Questions H240/03 - A-Level Maths

State the value of $c$. The car passes $Q$ at time $t = 5$ and at that instant its deceleration is $0.12 \mathrm {~ms} ^ { - 2 }$. The car passes $R$ at time $t = 18$ with velocity $2.96 \mathrm {~ms} ^ { - 1 }$.
Determine the values of $a$ and $b$.
Find, to the nearest metre, the distance between points $P$ and $R$.
\includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-08_469_798_251_244} One end of a light inextensible string is attached to a particle $A$ of mass 2 kg . The other end of the string is attached to a second particle $B$ of mass 2.5 kg . Particle $A$ is in contact with a rough plane inclined at $\theta$ to the horizontal, where $\cos \theta = \frac { 4 } { 5 }$. The string is taut and passes over a small smooth pulley $P$ at the top of the plane. The part of the string from $A$ to $P$ is parallel to a line of greatest slope of the plane. Particle $B$ hangs freely below $P$ at a distance 1.5 m above horizontal ground, as shown in the diagram. The coefficient of friction between $A$ and the plane is $\mu$. The system is released from rest and in the subsequent motion $B$ hits the ground before $A$ reaches $P$. The speed of $B$ at the instant that it hits the ground is $1.2 \mathrm {~ms} ^ { - 1 }$.

OCR H240/03 2020 November Q11

11
\includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-10_474_853_264_242} A particle $P$ moves freely under gravity in the plane of a fixed horizontal axis $O x$, which lies on horizontal ground, and a fixed vertical axis $O y . P$ is projected from $O$ with a velocity whose components along $O x$ and $O y$ are $U$ and $V$, respectively. $P$ returns to the ground at a point $C$.

Determine, in terms of $U , V$ and $g$, the distance $O C$.
\includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-10_478_851_1151_244}
$P$ passes through two points $A$ and $B$, each at a height $h$ above the ground and a distance $d$ apart, as shown in the diagram.
Write down the horizontal and vertical components of the velocity of $P$ at $A$.
Hence determine an expression for $d$ in terms of $U , V , g$ and $h$.
Given that the direction of motion of $P$ as it passes through $A$ is inclined to the horizontal at an angle $\theta$, where $\tan \theta = \frac { 1 } { 2 }$, determine an expression for $V$ in terms of $g , d$ and $h$.

OCR H240/03 2022 June Q1

1 Solve the equation $| 2 x - 3 | = 9$.

OCR H240/03 2022 June Q2

Give full details of the single transformation that transforms the graph of $y = x ^ { 3 }$ to the graph of $y = x ^ { 3 } - 8$. The function f is defined by $\mathrm { f } ( x ) = x ^ { 3 } - 8$.
Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
State how the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ are related geometrically.

OCR H240/03 2022 June Q3

3 The points $P$ and $Q$ have coordinates $( 2 , - 5 )$ and $( 3,1 )$ respectively.
Determine the equation of the circle that has $P Q$ as a diameter. Give your answer in the form $x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$, where $a , b$ and $c$ are integers.

OCR H240/03 2022 June Q4

4 The positive integers $x , y$ and $z$ are the first, second and third terms, respectively, of an arithmetic progression with common difference - 4 . Also, $x , \frac { 15 } { y }$ and $z$ are the first, second and third terms, respectively, of a geometric progression.

Show that $y$ satisfies the equation $y ^ { 4 } - 16 y ^ { 2 } - 225 = 0$.
Hence determine the sum to infinity of the geometric progression.

OCR H240/03 2022 June Q8

8
\includegraphics[max width=\textwidth, alt={}, center]{e69f8d73-764e-4f13-a126-faec02c4ad08-07_159_896_488_244} A child attempts to drag a sledge along horizontal ground by means of a rope attached to the sledge. The rope makes an angle of $15 ^ { \circ }$ with the horizontal (see diagram). Given that the sledge remains at rest and that the frictional force acting on the sledge is 60 N , find the tension in the rope.

OCR H240/03 2022 June Q9

9
\includegraphics[max width=\textwidth, alt={}, center]{e69f8d73-764e-4f13-a126-faec02c4ad08-07_579_848_1142_242} The diagram shows a velocity-time graph representing the motion of two cars $A$ and $B$ which are both travelling along a horizontal straight road. At time $t = 0$, car $B$, which is travelling with constant speed $12 \mathrm {~ms} ^ { - 1 }$, is overtaken by car $A$ which has initial speed $20 \mathrm {~ms} ^ { - 1 }$. From $t = 0 \operatorname { car } A$ travels with constant deceleration for 30 seconds. When $t = 30$ the speed of car $A$ is $8 \mathrm {~ms} ^ { - 1 }$ and the car maintains this speed in its subsequent motion.

