Questions - OCR - A-Level Maths

OCR M1 Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-02_508_501_274_822} A light inextensible string has its ends attached to two fixed points $A$ and $B$. The point $A$ is vertically above $B$. A smooth ring $R$ of mass $m \mathrm {~kg}$ is threaded on the string and is pulled by a force of magnitude 1.6 N acting upwards at $45 ^ { \circ }$ to the horizontal. The section $A R$ of the string makes an angle of $30 ^ { \circ }$ with the downward vertical and the section $B R$ is horizontal (see diagram). The ring is in equilibrium with the string taut.

Give a reason why the tension in the part $A R$ of the string is the same as that in the part $B R$.
Show that the tension in the string is 0.754 N , correct to 3 significant figures.
Find the value of $m$.

OCR M1 Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-02_643_289_1475_927} Particles $A$ and $B$, of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. Particle $A$ is held at rest at a fixed point and $B$ hangs vertically below $A$. Particle $A$ is now released. As the particles fall the air resistance acting on $A$ is 0.4 N and the air resistance acting on $B$ is 0.25 N (see diagram). The downward acceleration of each of the particles is $a \mathrm {~ms} ^ { - 2 }$ and the tension in the string is $T \mathrm {~N}$.

Write down two equations in $a$ and $T$ obtained by applying Newton's second law to $A$ and to $B$.
Find the values of $a$ and $T$. \section*{June 2005}

OCR M1 Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-03_697_579_1238_781} Two small rings $A$ and $B$ are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. $A$ is above $B$. A horizontal force is applied to a point $P$ of the string. Both parts $A P$ and $B P$ of the string are taut. The system is in equilibrium with angle $B A P = \alpha$ and angle $A B P = \beta$ (see diagram). The weight of $A$ is 2 N and the tensions in the parts $A P$ and $B P$ of the string are 7 N and $T \mathrm {~N}$ respectively. It is given that $\cos \alpha = 0.28$ and $\sin \alpha = 0.96$, and that $A$ is in limiting equilibrium.

Find the coefficient of friction between the wire and the ring $A$.
By considering the forces acting at $P$, show that $T \cos \beta = 1.96$.
Given that there is no frictional force acting on $B$, find the mass of $B$. \section*{June 2005}

OCR M1 Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-04_632_1121_936_511} A car $P$ starts from rest and travels along a straight road for 600 s . The ( $t , v$ ) graph for the journey is shown in the diagram. This graph consists of three straight line segments. Find

the distance travelled by $P$,
the deceleration of $P$ during the interval $500 < t < 600$. Another car $Q$ starts from rest at the same instant as $P$ and travels in the same direction along the same road for 600 s . At time $t \mathrm {~s}$ after starting the velocity of $Q$ is $\left( 600 t ^ { 2 } - t ^ { 3 } \right) \times 10 ^ { - 6 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find an expression in terms of $t$ for the acceleration of $Q$.
Find how much less $Q$ 's deceleration is than $P$ 's when $t = 550$.
Show that $Q$ has its maximum velocity when $t = 400$.
Find how much further $Q$ has travelled than $P$ when $t = 400$. \section*{Jan 2006} 1
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-05_547_455_269_845} Particles $P$ and $Q$, of masses 0.3 kg and 0.4 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in motion with the string taut and with each of the particles moving vertically. The downward acceleration of $P$ is $a \mathrm {~ms} ^ { - 2 }$ (see diagram).
Show that $a = - 1.4$. Initially $P$ and $Q$ are at the same horizontal level. $P$ 's initial velocity is vertically downwards and has magnitude $2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Assuming that $P$ does not reach the floor and that $Q$ does not reach the pulley, find the time taken for $P$ to return to its initial position. 2
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-05_616_809_1562_667} An object of mass 0.08 kg is attached to one end of a light inextensible string. The other end of the string is attached to the underside of the roof inside a furniture van. The van is moving horizontally with constant acceleration $1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The string makes a constant angle $\alpha$ with the downward vertical and the tension in the string is $T \mathrm {~N}$ (see diagram).
By applying Newton's second law horizontally to the object, find the value of $T \sin \alpha$.
Find the value of $T$. 3 A motorcyclist starts from rest at a point $O$ and travels in a straight line. His velocity after $t$ seconds is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, for $0 \leqslant t \leqslant T$, where $v = 7.2 t - 0.45 t ^ { 2 }$. The motorcyclist's acceleration is zero when $t = T$.
Find the value of $T$.
Show that $v = 28.8$ when $t = T$. For $t \geqslant T$ the motorcyclist travels in the same direction as before, but with constant speed $28.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the displacement of the motorcyclist from $O$ when $t = 31$. 4
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-06_225_623_813_762} A block of mass 2 kg is at rest on a rough horizontal plane, acted on by a force of magnitude 12 N at an angle of $15 ^ { \circ }$ upwards from the horizontal (see diagram).
Find the frictional component of the contact force exerted on the block by the plane.
Show that the normal component of the contact force exerted on the block by the plane has magnitude 16.5 N , correct to 3 significant figures. It is given that the block is on the point of sliding.
Find the coefficient of friction between the block and the plane. The force of magnitude 12 N is now replaced by a horizontal force of magnitude 20 N . The block starts to move.
Find the acceleration of the block. 5 A man drives a car on a horizontal straight road. At $t = 0$, where the time $t$ is in seconds, the car runs out of petrol. At this instant the car is moving at $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The car decelerates uniformly, coming to rest when $t = 8$. The man then walks back along the road at $0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ until he reaches a petrol station a distance of 420 m from his car. After his arrival at the petrol station it takes him 250 s to obtain a can of petrol. He is then given a lift back to his car on a motorcycle. The motorcycle starts from rest and accelerates uniformly until its speed is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$; it then decelerates uniformly, coming to rest at the stationary car at time $t = T$.
Sketch the shape of the $( t , v )$ graph for the man for $0 \leqslant t \leqslant T$. [Your sketch need not be drawn to scale; numerical values need not be shown.]
Find the deceleration of the car for $0 < t < 8$.
Find the value of $T$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-07_540_542_267_799} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A smooth ring $R$ of weight $W \mathrm {~N}$ is threaded on a light inextensible string. The ends of the string are attached to fixed points $A$ and $B$, where $A$ is vertically above $B$. A horizontal force of magnitude $P \mathrm {~N}$ acts on $R$. The system is in equilibrium with the string taut; $A R$ makes an angle $\alpha$ with the downward vertical and $B R$ makes an angle $\beta$ with the upward vertical (see Fig.1).
By considering the vertical components of the forces acting on $R$, show that $\alpha < \beta$.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-07_537_559_1302_833} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} It is given that when $P = 14 , A R = 0.4 \mathrm {~m} , B R = 0.3 \mathrm {~m}$ and the distance of $R$ from the vertical line $A B$ is 0.24 m (see Fig. 2). Find
(a) the tension in the string,
(b) the value of $W$.
For the case when $P = 0$,
(a) describe the position of $R$,
(b) state the tension in the string.
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-08_624_1077_267_536}
$P Q$ is a line of greatest slope, of length 4 m , on a smooth plane inclined at $30 ^ { \circ }$ to the horizontal. Particles $A$ and $B$, of masses 0.15 kg and 0.5 kg respectively, move along $P Q$ with $A$ below $B$. The particles are both moving upwards, $A$ with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $B$ with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when they collide at the mid-point of $P Q$ (see diagram). Particle $A$ is instantaneously at rest immediately after the collision.
Show that $B$ does not reach $Q$ in the subsequent motion.
Find the time interval between the instant of $A$ 's arrival at $P$ and the instant of $B$ 's arrival at $P$. \section*{June 2006} 1 Each of two wagons has an unloaded mass of 1200 kg . One of the wagons carries a load of mass $m \mathrm {~kg}$ and the other wagon is unloaded. The wagons are moving towards each other on the same rails, each with speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when they collide. Immediately after the collision the loaded wagon is at rest and the speed of the unloaded wagon is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the value of $m$. 2
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-09_622_716_541_715} Forces of magnitudes 6.5 N and 2.5 N act at a point in the directions shown. The resultant of the two forces has magnitude $R \mathrm {~N}$ and acts at right angles to the force of magnitude 2.5 N (see diagram).
Show that $\theta = 22.6 ^ { \circ }$, correct to 3 significant figures.
Find the value of $R$. 3 A man travels 360 m along a straight road. He walks for the first 120 m at $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, runs the next 180 m at $4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, and then walks the final 60 m at $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The man's displacement from his starting point after $t$ seconds is $x$ metres.
Sketch the $( t , x )$ graph for the journey, showing the values of $t$ for which $x = 120,300$ and 360 . A woman jogs the same 360 m route at constant speed, starting at the same instant as the man and finishing at the same instant as the man.
Draw a dotted line on your $( t , x )$ graph to represent the woman's journey.
Calculate the value of $t$ at which the man overtakes the woman. June 2006
4 A cyclist travels along a straight road. Her velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, at time $t$ seconds after starting from a point $O$, is given by $$\begin{aligned} & v = 2 \quad \text { for } 0 \leqslant t \leqslant 10 ,
& v = 0.03 t ^ { 2 } - 0.3 t + 2 \quad \text { for } t \geqslant 10 . \end{aligned}$$
Find the displacement of the cyclist from $O$ when $t = 10$.
Show that, for $t \geqslant 10$, the displacement of the cyclist from $O$ is given by the expression $0.01 t ^ { 3 } - 0.15 t ^ { 2 } + 2 t + 5$.
Find the time when the acceleration of the cyclist is $0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Hence find the displacement of the cyclist from $O$ when her acceleration is $0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. 5 A block of mass $m \mathrm {~kg}$ is at rest on a horizontal plane. The coefficient of friction between the block and the plane is 0.2 .
When a horizontal force of magnitude 5 N acts on the block, the block is on the point of slipping. Find the value of $m$.
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-10_312_711_1244_758} When a force of magnitude $P \mathrm {~N}$ acts downwards on the block at an angle $\alpha$ to the horizontal, as shown in the diagram, the frictional force on the block has magnitude 6 N and the block is again on the point of slipping. Find
(a) the value of $\alpha$ in degrees,
(b) the value of $P$. June 2006
6
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-11_317_999_264_575} A train of total mass 80000 kg consists of an engine $E$ and two trucks $A$ and $B$. The engine $E$ and truck $A$ are connected by a rigid coupling $X$, and trucks $A$ and $B$ are connected by another rigid coupling $Y$. The couplings are light and horizontal. The train is moving along a straight horizontal track. The resistances to motion acting on $E , A$ and $B$ are $10500 \mathrm {~N} , 3000 \mathrm {~N}$ and 1500 N respectively (see diagram).
By modelling the whole train as a single particle, show that it is decelerating when the driving force of the engine is less than 15000 N .
Show that, when the magnitude of the driving force is 35000 N , the acceleration of the train is $0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Hence find the mass of $E$, given that the tension in the coupling $X$ is 8500 N when the magnitude of the driving force is 35000 N . The driving force is replaced by a braking force of magnitude 15000 N acting on the engine. The force exerted by the coupling $Y$ is zero.
Find the mass of $B$.
Show that the coupling $X$ exerts a forward force of magnitude 1500 N on the engine. 7 A particle of mass 0.1 kg is at rest at a point $A$ on a rough plane inclined at $15 ^ { \circ }$ to the horizontal. The particle is given an initial velocity of $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after 1.5 s .
Find the coefficient of friction between the particle and the plane.
Show that, after coming to instantaneous rest, the particle moves down the plane.
Find the speed with which the particle passes through $A$ during its downward motion. 1 A trailer of mass 600 kg is attached to a car of mass 1100 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration $0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Given that the force exerted on the trailer by the tow-bar is 700 N , find the resistance to motion of the trailer.
Given also that the driving force of the car is 2100 N , find the resistance to motion of the car. 2
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-12_595_791_671_676} Three horizontal forces of magnitudes $15 \mathrm {~N} , 11 \mathrm {~N}$ and 13 N act on a particle $P$ in the directions shown in the diagram. The angles $\alpha$ and $\beta$ are such that $\sin \alpha = 0.28 , \cos \alpha = 0.96 , \sin \beta = 0.8$ and $\cos \beta = 0.6$.
Show that the component, in the $y$-direction, of the resultant of the three forces is zero.
Find the magnitude of the resultant of the three forces.
State the direction of the resultant of the three forces. 3
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-12_351_716_1804_715} A block $B$ of mass 0.4 kg and a particle $P$ of mass 0.3 kg are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. $B$ is in contact with the table and the part of the string between $B$ and the pulley is horizontal. $P$ hangs freely below the pulley (see diagram).
The system is in limiting equilibrium with the string taut and $P$ on the point of moving downwards. Find the coefficient of friction between $B$ and the table.
A horizontal force of magnitude $X \mathrm {~N}$, acting directly away from the pulley, is now applied to $B$. The system is again in limiting equilibrium with the string taut, and with $P$ now on the point of moving upwards. Find the value of $X$. 4
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-13_225_1155_264_497} Three uniform spheres $L , M$ and $N$ have masses $0.8 \mathrm {~kg} , 0.6 \mathrm {~kg}$ and 0.7 kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with $M$ between $L$ and $N$. The sphere $L$ is moving towards $M$ with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the spheres $M$ and $N$ are moving towards $L$ with speeds $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively (see diagram).
$L$ collides with $M$. As a result of this collision the direction of motion of $M$ is reversed, and its speed remains $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the speed of $L$ after the collision.
$M$ then collides with $N$.
(a) Find the total momentum of $M$ and $N$ in the direction of $M$ 's motion before this collision takes place, and deduce that the direction of motion of $N$ is reversed as a result of this collision.
(b) Given that $M$ is at rest immediately after this collision, find the speed of $N$ immediately after this collision. 5 A particle starts from rest at a point $A$ at time $t = 0$, where $t$ is in seconds. The particle moves in a straight line. For $0 \leqslant t \leqslant 4$ the acceleration is $1.8 t \mathrm {~m} \mathrm {~s} ^ { - 2 }$, and for $4 \leqslant t \leqslant 7$ the particle has constant acceleration $7.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Find an expression for the velocity of the particle in terms of $t$, valid for $0 \leqslant t \leqslant 4$.
Show that the displacement of the particle from $A$ is 19.2 m when $t = 4$.
Find the displacement of the particle from $A$ when $t = 7$. \section*{Jan 2007} 6
\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-14_558_1373_267_386} The diagram shows the ( $t , v$ ) graph for the motion of a hoist used to deliver materials to different levels at a building site. The hoist moves vertically. The graph consists of straight line segments. In the first stage the hoist travels upwards from ground level for 25 s , coming to rest 8 m above ground level.
Find the greatest speed reached by the hoist during this stage. The second stage consists of a 40 s wait at the level reached during the first stage. In the third stage the hoist continues upwards until it comes to rest 40 m above ground level, arriving 135 s after leaving ground level. The hoist accelerates at $0.02 \mathrm {~ms} ^ { - 2 }$ for the first 40 s of the third stage, reaching a speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find
the value of $V$,
the length of time during the third stage for which the hoist is moving at constant speed,
the deceleration of the hoist in the final part of the third stage. 7 A particle $P$ of mass 0.5 kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of $40 ^ { \circ }$ to the horizontal. $P$ reaches its highest point and then moves back down the plane. The coefficient of friction between $P$ and the plane is 0.6 .
Show that the magnitude of the frictional force acting on $P$ is 2.25 N , correct to 3 significant figures.
Find the acceleration of $P$ when it is moving
(a) up the plane,
(b) down the plane.
When $P$ is moving up the plane, it passes through a point $A$ with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
(a) Find the length of time before $P$ reaches its highest point.
(b) Find the total length of time for $P$ to travel from the point $A$ to its highest point and back to $A$.

