Questions H240/03 - A-Level Maths

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OCR H240/03 2022 June Q4

8 marks Standard +0.8

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 $y^4 - 16y^2 - 225 = 0$. [4]
Hence determine the sum to infinity of the geometric progression. [4]

OCR H240/03 2022 June Q5

14 marks Standard +0.3

In this question you must show detailed reasoning. \includegraphics{figure_5} The diagram shows the curve with equation $y = \frac{2x - 3}{4x^2 + 1}$. The tangent to the curve at the point $P$ has gradient 2.

Show that the $x$-coordinate of $P$ satisfies the equation $$4x^3 + 3x - 3 = 0.$$ [5]
Show by calculation that the $x$-coordinate of $P$ lies between 0.5 and 1. [2]
Show that the iteration $$x_{n+1} = \frac{3 - 4x_n^3}{3}$$ cannot converge to the $x$-coordinate of $P$ whatever starting value is used. [2]
Use the Newton-Raphson method, with initial value 0.5, to determine the coordinates of $P$ correct to 5 decimal places. [5]

OCR H240/03 2022 June Q6

8 marks Standard +0.3

In this question you must show detailed reasoning. \includegraphics{figure_6} The diagram shows the curves $y = \sqrt{2x + 9}$ and $y = 4\mathrm{e}^{-2x} - 1$ which intersect on the $y$-axis. The shaded region is bounded by the curves and the $x$-axis. Determine the area of the shaded region, giving your answer in the form $p + q \ln 2$ where $p$ and $q$ are constants to be determined. [8]

OCR H240/03 2022 June Q7

8 marks Standard +0.8

In this question you must show detailed reasoning.

Show that the equation $m \sec \theta + 3 \cos \theta = 4 \sin \theta$ can be expressed in the form $$m \tan^2 \theta - 4 \tan \theta + (m + 3) = 0.$$ [3]
It is given that there is only one value of $\theta$, for $0 < \theta < \pi$, satisfying the equation $m \sec \theta + 3 \cos \theta = 4 \sin \theta$. Given also that $m$ is a negative integer, find this value of $\theta$, correct to 3 significant figures. [5]

OCR H240/03 2022 June Q8

2 marks Easy -1.3

\includegraphics{figure_8} 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 $15°$ 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. [2]

OCR H240/03 2022 June Q9

6 marks Moderate -0.3

\includegraphics{figure_9} 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 $12 \mathrm{m s}^{-1}$, is overtaken by car $A$ which has initial speed $20 \mathrm{m s}^{-1}$. From $t = 0$ car $A$ travels with constant deceleration for 30 seconds. When $t = 30$ the speed of car $A$ is $8 \mathrm{m s}^{-1}$ and the car maintains this speed in subsequent motion.

Calculate the deceleration of car $A$. [2]
Determine the value of $t$ when $B$ overtakes $A$. [4]

OCR H240/03 2022 June Q10

8 marks Standard +0.3

\includegraphics{figure_10} 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.

Find the acceleration of $Q$ while $P$ and $B$ are in contact. [2]
Determine the coefficient of friction between $P$ and $B$. [3]
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 $B$. [3]

OCR H240/03 2022 June Q11

7 marks Challenging +1.2

\includegraphics{figure_11} A uniform rod $AB$ 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 $60°$ with the horizontal and is held in limiting equilibrium by a light inextensible string $CD$. $D$ is a fixed point vertically above $A$ and $CD$ makes an angle of $60°$ with the vertical. The distance $AC$ is $x$ m (see diagram).

Find, in terms of $g$ and $x$, the tension in the string. [3]

The coefficient of friction between the rod and the ground is $\frac{9\sqrt{3}}{35}$.

Determine the value of $x$. [4]

OCR H240/03 2022 June Q12

13 marks Standard +0.3

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 $t \geq 0$, the velocity $\mathbf{v} \mathrm{m s}^{-1}$ of $P$, relative to a fixed origin $O$, is given by $$\mathbf{v} = (1 - 2t)\mathbf{i} + (2t^2 + t - 13)\mathbf{j}.$$

Show that $P$ is never stationary. [2]
Find, in terms of $\mathbf{i}$ and $\mathbf{j}$, the acceleration of $P$ at time $t$. [1]

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 3 significant figures. [5]

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 $t = 1.5$. [5]

OCR H240/03 2022 June Q13

14 marks Standard +0.3

A small ball $B$ moves in the plane of a fixed horizontal axis $Ox$, which lies on horizontal ground, and a fixed vertically upwards axis $Oy$. $B$ is projected from $O$ with a velocity whose components along $Ox$ and $Oy$ are $U \mathrm{m s}^{-1}$ and $V \mathrm{m s}^{-1}$, 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 $2U^2 y = 2UVx - gx^2$. [3]

During its motion, $B$ just clears a vertical wall of height $\frac{1}{3}a$ m at a horizontal distance $a$ m from $O$. $B$ strikes the ground at a horizontal distance $3a$ m beyond the wall.