Calculate the deceleration of $\operatorname { car } A$.
Determine the value of $t$ when $B$ overtakes $A$.
\includegraphics[max width=\textwidth, alt={}, center]{e69f8d73-764e-4f13-a126-faec02c4ad08-08_293_773_354_246} A rectangular block $B$ is at rest on a horizontal surface. A particle $P$ of mass 2.5 kg is placed on the upper surface of $B$. The particle $P$ is attached to one end of a light inextensible string which passes over a smooth fixed pulley. A particle $Q$ of mass 3 kg is attached to the other end of the string and hangs freely below the pulley. The part of the string between $P$ and the pulley is horizontal (see diagram). The particles are released from rest with the string taut. It is given that $B$ remains in equilibrium while $P$ moves on the upper surface of $B$. The tension in the string while $P$ moves on $B$ is 16.8 N .

OCR H240/03 2022 June Q12

12 In this question the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are in the directions east and north respectively.
A particle $P$ is moving on a smooth horizontal surface under the action of a single force $\mathbf { F N }$. At time $t$ seconds, where $t \geqslant 0$, the velocity $\mathbf { v } \mathrm { ms } ^ { - 1 }$ of $P$, relative to a fixed origin $O$, is given by $\mathbf { v } = ( 1 - 2 t ) \mathbf { i } + \left( 2 t ^ { 2 } + t - 13 \right) \mathbf { j }$.

Show that $P$ is never stationary.
Find, in terms of $\mathbf { i }$ and $\mathbf { j }$, the acceleration of $P$ at time $t$. The mass of $P$ is 0.5 kg .
Determine the magnitude of $\mathbf { F }$ when $P$ is moving in the direction of the vector $- 2 \mathbf { i } + \mathbf { j }$. Give your answer correct to $\mathbf { 3 }$ significant figures. When $t = 1 , P$ is at the point with position vector $\frac { 1 } { 6 } \mathbf { j }$.
Determine the bearing of $P$ from $O$ at time $t = 1.5$.

OCR H240/03 2022 June Q13

13 A small ball $B$ moves in the plane of a fixed horizontal axis $O x$, which lies on horizontal ground, and a fixed vertically upwards axis $O y . B$ is projected from $O$ with a velocity whose components along $O x$ and $O y$ are $U \mathrm {~ms} ^ { - 1 }$ and $V \mathrm {~ms} ^ { - 1 }$, respectively. The units of $x$ and $y$ are metres.
$B$ is modelled as a particle moving freely under gravity.

Show that the path of $B$ has equation $2 U ^ { 2 } y = 2 U V x - g x ^ { 2 }$. During its motion, $B$ just clears a vertical wall of height $\frac { 1 } { 2 } a \mathrm {~m}$ at a horizontal distance $a \mathrm {~m}$ from $O$. $B$ strikes the ground at a horizontal distance $3 a \mathrm {~m}$ beyond the wall.
Determine the angle of projection of $B$. Give your answer in degrees correct to $\mathbf { 3 }$ significant figures.
Given that the speed of projection of $B$ is $54.6 \mathrm {~ms} ^ { - 1 }$, determine the value of $a$.
Hence find the maximum height of $B$ above the ground during its motion.
State one refinement of the model, other than including air resistance, that would make it more realistic. \section*{END OF QUESTION PAPER}

OCR H240/03 2023 June Q1

1 Using logarithms, solve the equation
$4 ^ { 2 x + 1 } = 5 ^ { x }$,
giving your answer correct to $\mathbf { 3 }$ significant figures.

OCR H240/03 2023 June Q2

Express $3 \sin x - 4 \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give the value of $\alpha$ correct to $\mathbf { 4 }$ significant figures.
Hence solve the equation $3 \sin x - 4 \cos x = 2$ for $0 ^ { \circ } < x < 90 ^ { \circ }$, giving your answer correct to 3 significant figures.

OCR H240/03 2023 June Q3

3 The cubic polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = x ^ { 3 } + p x + q$, where $p$ and $q$ are constants.

1. Given that $\mathrm { f } ^ { \prime } ( 2 ) = 13$, find the value of $p$.
2. Given also that ( $x - 2$ ) is a factor of $\mathrm { f } ( x )$, find the value of $q$. The curve $y = \mathrm { f } ( x )$ is translated by the vector $\binom { 2 } { - 3 }$.
Using the values from part (a), determine the equation of the curve after it has been translated. Give your answer in the form $y = x ^ { 3 } + a x ^ { 2 } + b x + c$, where $a , b$ and $c$ are integers to be found.

OCR H240/03 2023 June Q4

4 A circle $C$ has equation $x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + k = 0$.

Find the set of possible values of $k$.
It is given that $k = - 46$. Determine the coordinates of the two points on $C$ at which the gradient of the tangent is $\frac { 1 } { 2 }$.