OCR M2 Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-02_538_535_269_806} A uniform solid cone has vertical height 20 cm and base radius $r \mathrm {~cm}$. It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cone topples when the angle of inclination is $24 ^ { \circ }$ (see diagram).

Find $r$, correct to 1 decimal place. A uniform solid cone of vertical height 20 cm and base radius 2.5 cm is placed on the plane which is inclined at an angle of $24 ^ { \circ }$.
State, with justification, whether this cone will topple.

OCR M2 Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-02_451_533_1676_808} One end of a light inextensible string of length 1.6 m is attached to a point $P$. The other end is attached to the point $Q$, vertically below $P$, where $P Q = 0.8 \mathrm {~m}$. A small smooth bead $B$, of mass 0.01 kg , is threaded on the string and moves in a horizontal circle, with centre $Q$ and radius $0.6 \mathrm {~m} . Q B$ rotates with constant angular speed $\omega$ rad s $^ { - 1 }$ (see diagram).

Show that the tension in the string is 0.1225 N .
Find $\omega$.
Calculate the kinetic energy of the bead.

OCR M2 Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-03_168_959_246_593} Three smooth spheres $A , B$ and $C$, of equal radius and of masses $m \mathrm {~kg} , 2 m \mathrm {~kg}$ and $3 m \mathrm {~kg}$ respectively, lie in a straight line and are free to move on a smooth horizontal table. Sphere $A$ is moving with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it collides directly with sphere $B$ which is stationary. As a result of the collision $B$ starts to move with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the coefficient of restitution between $A$ and $B$.
Find, in terms of $m$, the magnitude of the impulse that $A$ exerts on $B$, and state the direction of this impulse. Sphere $B$ subsequently collides with sphere $C$ which is stationary. As a result of this impact $B$ and $C$ coalesce.
Show that there will be another collision.

OCR M2 Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-03_319_650_1219_749} A uniform $\operatorname { rod } A B$ of length 60 cm and weight 15 N is freely suspended from its end $A$. The end $B$ of the rod is attached to a light inextensible string of length 80 cm whose other end is fixed to a point $C$ which is at the same horizontal level as $A$. The rod is in equilibrium with the string at right angles to the rod (see diagram).

Show that the tension in the string is 4.5 N .
Find the magnitude and direction of the force acting on the rod at $A$.

OCR M2 Q6

6 A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of $5 ^ { \circ }$ to the horizontal. At a certain point $P$ on the hill the car's speed is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The point $Q$ is 400 m further up the hill from $P$, and at $Q$ the car's speed is $15 \mathrm {~ms} ^ { - 1 }$.

Calculate the work done by the car's engine as the car moves from $P$ to $Q$, assuming that any resistances to the car's motion may be neglected. Assume instead that the resistance to the car's motion between $P$ and $Q$ is a constant force of magnitude 200 N.
Given that the acceleration of the car at $Q$ is zero, show that the power of the engine as the car passes through $Q$ is 12.0 kW , correct to 3 significant figures.
Given that the power of the car's engine at $P$ is the same as at $Q$, calculate the car's retardation at $P$. \section*{June 2005}

OCR M2 Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-04_397_1431_264_358} A barrier is modelled as a uniform rectangular plank of wood, $A B C D$, rigidly joined to a uniform square metal plate, $D E F G$. The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m . The metal plate has mass 80 kg and side 0.5 m . The plank and plate are joined in such a way that $C D E$ is a straight line (see diagram). The barrier is smoothly pivoted at the point $D$. In the closed position, the barrier rests on a thin post at $H$. The distance $C H$ is 0.25 m .