Determine the angle of projection of $B$. Give your answer in degrees correct to 3 significant figures. [5]
Given that the speed of projection of $B$ is $54.6 \mathrm{m s}^{-1}$, determine the value of $a$. [2]
Hence find the maximum height of $B$ above the ground during its motion. [3]
State one refinement of the model, other than including air resistance, that would make it more realistic. [1]

OCR H240/03 2023 June Q1

3 marks Easy -1.2

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

OCR H240/03 2023 June Q2

5 marks Moderate -0.3

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

OCR H240/03 2023 June Q3

8 marks Moderate -0.3

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

1. Given that $\text{f}'(2) = 13$, find the value of $p$. [2]
2. Given also that $(x - 2)$ is a factor of $\text{f}(x)$, find the value of $q$. [2]
The curve $y = \text{f}(x)$ is translated by the vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$.
Using the values from part (a), determine the equation of the curve after it has been translated. Give your answer in the form $y = x^3 + ax^2 + bx + c$, where $a$, $b$ and $c$ are integers to be found. [4]

OCR H240/03 2023 June Q4

7 marks Standard +0.3

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

Find the set of possible values of $k$. [2]
It is given that $k = -46$. Determine the coordinates of the two points on $C$ at which the gradient of the tangent is $\frac{1}{2}$. [5]

OCR H240/03 2023 June Q5

9 marks Standard +0.8

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{figure_5} The emblem is modelled by the region between the $x$-axis and the curve with parametric equations $$x = 1 + 0.2t - \cos t, \quad y = k \sin^2 t,$$ where $k$ is a positive constant and $0 \leq t \leq \pi$. Lengths are in metres and the area of the emblem must be $1 \text{m}^2$.

Show that $k \int_0^\pi (0.2 + \sin t - 0.2 \cos^2 t - \sin t \cos^2 t) dt = 1$. [3]
Determine the exact value of $k$. [6]

OCR H240/03 2023 June Q6

6 marks Standard +0.8

The first, third and fourth terms of an arithmetic progression are $u_1$, $u_3$ and $u_4$ respectively, where $$u_1 = 2 \sin \theta, \quad u_3 = -\sqrt{3} \cos \theta, \quad u_4 = \frac{7}{3} \sin \theta,$$ and $\frac{1}{2}\pi < \theta < \pi$.

Determine the exact value of $\theta$. [3]
Hence determine the value of $\sum_{r=1}^{100} u_r$. [3]

OCR H240/03 2023 June Q7

12 marks Standard +0.8

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

In this question you must show detailed reasoning. Find the times when the acceleration of $C$ is zero. [3] At $t = 0$ the velocity of $C$ is $17.5 \text{ms}^{-1}$ and at $t = T$ the velocity of $C$ is $5 \text{ms}^{-1}$.
By setting up and solving a differential equation, show that $T$ satisfies the equation $$T = 2 \ln\left(\frac{7 + 4T^2}{2}\right).$$ [6]
Use an iterative formula, based on the equation in part (b), to find the value of $T$, giving your answer correct to 4 significant figures. Use an initial value of 11.25 and show the result of each step of the iteration process. [2]
The diagram below shows the velocity-time graph for the motion of $C$. \includegraphics{figure_7d} Find the time taken for $C$ to decelerate from travelling at its maximum speed until it is travelling at $5 \text{ms}^{-1}$. [1]

OCR H240/03 2023 June Q8

4 marks Moderate -0.8

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

OCR H240/03 2023 June Q9

6 marks Challenging +1.2

\includegraphics{figure_9} A block $B$ of weight $10 \text{N}$ lies at rest in equilibrium on a rough plane inclined at $\theta$ to the horizontal. A horizontal force of magnitude $2 \text{N}$, 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 $B$. [1]

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 $\tan \theta$. [5]

OCR H240/03 2023 June Q10

7 marks Standard +0.3

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

Show that $3a + b = 10$. [3]

It is given that $a = 6$ and that $P$ moves with an acceleration of magnitude $5\sqrt{10} \text{ms}^{-2}$.

Determine the value of $m$. [4]

OCR H240/03 2023 June Q11

8 marks Standard +0.3

\includegraphics{figure_11} A uniform rod $AB$, of weight $20 \text{N}$ and length $2.8 \text{m}$, rests in equilibrium with the end $A$ in contact with rough horizontal ground and the end $B$ resting against a smooth wall inclined at $55°$ to the horizontal. The rod, which rests in a vertical plane that is perpendicular to the wall, is inclined at $30°$ to the horizontal (see diagram).

Show that the magnitude of the force acting on the rod at $B$ is $9.56 \text{N}$, correct to 3 significant figures. [3]
Determine the magnitude of the contact force between the rod and the ground. Give your answer correct to 3 significant figures. [5]

OCR H240/03 2023 June Q12

13 marks Standard +0.8

In this question you should take the acceleration due to gravity to be $10 \text{ms^{-2}$.} \includegraphics{figure_12} A small ball $P$ is projected from a point $A$ with speed $39 \text{ms}^{-1}$ at an angle of elevation $\theta$, where $\sin \theta = \frac{5}{13}$ and $\cos \theta = \frac{12}{13}$. Point $A$ is $20 \text{m}$ vertically above a point $B$ on horizontal ground. The ball first lands at a point $C$ on the horizontal ground (see diagram). The ball $P$ is modelled as a particle moving freely under gravity.

Find the maximum height of $P$ above the ground during its motion. [3]

The time taken for $P$ to travel from $A$ to $C$ is $7$ seconds.

Determine the value of $T$. [3]
State one limitation of the model, other than air resistance or the wind, that could affect the answer to part (b). [1]

At the instant that $P$ is projected, a second small ball $Q$ is released from rest at $B$ and moves towards $C$ along the horizontal ground. At time $t$ seconds, where $t \geq 0$, the velocity $v \text{ms}^{-1}$ of $Q$ is given by $$v = kt^3 + 6t^2 + \frac{3}{2}t,$$ where $k$ is a positive constant.