OCR H240/03 2023 June Q5

5 A mathematics department is designing a new emblem to place on the walls outside its classrooms. The design for the emblem is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-05_453_1200_358_242} The emblem is modelled by the region between the $x$-axis and the curve with parametric equations $x = 1 + 0.2 t - \cos t , \quad y = k \sin ^ { 2 } t$, where $k$ is a positive constant and $0 \leqslant t \leqslant \pi$. Lengths are in metres and the area of the emblem must be $1 \mathrm {~m} ^ { 2 }$.

Show that $k \int _ { 0 } ^ { \pi } \left( 0.2 + \sin t - 0.2 \cos ^ { 2 } t - \sin t \cos ^ { 2 } t \right) \mathrm { d } t = 1$.
Determine the exact value of $k$.

OCR H240/03 2023 June Q6

6 The first, third and fourth terms of an arithmetic progression are $u _ { 1 } , u _ { 3 }$ and $u _ { 4 }$ respectively, where $u _ { 1 } = 2 \sin \theta , \quad u _ { 3 } = - \sqrt { 3 } \cos \theta , \quad u _ { 4 } = \frac { 7 } { 2 } \sin \theta$, and $\frac { 1 } { 2 } \pi < \theta < \pi$.

Determine the exact value of $\theta$.
Hence determine the value of $\sum _ { r = 1 } ^ { 100 } u _ { r }$.

OCR H240/03 2023 June Q7

7 A car $C$ is moving horizontally in a straight line with velocity $v \mathrm {~ms} ^ { - 1 }$ at time $t$ seconds, where $v > 0$ and $t \geqslant 0$. The acceleration, $a \mathrm {~ms} ^ { - 2 }$, of $C$ is modelled by the equation
$a = v \left( \frac { 8 t } { 7 + 4 t ^ { 2 } } - \frac { 1 } { 2 } \right)$.

In this question you must show detailed reasoning. Find the times when the acceleration of $C$ is zero. At $t = 0$ the velocity of $C$ is $17.5 \mathrm {~ms} ^ { - 1 }$ and at $t = T$ the velocity of $C$ is $5 \mathrm {~ms} ^ { - 1 }$.
By setting up and solving a differential equation, show that $T$ satisfies the equation $$T = 2 \ln \left( \frac { 7 + 4 T ^ { 2 } } { 2 } \right)$$
Use an iterative formula, based on the equation in part (b), to find the value of $T$, giving your answer correct to $\mathbf { 4 }$ significant figures. Use an initial value of 11.25 and show the result of each step of the iteration process.
The diagram below shows the velocity-time graph for the motion of $C$.
\includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-06_751_878_1372_322} Find the time taken for $C$ to decelerate from travelling at its maximum speed until it is travelling at $5 \mathrm {~ms} ^ { - 1 }$.

OCR H240/03 2023 June Q8

8 A particle $P$ moves with constant acceleration $( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 2 }$. At time $t = 4$ seconds, $P$ has velocity $6 \mathbf { i } \mathrm {~ms} ^ { - 1 }$. Determine the speed of $P$ at time $t = 0$ seconds.

OCR H240/03 2023 June Q9

9
\includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-07_273_569_781_242} A block $B$ of weight 10 N lies at rest in equilibrium on a rough plane inclined at $\theta$ to the horizontal. A horizontal force of magnitude 2 N , acting above a line of greatest slope, is applied to $B$ (see diagram).

Complete the diagram in the Printed Answer Booklet to show all the forces acting on $B$. It is given that $B$ remains at rest and the coefficient of friction between $B$ and the plane is 0.8 .
Determine the greatest possible value of $\tan \theta$.

OCR H240/03 2023 June Q10

10 A particle $P$ of mass $m \mathrm {~kg}$ is moving on a smooth horizontal surface under the action of two constant horizontal forces $( - 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }$ and $( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }$. The resultant of these two forces is $\mathbf { R } \mathrm { N }$. It is given that $\mathbf { R }$ acts in a direction which is parallel to the vector $- \mathbf { i } + 3 \mathbf { j }$.

Show that $3 a + b = 10$. It is given that $a = 6$ and that $P$ moves with an acceleration of magnitude $5 \sqrt { 10 } \mathrm {~ms} ^ { - 2 }$.
Determine the value of $m$.
\includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-08_515_889_248_242} A uniform rod $A B$, of weight 20 N and length 2.8 m , rests in equilibrium with the end $A$ in contact with rough horizontal ground and the end $B$ resting against a smooth wall inclined at $55 ^ { \circ }$ to the horizontal. The rod, which rests in a vertical plane that is perpendicular to the wall, is inclined at $30 ^ { \circ }$ to the horizontal (see diagram).
Show that the magnitude of the force acting on the rod at $B$ is 9.56 N , correct to $\mathbf { 3 }$ significant figures.
Determine the magnitude of the contact force between the rod and the ground. Give your answer correct to $\mathbf { 3 }$ significant figures.