Calculate the contact force at $H$ when the barrier is in the closed position. In the open position, the centre of mass of the barrier is vertically above $D$.
Calculate the angle between $A B$ and the horizontal when the barrier is in the open position.

OCR M2 Q8

17 marks

8 A particle is projected with speed $49 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation $\theta$ from a point $O$ on a horizontal plane, and moves freely under gravity. The horizontal and upward vertical displacements of the particle from $O$ at time $t$ seconds after projection are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively.

Express $x$ and $y$ in terms of $\theta$ and $t$, and hence show that $$y = x \tan \theta - \frac { x ^ { 2 } \left( 1 + \tan ^ { 2 } \theta \right) } { 490 } .$$
\includegraphics[max width=\textwidth, alt={}]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-04_638_1259_1695_443}
The particle passes through the point where $x = 70$ and $y = 30$. The two possible values of $\theta$ are $\theta _ { 1 }$ and $\theta _ { 2 }$, and the corresponding points where the particle returns to the plane are $A _ { 1 }$ and $A _ { 2 }$ respectively (see diagram).
Find $\theta _ { 1 }$ and $\theta _ { 2 }$.
Calculate the distance between $A _ { 1 }$ and $A _ { 2 }$.
Jan 2006 1 A uniform rod $A B$ has weight 20 N and length 3 m . The end $A$ is freely hinged to a point on a vertical wall. The rod is held horizontally and in equilibrium by a light inextensible string. One end of the string is attached to the rod at $B$. The other end of the string is attached to a point $C$, which is 1 m directly above $A$ (see diagram). Calculate the tension in the string. 2 A golfer hits a ball from a point $O$ on horizontal ground with a velocity of $50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $25 ^ { \circ }$ above the horizontal. The ball first hits the ground at a point $A$. Assuming that there is no air resistance, calculate
the time taken for the ball to travel from $O$ to $A$,
the distance $O A$. 3 A box of mass 50 kg is dragged along a horizontal floor by a constant force of magnitude 400 N acting at an angle of $\alpha$ above the horizontal. The total resistance to the motion of the box has magnitude 300 N . The box starts from rest at the point $O$, and passes the point $P , 25 \mathrm {~m}$ from $O$, with a speed of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
For the box's motion from $O$ to $P$, find
(a) the increase in kinetic energy of the box,
(b) the work done against the resistance to motion of the box.
Hence calculate $\alpha$. 4
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-06_490_753_264_696} A rectangular frame consists of four uniform metal rods. $A B$ and $C D$ are vertical and each is 40 cm long and has mass $0.2 \mathrm {~kg} . A D$ and $B C$ are horizontal and each is 60 cm long. $A D$ has mass 0.7 kg and $B C$ has mass 0.5 kg . The frame is freely hinged at $E$ and $F$, where $E$ is 10 cm above $A$, and $F$ is 10 cm below $B$ (see diagram).
Sketch a diagram showing the directions of the horizontal components of the forces acting on the frame at $E$ and $F$.
Calculate the magnitude of the horizontal component of the force acting on the frame at $E$.
Calculate the distance from $A D$ of the centre of mass of the frame. 5 Three smooth spheres $A , B$ and $C$, of equal radius and of masses $3 m \mathrm {~kg} , 2 m \mathrm {~kg}$ and $m \mathrm {~kg}$ respectively, are free to move in a straight line on a smooth horizontal table. Spheres $B$ and $C$ are stationary. Sphere $A$ is moving with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it collides directly with sphere $B$. The collision is perfectly elastic.
Find the velocities of $A$ and $B$ after the collision.
Find, in terms of $m$, the magnitude of the impulse that $A$ exerts on $B$, and state the direction of this impulse. Sphere $B$ continues its motion and hits $C$. After the collision, $B$ continues in the same direction with speed $1.0 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $C$ moves with speed $2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the coefficient of restitution between $B$ and $C$. 6 A stone is projected horizontally with speed $7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point $O$ on the edge of a vertical cliff. The horizontal and upward vertical displacements of the stone from $O$ at any subsequent time, $t$ seconds, are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively. Assume that there is no air resistance.
Express $x$ and $y$ in terms of $t$, and hence show that $y = - \frac { 1 } { 10 } x ^ { 2 }$. The stone hits the sea at a point which is 20 m below the level of $O$.
Find the distance between the foot of the cliff and the point where the stone hits the sea.
Find the speed and direction of motion of the stone immediately before it hits the sea. Jan 2006
7 Marco is riding his bicycle at a constant speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along a horizontal road, working at a constant rate of 300 W . Marco and his bicycle have a combined mass of 75 kg .
Calculate the wind resistance acting on Marco and his bicycle. Nicolas is riding his bicycle at the same speed as Marco and directly behind him. Nicolas experiences $30 \%$ less wind resistance than Marco.
Calculate the power output of Nicolas. The two cyclists arrive at the bottom of a hill which is at an angle of $1 ^ { \circ }$ to the horizontal. Marco increases his power output to 500 W .
Assuming Marco's wind resistance is unchanged, calculate his instantaneous acceleration immediately after starting to climb the hill. Marco reaches the top of the hill at a speed of $13 \mathrm {~ms} ^ { - 1 }$. He then freewheels down a hill of length 200 m which is at a constant angle of $10 ^ { \circ }$ to the horizontal. He experiences a constant wind resistance of 120 N .
Calculate Marco's speed at the bottom of this hill. 8 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-08_282_711_264_717} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A particle $P$ of mass 0.1 kg is moving with constant angular speed $\omega \mathrm { rads } ^ { - 1 }$ in a horizontal circle on the smooth inner surface of a cone which is fixed with its axis vertical and its vertex $A$ at its lowest point. The semi-vertical angle of the cone is $60 ^ { \circ }$ and the distance $A P$ is 0.8 m (see Fig.1).
Calculate the magnitude of the force exerted by the cone on the particle.
Calculate $\omega$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-08_407_711_1103_717} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The particle $P$ is now attached to one end of a light inextensible string which passes through a small smooth hole at $A$. The lower end of the string is attached to a particle $Q$ of mass 0.2 kg . $Q$ is in equilibrium with the string taut and $A P = 0.8 \mathrm {~m} . P$ moves in a horizontal circle with constant speed $\nu \mathrm { m } \mathrm { s } ^ { - 1 }$ (see Fig. 2).
State the tension in the string.
Find $v$. \section*{June 2006} 1 A child of mass 35 kg runs up a flight of stairs in 10 seconds. The vertical distance climbed is 4 m . Assuming that the child's speed is constant, calculate the power output. 2 A small sphere of mass 0.3 kg is dropped from rest at a height of 2 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.4 m above the ground. Ignoring air resistance, calculate the magnitude of the impulse which the ground exerts on the sphere when it rebounds. 3
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-09_718_588_717_776} A uniform solid hemisphere of weight 12 N and radius 6 cm is suspended by two vertical strings. One string is attached to the point $O$, the centre of the plane face, and the other string is attached to the point $A$ on the rim of the plane face. The hemisphere hangs in equilibrium and $O A$ makes an angle of $60 ^ { \circ }$ with the vertical (see diagram).
Find the horizontal distance from the centre of mass of the hemisphere to the vertical through $O$.
Calculate the tensions in the strings. \section*{June 2006} 4 A car of mass 900 kg is travelling at a constant speed of $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a level road. The total resistance to motion is 450 N .
Calculate the power output of the car's engine. A roof box of mass 50 kg is mounted on the roof of the car. The total resistance to motion of the vehicle increases to 500 N .
The car's engine continues to work at the same rate. Calculate the maximum speed of the car on the level road. The power output of the car's engine increases to 15000 W . The resistance to motion of the car, with roof box, remains 500 N .
Calculate the instantaneous acceleration of the car on the level road when its speed is $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
The car climbs a hill which is at an angle of $5 ^ { \circ }$ to the horizontal. Calculate the instantaneous retardation of the car when its speed is $26 \mathrm {~ms} ^ { - 1 }$. 5
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-10_668_565_1213_790} A uniform lamina $A B C D E$ consists of a square and an isosceles triangle. The square has sides of 18 cm and $B C = C D = 15 \mathrm {~cm}$ (see diagram).
Taking $x$ - and $y$-axes along $A E$ and $A B$ respectively, find the coordinates of the centre of mass of the lamina.
The lamina is freely suspended from $B$. Calculate the angle that $B D$ makes with the vertical. 6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-11_446_1358_262_391} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A light inextensible string of length 1 m passes through a small smooth hole $A$ in a fixed smooth horizontal plane. One end of the string is attached to a particle $P$, of mass 0.5 kg , which hangs in equilibrium below the plane. The other end of the string is attached to a particle $Q$, of mass 0.3 kg , which rotates with constant angular speed in a circle of radius 0.2 m on the surface of the plane (see Fig. 1).
Calculate the tension in the string and hence find the angular speed of $Q$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-11_489_1358_1286_392} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The particle $Q$ on the plane is now fixed to a point 0.2 m from the hole at $A$ and the particle $P$ rotates in a horizontal circle of radius 0.2 m (see Fig. 2).
Calculate the tension in the string.
Calculate the speed of $P$. \section*{June 2006} 7 A small ball is projected at an angle of $50 ^ { \circ }$ above the horizontal, from a point $A$, which is 2 m above ground level. The highest point of the path of the ball is 15 m above the ground, which is horizontal. Air resistance may be ignored.
Find the speed with which the ball is projected from $A$. The ball hits a net at a point $B$ when it has travelled a horizontal distance of 45 m .
Find the height of $B$ above the ground.
Find the speed of the ball immediately before it hits the net. 8 Two uniform smooth spheres, $A$ and $B$, have the same radius. The mass of $A$ is 2 kg and the mass of $B$ is $m \mathrm {~kg}$. Sphere $A$ is travelling in a straight line on a smooth horizontal surface, with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when it collides directly with sphere $B$, which is at rest. As a result of the collision, sphere $A$ continues in the same direction with a speed of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the greatest possible value of $m$. It is given that $m = 1$.
Find the coefficient of restitution between $A$ and $B$. On another occasion $A$ and $B$ are travelling towards each other, each with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when they collide directly.
Find the kinetic energy lost due to the collision. 1 A uniform solid cylinder has height 20 cm and diameter 12 cm . It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cylinder topples when the angle of inclination is $\alpha$. Find $\alpha$. 2 Two smooth spheres $A$ and $B$, of equal radius and of masses 0.2 kg and 0.1 kg respectively, are free to move on a smooth horizontal table. $A$ is moving with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it collides directly with $B$, which is stationary. The collision is perfectly elastic. Calculate the speed of $A$ after the impact. [4] 3 A small sphere of mass 0.2 kg is projected vertically downwards with speed $21 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point at a height of 40 m above horizontal ground. It hits the ground and rebounds vertically upwards, coming to instantaneous rest at its initial point of projection. Ignoring air resistance, calculate
the coefficient of restitution between the sphere and the ground,
the magnitude of the impulse which the ground exerts on the sphere. 4 A skier of mass 80 kg is pulled up a slope which makes an angle of $20 ^ { \circ }$ with the horizontal. The skier is subject to a constant frictional force of magnitude 70 N . The speed of the skier increases from $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at the point $A$ to $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at the point $B$, and the distance $A B$ is 25 m .
By modelling the skier as a small object, calculate the work done by the pulling force as the skier moves from $A$ to $B$.
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-13_458_1027_1420_598} It is given that the pulling force has constant magnitude $P \mathrm {~N}$, and that it acts at a constant angle of $30 ^ { \circ }$ above the slope (see diagram). Calculate $P$. 5 A model train has mass 100 kg . When the train is moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the resistance to its motion is $3 v ^ { 2 } \mathrm {~N}$ and the power output of the train is $\frac { 3000 } { v } \mathrm {~W}$.
Show that the driving force acting on the train is 120 N at an instant when the train is moving with speed $5 \mathrm {~ms} ^ { - 1 }$.
Find the acceleration of the train at an instant when it is moving horizontally with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The train moves with constant speed up a straight hill inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 98 }$.
Calculate the speed of the train. 6
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-14_540_894_989_628} A uniform lamina $A B C D E$ of weight 30 N consists of a rectangle and a right-angled triangle. The dimensions are as shown in the diagram.
Taking $x$ - and $y$-axes along $A E$ and $A B$ respectively, find the coordinates of the centre of mass of the lamina. The lamina is freely suspended from a hinge at $B$.
Calculate the angle that $A B$ makes with the vertical. The lamina is now held in a position such that $B D$ is horizontal. This is achieved by means of a string attached to $D$ and to a fixed point 15 cm directly above the hinge at $B$.
Calculate the tension in the string. 7
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-15_787_1009_269_568} One end of a light inextensible string of length 0.8 m is attached to a fixed point $A$ which lies above a smooth horizontal table. The other end of the string is attached to a particle $P$, of mass 0.3 kg , which moves in a horizontal circle on the table with constant angular speed $2 \mathrm { rad } \mathrm { s } ^ { - 1 } . A P$ makes an angle of $30 ^ { \circ }$ with the vertical (see diagram).
Calculate the tension in the string.
Calculate the normal contact force between the particle and the table. The particle now moves with constant speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is on the point of leaving the surface of the table.
Calculate $v$. 8 A missile is projected with initial speed $42 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ above the horizontal. Ignoring air resistance, calculate
the maximum height of the missile above the level of the point of projection,
the distance of the missile from the point of projection at the instant when it is moving downwards at an angle of $10 ^ { \circ }$ to the horizontal. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }\section*{June 2007} 1 A man drags a sack at constant speed in a straight line along horizontal ground by means of a rope attached to the sack. The rope makes an angle of $35 ^ { \circ }$ with the horizontal and the tension in the rope is 40 N . Calculate the work done in moving the sack 100 m . 2 Calculate the range on a horizontal plane of a small stone projected from a point on the plane with speed $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation of $27 ^ { \circ }$. 3 A rocket of mass 250 kg is moving in a straight line in space. There is no resistance to motion, and the mass of the rocket is assumed to be constant. With its motor working at a constant rate of 450 kW the rocket's speed increases from $100 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $150 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a time $t$ seconds.
Calculate the value of $t$.
Calculate the acceleration of the rocket at the instant when its speed is $120 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. 4 A ball is projected from a point $O$ on the edge of a vertical cliff. The horizontal and vertically upward components of the initial velocity are $7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $21 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. At time $t$ seconds after projection the ball is at the point $( x , y )$ referred to horizontal and vertically upward axes through $O$. Air resistance may be neglected.
Express $x$ and $y$ in terms of $t$, and hence show that $y = 3 x - \frac { 1 } { 10 } x ^ { 2 }$. The ball hits the sea at a point which is 25 m below the level of $O$.
Find the horizontal distance between the cliff and the point where the ball hits the sea. 5 A cyclist and her bicycle have a combined mass of 70 kg . The cyclist ascends a straight hill $A B$ of constant slope, starting from rest at $A$ and reaching a speed of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $B$. The level of $B$ is 6 m above the level of $A$. For the cyclist's motion from $A$ to $B$, find
the increase in kinetic energy,
the increase in gravitational potential energy. During the ascent the resistance to motion is constant and has magnitude 60 N . The work done by the cyclist in moving from $A$ to $B$ is 8000 J .
Calculate the distance $A B$. 6
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-17_679_621_267_762} A particle $P$ of mass 0.3 kg is attached to one end of each of two light inextensible strings. The other end of the longer string is attached to a fixed point $A$ and the other end of the shorter string is attached to a fixed point $B$, which is vertically below $A$. $A P$ makes an angle of $30 ^ { \circ }$ with the vertical and is 0.4 m long. $P B$ makes an angle of $60 ^ { \circ }$ with the vertical. The particle moves in a horizontal circle with constant angular speed and with both strings taut (see diagram). The tension in the string $A P$ is 5 N . Calculate
the tension in the string $P B$,
the angular speed of $P$,
the kinetic energy of $P$. 7 Two small spheres $A$ and $B$, with masses 0.3 kg and $m \mathrm {~kg}$ respectively, lie at rest on a smooth horizontal surface. $A$ is projected directly towards $B$ with speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and hits $B$. The direction of motion of $A$ is reversed in the collision. The speeds of $A$ and $B$ after the collision are $1 \mathrm {~ms} ^ { - 1 }$ and $3 \mathrm {~ms} ^ { - 1 }$ respectively. The coefficient of restitution between $A$ and $B$ is $e$.
Show that $m = 0.7$.
Find $e$.
$B$ continues to move at $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and strikes a vertical wall at right angles. The coefficient of restitution between $B$ and the wall is $f$.
Find the range of values of $f$ for which there will be a second collision between $A$ and $B$.
Find, in terms of $f$, the magnitude of the impulse that the wall exerts on $B$.
Given that $f = \frac { 3 } { 4 }$, calculate the final speeds of $A$ and $B$, correct to 1 decimal place. 8 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-18_460_495_269_826} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} An object consists of a uniform solid hemisphere of weight 40 N and a uniform solid cylinder of weight 5 N . The cylinder has height $h \mathrm {~m}$. The solids have the same base radius 0.8 m and are joined so that the hemisphere's plane face coincides with one of the cylinder's faces. The centre of the common face is the point $O$ (see Fig.1). The centre of mass of the object lies inside the hemisphere and is at a distance of 0.2 m from $O$.
Show that $h = 1.2$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-18_630_1067_1292_539} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} One end of a light inextensible string is attached to a point on the circumference of the upper face of the cylinder. The string is horizontal and its other end is tied to a fixed point on a rough plane. The object rests in equilibrium on the plane with its axis of symmetry vertical. The plane makes an angle of $30 ^ { \circ }$ with the horizontal (see Fig. 2). The tension in the string is $T \mathrm {~N}$ and the frictional force acting on the object is $F \mathrm {~N}$.
By taking moments about $O$, express $F$ in terms of $T$.
Find another equation connecting $T$ and $F$. Hence calculate the tension and the frictional force.

OCR FP2 Q1

1

Write down and simplify the first three non-zero terms of the Maclaurin series for $\ln ( 1 + 3 x )$.
Hence find the first three non-zero terms of the Maclaurin series for $$\mathrm { e } ^ { x } \ln ( 1 + 3 x )$$ simplifying the coefficients.

