $u$ m s$^{-1}$ from the highest point $A$ of the hemisphere. The particle leaves the hemisphere at the point $B$, which is a vertical distance of 0.2 m below the level of $A$. The speed of the particle at $B$ is $v$ m s$^{-1}$ and the angle between $OA$ and $OB$ is $\theta$, as shown in Fig. 1.

Find the value of $\cos \theta$. [1]
Show that $v^2 = 5.88$. [3]
Find the value of $u$. [3]

Edexcel M3 2001 June Q3

10 marks Standard +0.3

\includegraphics{figure_2} A light horizontal spring, of natural length 0.25 m and modulus of elasticity 52 N, is fastened at one end to a point $A$. The other end of the spring is fastened to a small wooden block $B$ of mass 1.5 kg which is on a horizontal table, as shown in Fig. 2. The block is modelled as a particle. The table is initially assumed to be smooth. The block is released from rest when it is a distance 0.3 m from $A$. By using the principle of the conservation of energy,

find, to 3 significant figures, the speed of $B$ when it is a distance 0.25 m from $A$. [5]

It is now assumed that the table is rough and the coefficient of friction between $B$ and the table is 0.6.

Find, to 3 significant figures, the minimum distance from $A$ at which $B$ can rest in equilibrium. [5]

Edexcel M3 2001 June Q4

10 marks Standard +0.8

A projectile $P$ is fired vertically upwards from a point on the earth's surface. When $P$ is at a distance $x$ from the centre of the earth its speed is $v$. Its acceleration is directed towards the centre of the earth and has magnitude $\frac{k}{x^2}$, where $k$ is a constant. The earth may be assumed to be a sphere of radius $R$.

Show that the motion of $P$ may be modelled by the differential equation $$v \frac{dv}{dx} = -\frac{gR^2}{x^2}.$$ [4]

The initial speed of $P$ is $U$, where $U^2 < 2gR$. The greatest distance of $P$ from the centre of the earth is $X$.

Find $X$ in terms of $U$, $R$ and $g$. [6]

Edexcel M3 2001 June Q5

11 marks Standard +0.3

\includegraphics{figure_3} An ornament $S$ is formed by removing a solid right circular cone, of radius $r$ and height $\frac{1}{2}h$, from a solid uniform cylinder, of radius $r$ and height $h$, as shown in Fig. 3.

Show that the distance of the centre of mass $S$ from its plane face is $\frac{19}{30}h$. [7]

The ornament is suspended from a point on the circular rim of its open end. It hangs in equilibrium with its axis of symmetry inclined at an angle $\alpha$ to the horizontal. Given that $h = 4r$,

find, in degrees to one decimal place, the value of $\alpha$. [4]

Edexcel M3 2001 June Q6

14 marks Standard +0.3

\includegraphics{figure_4} A particle $P$ of mass $m$ is attached to two light inextensible strings. The other ends of the string are attached to fixed points $A$ and $B$. The point $A$ is a distance $h$ vertically above $B$. The system rotates about the line $AB$ with constant angular speed $\omega$. Both strings are taut and inclined at $60°$ to $AB$, as shown in Fig. 4. The particle moves in a circle of radius $r$.

Show that $r = \frac{\sqrt{3}}{2}h$. [2]
Find, in terms of $m$, $g$, $h$ and $\omega$, the tension in $AP$ and the tension in $BP$. [8]

The time taken for $P$ to complete one circle is $T$.

Show that $T < \pi\sqrt{\left(\frac{2h}{g}\right)}$. [4]

Edexcel M3 2001 June Q7

16 marks Challenging +1.2

\includegraphics{figure_5} A small ring $R$ of mass $m$ is free to slide on a smooth straight wire which is fixed at an angle of $30°$ to the horizontal. The ring is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $\lambda$. The other end of the string is attached to a fixed point $A$ of the wire, as shown in Fig. 5. The ring rests in equilibrium at the point $B$, where $AB = \frac{a}{2}$.

Show that $\lambda = 4mg$. [3]

The ring is pulled down to the point $C$, where $BC = \frac{1}{4}a$, and released from rest. At time $t$ after $R$ is released the extension of the string is $(\frac{1}{4}a + x)$.

Obtain a differential equation for the motion of $R$ while the string remains taut, and show that it represents simple harmonic motion with period $\pi\sqrt{\left(\frac{a}{g}\right)}$. [6]
Find, in terms of $g$, the greatest magnitude of the acceleration of $R$ while the string remains taut. [2]
Find, in terms of $a$ and $g$, the time taken for $R$ to move from the point at which it first reaches maximum speed to the point where the string becomes slack for the first time. [5]