OCR H240/03 2019 June Q5

Prove that $( \cot \theta + \operatorname { cosec } \theta ) ^ { 2 } = \frac { 1 + \cos \theta } { 1 - \cos \theta }$.
Hence solve, for $0 < \theta < 2 \pi , 3 ( \cot \theta + \operatorname { cosec } \theta ) ^ { 2 } = 2 \sec \theta$.
\includegraphics[max width=\textwidth, alt={}]{7d1b7598-8f97-43a0-8366-efa8192d549e-06_574_695_306_258}
The diagram shows part of the curve $y = \frac { 2 x - 1 } { ( 2 x + 3 ) ( x + 1 ) ^ { 2 } }$.
Find the exact area of the shaded region, giving your answer in the form $p + q \ln r$, where $p$ and $q$ are positive integers and $r$ is a positive rational number.

OCR H240/03 2020 November Q9

For the motion before $B$ hits the ground, show that the acceleration of $B$ is $0.48 \mathrm {~ms} ^ { - 2 }$.
For the motion before $B$ hits the ground, show that the tension in the string is 23.3 N .
Determine the value of $\mu$. After $B$ hits the ground, $A$ continues to travel up the plane before coming to instantaneous rest before it reaches $P$.
Determine the distance that $A$ travels from the instant that $B$ hits the ground until $A$ comes to instantaneous rest.
\includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-09_917_784_244_242} The diagram shows a wall-mounted light. It consists of a rod $A B$ of mass 0.25 kg and length 0.8 m which is freely hinged to a vertical wall at $A$, and a lamp of mass 0.5 kg fixed at $B$. The system is held in equilibrium by a chain $C D$ whose end $C$ is attached to the midpoint of $A B$. The end $D$ is fixed to the wall a distance 0.4 m vertically above $A$. The rod $A B$ makes an angle of $60 ^ { \circ }$ with the downward vertical. The chain is modelled as a light inextensible string, the rod is modelled as uniform and the lamp is modelled as a particle.
By taking moments about $A$, determine the tension in the chain.
1. Determine the magnitude of the force exerted on the rod at $A$.
2. Calculate the direction of the force exerted on the rod at $A$.
Suggest one improvement that could be made to the model to make it more realistic.

OCR H240/03 2022 June Q5

Show that the $x$-coordinate of $P$ satisfies the equation $$4 x ^ { 3 } + 3 x - 3 = 0 .$$
Show by calculation that the $x$-coordinate of $P$ lies between 0.5 and 1 .
Show that the iteration $$x _ { n + 1 } = \frac { 3 - 4 x _ { n } ^ { 3 } } { 3 }$$ cannot converge to the $x$-coordinate of $P$ whatever starting value is used.
Use the Newton-Raphson method, with initial value 0.5 , to determine the coordinates of $P$ correct to $\mathbf { 5 }$ decimal places.

OCR H240/03 2022 June Q10

Find the acceleration of $Q$ while $P$ and $B$ are in contact.
Determine the coefficient of friction between $P$ and $B$.
Given that the coefficient of friction between $B$ and the horizontal surface is $\frac { 5 } { 49 }$, determine the least possible value for the mass of $B$.
\includegraphics[max width=\textwidth, alt={}, center]{e69f8d73-764e-4f13-a126-faec02c4ad08-09_634_625_255_246} A uniform rod $A B$ of mass 4 kg and length 3 m rests in a vertical plane with $A$ on rough horizontal ground. A particle of mass 1 kg is attached to the rod at $B$. The rod makes an angle of $60 ^ { \circ }$ with the horizontal and is held in limiting equilibrium by a light inextensible string $C D . D$ is a fixed point vertically above $A$ and $C D$ makes an angle of $60 ^ { \circ }$ with the vertical. The distance $A C$ is $x \mathrm {~m}$ (see diagram).
Find, in terms of $g$ and $x$, the tension in the string. The coefficient of friction between the rod and the ground is $\frac { 9 \sqrt { 3 } } { 35 }$.
Determine the value of $x$.

OCR H240/03 2018 June Q3

3 In this question you must show detailed reasoning. A gardener is planning the design for a rectangular flower bed. The requirements are:

the length of the flower bed is to be 3 m longer than the width,
the length of the flower bed must be at least 14.5 m ,
the area of the flower bed must be less than $180 \mathrm {~m} ^ { 2 }$.

The width of the flower bed is $x \mathrm {~m}$.
By writing down and solving appropriate inequalities in $x$, determine the set of possible values for the width of the flower bed.

Questions H240/03 (80 questions)

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