OCR FP2 Q2

2 Use the Newton-Raphson method to find the root of the equation $\mathrm { e } ^ { - x } = x$ which is close to $x = 0.5$. Give the root correct to 3 decimal places.

OCR FP2 Q3

3 Express $\frac { x + 6 } { x \left( x ^ { 2 } + 2 \right) }$ in partial fractions.

OCR FP2 Q4

4 Answer the whole of this question on the insert provided.

\includegraphics[max width=\textwidth, alt={}]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-02_887_1273_1137_438}

The sketch shows the curve with equation $y = \mathrm { F } ( x )$ and the line $y = x$. The equation $x = \mathrm { F } ( x )$ has roots $x = \alpha$ and $x = \beta$ as shown.

Use the copy of the sketch on the insert to show how an iteration of the form $x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)$, with starting value $x _ { 1 }$ such that $0 < x _ { 1 } < \alpha$ as shown, converges to the root $x = \alpha$.
State what happens in the iteration in the following two cases.
(a) $x _ { 1 }$ is chosen such that $\alpha < x _ { 1 } < \beta$.
(b) $x _ { 1 }$ is chosen such that $x _ { 1 } > \beta$. \section*{Jan 2006} 4
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-03_873_1259_274_484}
(a) $\_\_\_\_$
(b) $\_\_\_\_$ \section*{Jan 2006}

OCR FP2 Q5

5

Find the equations of the asymptotes of the curve with equation $$y = \frac { x ^ { 2 } + 3 x + 3 } { x + 2 }$$
Show that $y$ cannot take values between - 3 and 1 .

OCR FP2 Q6

6

It is given that, for non-negative integers $n$, $$I _ { n } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { - x } x ^ { n } \mathrm {~d} x$$ Prove that, for $n \geqslant 1$, $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
Evaluate $I _ { 3 }$, giving the answer in terms of e.

OCR FP2 Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-04_673_1285_1176_429} The diagram shows the curve with equation $y = \sqrt { x }$. A set of $N$ rectangles of unit width is drawn, starting at $x = 1$ and ending at $x = N + 1$, where $N$ is an integer (see diagram).

By considering the areas of these rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } < \int _ { 1 } ^ { N + 1 } \sqrt { x } \mathrm {~d} x$$
By considering the areas of another set of rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } > \int _ { 0 } ^ { N } \sqrt { x } \mathrm {~d} x$$
Hence find, in terms of $N$, limits between which $\sum _ { r = 1 } ^ { N } \sqrt { r }$ lies. \section*{Jan 2006}

OCR FP2 Q8

8 The equation of a curve, in polar coordinates, is $$r = 1 + \cos 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$

State the greatest value of $r$ and the corresponding values of $\theta$.
Find the equations of the tangents at the pole.
Find the exact area enclosed by the curve and the lines $\theta = 0$ and $\theta = \frac { 1 } { 2 } \pi$.
Find, in simplified form, the cartesian equation of the curve.

OCR FP2 Q9

9 marks

9

Using the definitions of $\cosh x$ and $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, prove that $$\sinh 2 x = 2 \sinh x \cosh x$$
Show that the curve with equation $$y = \cosh 2 x - 6 \sinh x$$ has just one stationary point, and find its $x$-coordinate in logarithmic form. Determine the nature of the stationary point. \section*{June 2006} 1 Find the first three non-zero terms of the Maclaurin series for $$( 1 + x ) \sin x$$ simplifying the coefficients. 2
Given that $y = \tan ^ { - 1 } x$, prove that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }$.
Verify that $y = \tan ^ { - 1 } x$ satisfies the equation $$\left( 1 + x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 0$$ 3 The equation of a curve is $y = \frac { x + 1 } { x ^ { 2 } + 3 }$.
State the equation of the asymptote of the curve.
Show that $- \frac { 1 } { 6 } \leqslant y \leqslant \frac { 1 } { 2 }$. 4
Using the definition of $\cosh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, prove that $$\cosh 2 x = 2 \cosh ^ { 2 } x - 1$$
Hence solve the equation $$\cosh 2 x - 7 \cosh x = 3$$ giving your answer in logarithmic form. 5
Express $t ^ { 2 } + t + 1$ in the form $( t + a ) ^ { 2 } + b$.
By using the substitution $\tan \frac { 1 } { 2 } x = t$, show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 2 + \sin x } \mathrm {~d} x = \frac { \sqrt { 3 } } { 9 } \pi$$ 6
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-07_627_1356_260_392} The diagram shows the curve with equation $y = 3 ^ { x }$ for $0 \leqslant x \leqslant 1$. The area $A$ under the curve between these limits is divided into $n$ strips, each of width $h$ where $n h = 1$.
By using the set of rectangles indicated on the diagram, show that $A > \frac { 2 h } { 3 ^ { h } - 1 }$.
By considering another set of rectangles, show that $A < \frac { ( 2 h ) 3 ^ { h } } { 3 ^ { h } - 1 }$.
Given that $h = 0.001$, use these inequalities to find values between which $A$ lies. 7 The equation of a curve, in polar coordinates, is $$r = \sqrt { 3 } + \tan \theta , \quad \text { for } - \frac { 1 } { 3 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$$
Find the equation of the tangent at the pole.
State the greatest value of $r$ and the corresponding value of $\theta$.
Sketch the curve.
Find the exact area of the region enclosed by the curve and the lines $\theta = 0$ and $\theta = \frac { 1 } { 4 } \pi$. 8 The curve with equation $y = \frac { \sinh x } { x ^ { 2 } }$, for $x > 0$, has one turning point.
Show that the $x$-coordinate of the turning point satisfies the equation $x - 2 \tanh x = 0$.
Use the Newton-Raphson method, with a first approximation $x _ { 1 } = 2$, to find the next two approximations, $x _ { 2 }$ and $x _ { 3 }$, to the positive root of $x - 2 \tanh x = 0$.
By considering the approximate errors in $x _ { 1 }$ and $x _ { 2 }$, estimate the error in $x _ { 3 }$. (You are not expected to evaluate $x _ { 4 }$.) June 2006
9
Given that $y = \sinh ^ { - 1 } x$, prove that $y = \ln \left( x + \sqrt { x ^ { 2 } + 1 } \right)$.
It is given that, for non-negative integers $n$, $$I _ { n } = \int _ { 0 } ^ { \alpha } \sinh ^ { n } \theta \mathrm {~d} \theta$$ where $\alpha = \sinh ^ { - 1 } 1$. Show that $$n I _ { n } = \sqrt { 2 } - ( n - 1 ) I _ { n - 2 } , \quad \text { for } n \geqslant 2$$
Evaluate $I _ { 4 }$, giving your answer in terms of $\sqrt { 2 }$ and logarithms. \section*{Jan 2007} 1 It is given that $\mathrm { f } ( x ) = \ln ( 3 + x )$.
Find the exact values of $\mathrm { f } ( 0 )$ and $\mathrm { f } ^ { \prime } ( 0 )$, and show that $\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 9 }$.
Hence write down the first three terms of the Maclaurin series for $\mathrm { f } ( x )$, given that $- 3 < x \leqslant 3$. 2 It is given that $\mathrm { f } ( x ) = x ^ { 2 } - \tan ^ { - 1 } x$.
Show by calculation that the equation $\mathrm { f } ( x ) = 0$ has a root in the interval $0.8 < x < 0.9$.
Use the Newton-Raphson method, with a first approximation 0.8 , to find the next approximation to this root. Give your answer correct to 3 decimal places. 3
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-09_714_997_991_571} The diagram shows the curve with equation $y = \mathrm { e } ^ { x ^ { 2 } }$, for $0 \leqslant x \leqslant 1$. The region under the curve between these limits is divided into four strips of equal width. The area of this region under the curve is $A$.
By considering the set of rectangles indicated in the diagram, show that an upper bound for $A$ is 1.71 .
By considering an appropriate set of four rectangles, find a lower bound for $A$. 4
On separate diagrams, sketch the graphs of $y = \sinh x$ and $y = \operatorname { cosech } x$.
Show that $\operatorname { cosech } x = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } - 1 }$, and hence, using the substitution $u = \mathrm { e } ^ { x }$, find $\int \operatorname { cosech } x \mathrm {~d} x$. \section*{Jan 2007} 5 It is given that, for non-negative integers $n$, $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \cos x \mathrm {~d} x$$
Prove that, for $n \geqslant 2$, $$I _ { n } = \left( \frac { 1 } { 2 } \pi \right) ^ { n } - n ( n - 1 ) I _ { n - 2 } .$$
Find $I _ { 4 }$ in terms of $\pi$. 6
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-10_719_1435_849_354} The diagram shows the curve with equation $y = \frac { 2 x ^ { 2 } - 3 a x } { x ^ { 2 } - a ^ { 2 } }$, where $a$ is a positive constant.
Find the equations of the asymptotes of the curve.
Sketch the curve with equation $$y ^ { 2 } = \frac { 2 x ^ { 2 } - 3 a x } { x ^ { 2 } - a ^ { 2 } }$$ State the coordinates of any points where the curve crosses the axes, and give the equations of any asymptotes. 7
Express $\frac { 1 - t ^ { 2 } } { t ^ { 2 } \left( 1 + t ^ { 2 } \right) }$ in partial fractions.
Use the substitution $t = \tan \frac { 1 } { 2 } x$ to show that $$\int _ { \frac { 1 } { 3 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \cos x } { 1 - \cos x } \mathrm {~d} x = \sqrt { 3 } - 1 - \frac { 1 } { 6 } \pi$$ 8
Define tanh $y$ in terms of $\mathrm { e } ^ { y }$ and $\mathrm { e } ^ { - y }$.
Given that $y = \tanh ^ { - 1 } x$, where $- 1 < x < 1$, prove that $y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$.
Find the exact solution of the equation $3 \cosh x = 4 \sinh x$, giving the answer in terms of a logarithm.
Solve the equation $$\tanh ^ { - 1 } x + \ln ( 1 - x ) = \ln \left( \frac { 4 } { 5 } \right)$$ 9 The equation of a curve, in polar coordinates, is $$r = \sec \theta + \tan \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi$$
Sketch the curve.
Find the exact area of the region bounded by the curve and the lines $\theta = 0$ and $\theta = \frac { 1 } { 3 } \pi$.
Find a cartesian equation of the curve. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }\section*{June 2007} 1 The equation of a curve, in polar coordinates, is $$r = 2 \sin 3 \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi .$$ Find the exact area of the region enclosed by the curve between $\theta = 0$ and $\theta = \frac { 1 } { 3 } \pi$. 2
Given that $\mathrm { f } ( x ) = \sin \left( 2 x + \frac { 1 } { 4 } \pi \right)$, show that $\mathrm { f } ( x ) = \frac { 1 } { 2 } \sqrt { 2 } ( \sin 2 x + \cos 2 x )$.
Hence find the first four terms of the Maclaurin series for $\mathrm { f } ( x )$. [You may use appropriate results given in the List of Formulae.] 3 It is given that $\mathrm { f } ( x ) = \frac { x ^ { 2 } + 9 x } { ( x - 1 ) \left( x ^ { 2 } + 9 \right) }$.
Express $\mathrm { f } ( x )$ in partial fractions.
Hence find $\int f ( x ) \mathrm { d } x$. 4
Given that $$y = x \sqrt { 1 - x ^ { 2 } } - \cos ^ { - 1 } x$$ find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in a simplified form.
Hence, or otherwise, find the exact value of $\int _ { 0 } ^ { 1 } 2 \sqrt { 1 - x ^ { 2 } } \mathrm {~d} x$. 5 It is given that, for non-negative integers $n$, $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x$$
Show that, for $n \geqslant 1$, $$I _ { n } = \mathrm { e } - n I _ { n - 1 } .$$
Find $I _ { 3 }$ in terms of e. 6
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-13_816_1369_267_388} The diagram shows the curve with equation $y = \frac { 1 } { x ^ { 2 } }$ for $x > 0$, together with a set of $n$ rectangles of unit width, starting at $x = 1$.
By considering the areas of these rectangles, explain why $$\frac { 1 } { 1 ^ { 2 } } + \frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \ldots + \frac { 1 } { n ^ { 2 } } > \int _ { 1 } ^ { n + 1 } \frac { 1 } { x ^ { 2 } } \mathrm {~d} x$$
By considering the areas of another set of rectangles, explain why $$\frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \frac { 1 } { 4 ^ { 2 } } + \ldots + \frac { 1 } { n ^ { 2 } } < \int _ { 1 } ^ { n } \frac { 1 } { x ^ { 2 } } \mathrm {~d} x$$
Hence show that $$1 - \frac { 1 } { n + 1 } < \sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } } < 2 - \frac { 1 } { n }$$
Hence give bounds between which $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }$ lies. 7
Using the definitions of hyperbolic functions in terms of exponentials, prove that $$\cosh x \cosh y - \sinh x \sinh y = \cosh ( x - y )$$
Given that $\cosh x \cosh y = 9$ and $\sinh x \sinh y = 8$, show that $x = y$.
Hence find the values of $x$ and $y$ which satisfy the equations given in part (ii), giving the answers in logarithmic form. \section*{June 2007} 8 The iteration $x _ { n + 1 } = \frac { 1 } { \left( x _ { n } + 2 \right) ^ { 2 } }$, with $x _ { 1 } = 0.3$, is to be used to find the real root, $\alpha$, of the equation $x ( x + 2 ) ^ { 2 } = 1$.
Find the value of $\alpha$, correct to 4 decimal places. You should show the result of each step of the iteration process.
Given that $\mathrm { f } ( x ) = \frac { 1 } { ( x + 2 ) ^ { 2 } }$, show that $\mathrm { f } ^ { \prime } ( \alpha ) \neq 0$.
The difference, $\delta _ { r }$, between successive approximations is given by $\delta _ { r } = x _ { r + 1 } - x _ { r }$. Find $\delta _ { 3 }$.
Given that $\delta _ { r + 1 } \approx \mathrm { f } ^ { \prime } ( \alpha ) \delta _ { r }$, find an estimate for $\delta _ { 10 }$. 9 It is given that the equation of a curve is $$y = \frac { x ^ { 2 } - 2 a x } { x - a }$$ where $a$ is a positive constant.
Find the equations of the asymptotes of the curve.
Show that $y$ takes all real values.
Sketch the curve $y = \frac { x ^ { 2 } - 2 a x } { x - a }$. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }1 It is given that $\mathrm { f } ( x ) = \ln ( 1 + \cos x )$.
Find the exact values of $f ( 0 ) , f ^ { \prime } ( 0 )$ and $f ^ { \prime \prime } ( 0 )$.
Hence find the first two non-zero terms of the Maclaurin series for $\mathrm { f } ( x )$. 2
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-15_584_707_575_721} The diagram shows parts of the curves with equations $y = \cos ^ { - 1 } x$ and $y = \frac { 1 } { 2 } \sin ^ { - 1 } x$, and their point of intersection $P$.
Verify that the coordinates of $P$ are $\left( \frac { 1 } { 2 } \sqrt { 3 } , \frac { 1 } { 6 } \pi \right)$.
Find the gradient of each curve at $P$. 3
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-15_650_791_1619_678} The diagram shows the curve with equation $y = \sqrt { 1 + x ^ { 3 } }$, for $2 \leqslant x \leqslant 3$. The region under the curve between these limits has area $A$.
Explain why $3 < A < \sqrt { 28 }$.
The region is divided into 5 strips, each of width 0.2 . By using suitable rectangles, find improved lower and upper bounds between which $A$ lies. Give your answers correct to 3 significant figures. 4 The equation of a curve, in polar coordinates, is $$r = 1 + 2 \sec \theta , \quad \text { for } - \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi .$$
Find the exact area of the region bounded by the curve and the lines $\theta = 0$ and $\theta = \frac { 1 } { 6 } \pi$. [The result $\int \sec \theta \mathrm { d } \theta = \ln | \sec \theta + \tan \theta |$ may be assumed.]
Show that a cartesian equation of the curve is $( x - 2 ) \sqrt { x ^ { 2 } + y ^ { 2 } } = x$. 5
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-16_609_892_815_628} The diagram shows the curve with equation $y = x \mathrm { e } ^ { - x } + 1$. The curve crosses the $x$-axis at $x = \alpha$.
Use differentiation to show that the $x$-coordinate of the stationary point is 1 . $\alpha$ is to be found using the Newton-Raphson method, with $\mathrm { f } ( x ) = x \mathrm { e } ^ { - x } + 1$.
Explain why this method will not converge to $\alpha$ if an initial approximation $x _ { 1 }$ is chosen such that $x _ { 1 } > 1$.
Use this method, with a first approximation $x _ { 1 } = 0$, to find the next three approximations $x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$. Find $\alpha$, correct to 3 decimal places. 6 The equation of a curve is $y = \frac { 2 x ^ { 2 } - 11 x - 6 } { x - 1 }$.
Find the equations of the asymptotes of the curve.
Show that $y$ takes all real values. \section*{Jan 2008} 7 It is given that, for integers $n \geqslant 1$, $$I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { n } } \mathrm {~d} x$$
Use integration by parts to show that $I _ { n } = 2 ^ { - n } + 2 n \int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \left( 1 + x ^ { 2 } \right) ^ { n + 1 } } \mathrm {~d} x$.
Show that $2 n I _ { n + 1 } = 2 ^ { - n } + ( 2 n - 1 ) I _ { n }$.
Find $I _ { 2 }$ in terms of $\pi$. 8
By using the definition of $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, show that $$\sinh ^ { 3 } x = \frac { 1 } { 4 } \sinh 3 x - \frac { 3 } { 4 } \sinh x$$
Find the range of values of the constant $k$ for which the equation $$\sinh 3 x = k \sinh x$$ has real solutions other than $x = 0$.
Given that $k = 4$, solve the equation in part (ii), giving the non-zero answers in logarithmic form. 9
Prove that $\frac { \mathrm { d } } { \mathrm { d } x } \left( \cosh ^ { - 1 } x \right) = \frac { 1 } { \sqrt { x ^ { 2 } - 1 } }$.
Hence, or otherwise, find $\int \frac { 1 } { \sqrt { 4 x ^ { 2 } - 1 } } \mathrm {~d} x$.
By means of a suitable substitution, find $\int \sqrt { 4 x ^ { 2 } - 1 } \mathrm {~d} x$. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }June 2008
$1 \frac { 2 a x } { \text { It is given that } \mathrm { f } ( x ) = \frac { \text { where } a \text { is a non-zero constant. Express } \mathrm { f } ( x ) \text { in partial } } { ( x - 2 a ) \left( x ^ { 2 } + a ^ { 2 } \right) } \text {, whent } }$ fractions. 2
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-18_341_1043_466_552} The diagram shows the curve $y = \mathrm { f } ( x )$. The curve has a maximum point at ( 0,5 ) and crosses the $x$-axis at $( - 2,0 ) , ( 3,0 )$ and $( 4,0 )$. Sketch the curve $y ^ { 2 } = \mathrm { f } ( x )$, showing clearly the coordinates of any turning points and of any points where this curve crosses the axes. 3 By using the substitution $t = \tan \frac { 1 } { 2 } x$, find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 2 - \cos x } \mathrm {~d} x$$ giving the answer in terms of $\pi$. 4
Sketch, on the same diagram, the curves with equations $y = \operatorname { sech } x$ and $y = x ^ { 2 }$.
By using the definition of $\operatorname { sech } x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, show that the $x$-coordinates of the points at which these curves meet are solutions of the equation $$x ^ { 2 } = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } + 1 } .$$
The iteration $$x _ { n + 1 } = \sqrt { \frac { 2 \mathrm { e } ^ { x _ { n } } } { \mathrm { e } ^ { 2 x _ { n } } + 1 } }$$ can be used to find the positive root of the equation in part (ii). With initial value $x _ { 1 } = 1$, the approximations $x _ { 2 } = 0.8050 , x _ { 3 } = 0.8633 , x _ { 4 } = 0.8463$ and $x _ { 5 } = 0.8513$ are obtained, correct to 4 decimal places. State with a reason whether, in this case, the iteration produces a 'staircase' or a ‘cobweb’ diagram. 5 It is given that, for $n \geqslant 0$, $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { n } x \mathrm {~d} x$$
By considering $I _ { n } + I _ { n - 2 }$, or otherwise, show that, for $n \geqslant 2$, $$( n - 1 ) \left( I _ { n } + I _ { n - 2 } \right) = 1 .$$
Find $I _ { 4 }$ in terms of $\pi$. \section*{June 2008} 6 It is given that $\mathrm { f } ( x ) = 1 - \frac { 7 } { x ^ { 2 } }$.
Use the Newton-Raphson method, with a first approximation $x _ { 1 } = 2.5$, to find the next approximations $x _ { 2 }$ and $x _ { 3 }$ to a root of $\mathrm { f } ( x ) = 0$. Give the answers correct to 6 decimal places. [3]
The root of $\mathrm { f } ( x ) = 0$ for which $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ are approximations is denoted by $\alpha$. Write down the exact value of $\alpha$.
The error $e _ { n }$ is defined by $e _ { n } = \alpha - x _ { n }$. Find $e _ { 1 } , e _ { 2 }$ and $e _ { 3 }$, giving your answers correct to 5 decimal places. Verify that $e _ { 3 } \approx \frac { e _ { 2 } ^ { 3 } } { e _ { 1 } ^ { 2 } }$. 7 It is given that $\mathrm { f } ( x ) = \tanh ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right)$, for $x > - \frac { 1 } { 2 }$.
Show that $\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { 1 + 2 x }$, and find $\mathrm { f } ^ { \prime \prime } ( x )$.
Show that the first three terms of the Maclaurin series for $\mathrm { f } ( x )$ can be written as $\ln a + b x + c x ^ { 2 }$, for constants $a , b$ and $c$ to be found. 8 The equation of a curve, in polar coordinates, is $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-19_268_793_1567_717} The diagram shows the part of the curve for which $0 \leqslant \theta \leqslant \alpha$, where $\theta = \alpha$ is the equation of the tangent to the curve at $O$. Find $\alpha$ in terms of $\pi$.
(a) If $\mathrm { f } ( \theta ) = 1 - \sin 2 \theta$, show that $\mathrm { f } \left( \frac { 1 } { 2 } ( 2 k + 1 ) \pi - \theta \right) = \mathrm { f } ( \theta )$ for all $\theta$, where $k$ is an integer.
(b) Hence state the equations of the lines of symmetry of the curve $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$
Sketch the curve with equation $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$ State the maximum value of $r$ and the corresponding values of $\theta$. \section*{June 2008} 9
Prove that $\int _ { 0 } ^ { N } \ln ( 1 + x ) \mathrm { d } x = ( N + 1 ) \ln ( N + 1 ) - N$, where $N$ is a positive constant.
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-20_616_1261_406_482} The diagram shows the curve $y = \ln ( 1 + x )$, for $0 \leqslant x \leqslant 70$, together with a set of rectangles of unit width.
(a) By considering the areas of these rectangles, explain why $$\ln 2 + \ln 3 + \ln 4 + \ldots + \ln 70 < \int _ { 0 } ^ { 70 } \ln ( 1 + x ) d x$$ (b) By considering the areas of another set of rectangles, show that $$\ln 2 + \ln 3 + \ln 4 + \ldots + \ln 70 > \int _ { 0 } ^ { 69 } \ln ( 1 + x ) d x$$ (c) Hence find bounds between which $\ln ( 70 ! )$ lies. Give the answers correct to 1 decimal place.

OCR FP3 Q3

3 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 8 y = \mathrm { e } ^ { 3 x }$$

OCR FP3 Q5

5

Use de Moivre's theorem to prove that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
Hence find the largest positive root of the equation $$64 x ^ { 6 } - 96 x ^ { 4 } + 36 x ^ { 2 } - 3 = 0$$ giving your answer in trigonometrical form.

OCR FP3 Q6

6 Lines $l _ { 1 }$ and $l _ { 2 }$ have equations $$\frac { x - 3 } { 2 } = \frac { y - 4 } { - 1 } = \frac { z + 1 } { 1 } \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { 2 }$$ respectively.

Find the equation of the plane $\Pi _ { 1 }$ which contains $l _ { 1 }$ and is parallel to $l _ { 2 }$, giving your answer in the form r.n $= p$.
Find the equation of the plane $\Pi _ { 2 }$ which contains $l _ { 2 }$ and is parallel to $l _ { 1 }$, giving your answer in the form r.n $= p$.
Find the distance between the planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
State the relationship between the answer to part (iii) and the lines $l _ { 1 }$ and $l _ { 2 }$.
Show that $\left( z - \mathrm { e } ^ { \mathrm { i } \phi } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \phi } \right) \equiv z ^ { 2 } - ( 2 \cos \phi ) z + 1$.
Write down the seven roots of the equation $z ^ { 7 } = 1$ in the form $\mathrm { e } ^ { \mathrm { i } \theta }$ and show their positions in an Argand diagram.
Hence express $z ^ { 7 } - 1$ as the product of one real linear factor and three real quadratic factors. 8
Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \tan x = \cos ^ { 3 } x$$ expressing $y$ in terms of $x$ in your answer.
Find the particular solution for which $y = 2$ when $x = \pi$. 9 The set $S$ consists of the numbers $3 ^ { n }$, where $n \in \mathbb { Z }$. ( $\mathbb { Z }$ denotes the set of integers $\{ 0 , \pm 1 , \pm 2 , \ldots \}$.)
Prove that the elements of $S$, under multiplication, form a commutative group $G$. (You may assume that addition of integers is associative and commutative.)

Determine whether or not each of the following subsets of $S$, under multiplication, forms a subgroup of $G$, justifying your answers.
(a) The numbers $3 ^ { 2 n }$, where $n \in \mathbb { Z }$.
(b) The numbers $3 ^ { n }$, where $n \in \mathbb { Z }$ and $n \geqslant 0$.
(c) The numbers $3 ^ { \left( \pm n ^ { 2 } \right) }$, where $n \in \mathbb { Z }$. 1 (a) A group $G$ of order 6 has the combination table shown below.

	$e$	$a$	$b$	$p$	$q$	$r$
$e$	$e$	$a$	$b$	$p$	$q$	$r$
$a$	$a$	$b$	$e$	$r$	$p$	$q$
$b$	$b$	$e$	$a$	$q$	$r$	$p$
$p$	$p$	$q$	$r$	$e$	$a$	$b$
$q$	$q$	$r$	$p$	$b$	$e$	$a$
$r$	$r$	$p$	$q$	$a$	$b$	$e$

State, with a reason, whether or not $G$ is commutative.
State the number of subgroups of $G$ which are of order 2 .
List the elements of the subgroup of $G$ which is of order 3 .
(b) A multiplicative group $H$ of order 6 has elements $e , c , c ^ { 2 } , c ^ { 3 } , c ^ { 4 } , c ^ { 5 }$, where $e$ is the identity. Write down the order of each of the elements $c ^ { 3 } , c ^ { 4 }$ and $c ^ { 5 }$. 2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 4 x$$ 3 Two fixed points, $A$ and $B$, have position vectors $\mathbf { a }$ and $\mathbf { b }$ relative to the origin $O$, and a variable point $P$ has position vector $\mathbf { r }$.
Give a geometrical description of the locus of $P$ when $\mathbf { r }$ satisfies the equation $\mathbf { r } = \lambda \mathbf { a }$, where $0 \leqslant \lambda \leqslant 1$.
Given that $P$ is a point on the line $A B$, use a property of the vector product to explain why $( \mathbf { r } - \mathbf { a } ) \times ( \mathbf { r } - \mathbf { b } ) = \mathbf { 0 }$.
Give a geometrical description of the locus of $P$ when $\mathbf { r }$ satisfies the equation $\mathbf { r } \times ( \mathbf { a } - \mathbf { b } ) = \mathbf { 0 }$. 4 The integrals $C$ and $S$ are defined by $$C = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \cos 3 x \mathrm {~d} x \quad \text { and } \quad S = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \sin 3 x \mathrm {~d} x$$ By considering $C + \mathrm { i } S$ as a single integral, show that $$C = - \frac { 1 } { 13 } \left( 2 + 3 \mathrm { e } ^ { \pi } \right)$$ and obtain a similar expression for $S$.
(You may assume that the standard result for $\int \mathrm { e } ^ { k x } \mathrm {~d} x$ remains true when $k$ is a complex constant, so that $\left. \int \mathrm { e } ^ { ( a + \mathrm { i } b ) x } \mathrm {~d} x = \frac { 1 } { a + \mathrm { i } b } \mathrm { e } ^ { ( a + \mathrm { i } b ) x } .\right)$ 5
Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { y } { x } = \sin 2 x$$ expressing $y$ in terms of $x$ in your answer. In a particular case, it is given that $y = \frac { 2 } { \pi }$ when $x = \frac { 1 } { 4 } \pi$.
Find the solution of the differential equation in this case.
Write down a function to which $y$ approximates when $x$ is large and positive. 6 A tetrahedron $A B C D$ is such that $A B$ is perpendicular to the base $B C D$. The coordinates of the points $A , C$ and $D$ are $( - 1 , - 7,2 ) , ( 5,0,3 )$ and $( - 1,3,3 )$ respectively, and the equation of the plane $B C D$ is $x + 2 y - 2 z = - 1$.
Find, in either order, the coordinates of $B$ and the length of $A B$.
Find the acute angle between the planes $A C D$ and $B C D$.
(a) Verify, without using a calculator, that $\theta = \frac { 1 } { 8 } \pi$ is a solution of the equation $\sin 6 \theta = \sin 2 \theta$.
(b) By sketching the graphs of $y = \sin 6 \theta$ and $y = \sin 2 \theta$ for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$, or otherwise, find the other solution of the equation $\sin 6 \theta = \sin 2 \theta$ in the interval $0 < \theta < \frac { 1 } { 2 } \pi$.
Use de Moivre's theorem to prove that $$\sin 6 \theta \equiv \sin 2 \theta \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 3 \right)$$
Hence show that one of the solutions obtained in part (i) satisfies $\cos ^ { 2 } \theta = \frac { 1 } { 4 } ( 2 - \sqrt { 2 } )$, and justify which solution it is. \section*{Jan 2008} 8 Groups $A , B , C$ and $D$ are defined as follows:
A: the set of numbers $\{ 2,4,6,8 \}$ under multiplication modulo 10 ,
$B$ : the set of numbers $\{ 1,5,7,11 \}$ under multiplication modulo 12 ,
$C$ : the set of numbers $\left\{ 2 ^ { 0 } , 2 ^ { 1 } , 2 ^ { 2 } , 2 ^ { 3 } \right\}$ under multiplication modulo 15,
$D$ : the set of numbers $\left\{ \frac { 1 + 2 m } { 1 + 2 n } \right.$, where $m$ and $n$ are integers $\}$ under multiplication.
Write down the identity element for each of groups $A , B , C$ and $D$.
Determine in each case whether the groups $$\begin{aligned} & A \text { and } B ,
& B \text { and } C ,
& A \text { and } C \end{aligned}$$ are isomorphic or non-isomorphic. Give sufficient reasons for your answers.
Prove the closure property for group $D$.
Elements of the set $\left\{ \frac { 1 + 2 m } { 1 + 2 n } \right.$, where $m$ and $n$ are integers $\}$ are combined under addition. State which of the four basic group properties are not satisfied. (Justification is not required.) \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }1 (a) A cyclic multiplicative group $G$ has order 12. The identity element of $G$ is $e$ and another element is $r$, with order 12.
Write down, in terms of $e$ and $r$, the elements of the subgroup of $G$ which is of order 4.
Explain briefly why there is no proper subgroup of $G$ in which two of the elements are $e$ and $r$.
(b) A group $H$ has order $m n p$, where $m , n$ and $p$ are prime. State the possible orders of proper subgroups of $H$. 2 Find the acute angle between the line with equation $\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { k } + t ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )$ and the plane with equation $\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) + \mu ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )$. 3
Use the substitution $z = x + y$ to show that the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x + y + 3 } { x + y - 1 }$$ may be written in the form $\frac { \mathrm { d } z } { \mathrm {~d} x } = \frac { 2 ( z + 1 ) } { z - 1 }$.
Hence find the general solution of the differential equation (A). 4
By expressing $\cos \theta$ in terms of $\mathrm { e } ^ { \mathrm { i } \theta }$ and $\mathrm { e } ^ { - \mathrm { i } \theta }$, show that $$\cos ^ { 5 } \theta \equiv \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
Hence solve the equation $\cos 5 \theta + 5 \cos 3 \theta + 9 \cos \theta = 0$ for $0 \leqslant \theta \leqslant \pi$. 5 Two lines have equations $$\frac { x - k } { 2 } = \frac { y + 1 } { - 5 } = \frac { z - 1 } { - 3 } \quad \text { and } \quad \frac { x - k } { 1 } = \frac { y + 4 } { - 4 } = \frac { z } { - 2 }$$ where $k$ is a constant.
Show that, for all values of $k$, the lines intersect, and find their point of intersection in terms of $k$.
For the case $k = 1$, find the equation of the plane in which the lines lie, giving your answer in the form $a x + b y + c z = d$. 6 The operation ○ on real numbers is defined by $a \circ b = a | b |$.
Show that ∘ is not commutative.
Prove that ∘ is associative.
Determine whether the set of real numbers, under the operation ∘, forms a group. \section*{June 2008}

OCR FP3 Q9

9 The set $S$ consists of the numbers $3 ^ { n }$, where $n \in \mathbb { Z }$. ( $\mathbb { Z }$ denotes the set of integers $\{ 0 , \pm 1 , \pm 2 , \ldots \}$.)

Prove that the elements of $S$, under multiplication, form a commutative group $G$. (You may assume that addition of integers is associative and commutative.)
Determine whether or not each of the following subsets of $S$, under multiplication, forms a subgroup of $G$, justifying your answers.
(a) The numbers $3 ^ { 2 n }$, where $n \in \mathbb { Z }$.
(b) The numbers $3 ^ { n }$, where $n \in \mathbb { Z }$ and $n \geqslant 0$.
(c) The numbers $3 ^ { \left( \pm n ^ { 2 } \right) }$, where $n \in \mathbb { Z }$. 1 (a) A group $G$ of order 6 has the combination table shown below. $G$ and $H$ are the non-cyclic groups of order 4 and 6 respectively.
Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups $G$ and $H$. Hence explain why there are no non-cyclic proper subgroups of $Q$. 7 Three planes $\Pi _ { 1 } , \Pi _ { 2 }$ and $\Pi _ { 3 }$ have equations $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 5 , \quad \mathbf { r } . ( \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 6 , \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } - 12 \mathbf { k } ) = 12 ,$$ respectively. Planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$ intersect in a line $l$; planes $\Pi _ { 2 }$ and $\Pi _ { 3 }$ intersect in a line $m$.
Show that $l$ and $m$ are in the same direction.
Write down what you can deduce about the line of intersection of planes $\Pi _ { 1 }$ and $\Pi _ { 3 }$.
By considering the cartesian equations of $\Pi _ { 1 } , \Pi _ { 2 }$ and $\Pi _ { 3 }$, or otherwise, determine whether or not the three planes have a common line of intersection. 8 The operation $*$ is defined on the elements $( x , y )$, where $x , y \in \mathbb { R }$, by $$( a , b ) * ( c , d ) = ( a c , a d + b ) .$$ It is given that the identity element is $( 1,0 )$.
Prove that $*$ is associative.
Find all the elements which commute with $( 1,1 )$.
It is given that the particular element $( m , n )$ has an inverse denoted by $( p , q )$, where $$( m , n ) * ( p , q ) = ( p , q ) * ( m , n ) = ( 1,0 ) .$$ Find $( p , q )$ in terms of $m$ and $n$.
Find all self-inverse elements.
Give a reason why the elements $( x , y )$, under the operation $*$, do not form a group.

OCR D2 2006 June Q1

1 The network represents a system of pipes along which fluid can flow from $S$ to $T$. The values on the arcs are lower and upper capacities in litres per second.
\includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-02_696_1292_376_424}

Calculate the capacity of the cut with $\mathrm { X } = \{ S , A , B , C \} , \mathrm { Y } = \{ D , E , F , G , H , I , T \}$.
Show that the capacity of the cut $\alpha$, shown on the diagram, is 12 litres per second and calculate the minimum flow across the cut $\alpha$, from $S$ to $T$, (without regard to the remainder of the diagram).
Explain why the arc SC must have at least 5 litres per second flowing through it. By considering the flow through $A$, explain why $A D$ cannot be full to capacity.
Show that it is possible for 11 litres per second to flow through the system.
From your previous answers, what can be deduced about the maximum flow through the system?

	\(e\)	\(a\)	\(b\)	\(p\)	\(q\)	\(r\)
\(e\)	\(e\)	\(a\)	\(b\)	\(p\)	\(q\)	\(r\)
\(a\)	\(a\)	\(b\)	\(e\)	\(r\)	\(p\)	\(q\)
\(b\)	\(b\)	\(e\)	\(a\)	\(q\)	\(r\)	\(p\)
\(p\)	\(p\)	\(q\)	\(r\)	\(e\)	\(a\)	\(b\)
\(q\)	\(q\)	\(r\)	\(p\)	\(b\)	\(e\)	\(a\)
\(r\)	\(r\)	\(p\)	\(q\)	\(a\)	\(b\)	\(e\)

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