Questions H240/03 - A-Level Maths

OCR H240/03 2020 November Q6

6 In this question you must show detailed reasoning.

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The diagram shows the curve with equation $4 x y = 2 \left( x ^ { 2 } + 4 y ^ { 2 } \right) - 9 x$.

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x - 4 y - 9 } { 4 x - 16 y }$. At the point $P$ on the curve the tangent to the curve is parallel to the $y$-axis and at the point $Q$ on the curve the tangent to the curve is parallel to the $x$-axis.
Show that the distance $P Q$ is $k \sqrt { 5 }$, where $k$ is a rational number to be determined.

OCR H240/03 2022 June Q6

6 In this question you must show detailed reasoning.

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The diagram shows the curves $y = \sqrt { 2 x + 9 }$ and $y = 4 \mathrm { e } ^ { - 2 x } - 1$ 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.

OCR H240/03 2022 June Q7

7 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 $$m \tan ^ { 2 } \theta - 4 \tan \theta + ( m + 3 ) = 0 .$$
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 $\mathbf { 3 }$ significant figures.

OCR H240/03 2023 June Q12

12 In this question you should take the acceleration due to gravity to be $10 \mathrm {~ms ^ { - 2 }$.}

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A small ball $P$ is projected from a point $A$ with speed $39 \mathrm {~ms} ^ { - 1 }$ at an angle of elevation $\theta$, where $\sin \theta = \frac { 5 } { 13 }$ and $\cos \theta = \frac { 12 } { 13 }$. Point $A$ is 20 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. The time taken for $P$ to travel from $A$ to $C$ is $T$ seconds.
Determine the value of $T$.
State one limitation of the model, other than air resistance or the wind, that could affect the answer to part (b). 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 $t \geqslant 0$, the velocity $v \mathrm {~ms} ^ { - 1 }$ of $Q$ is given by
$v = k t ^ { 3 } + 6 t ^ { 2 } + \frac { 3 } { 2 } t$,
where $k$ is a positive constant.
Given that $P$ and $Q$ collide at $C$, determine the acceleration of $Q$ immediately before this collision.
\includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-10_607_803_303_246} The diagram shows a small block $B$, of mass 2 kg , and a particle $P$, of mass 4 kg , which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The particle can move on the inclined plane, which is rough, and which makes an angle of $60 ^ { \circ }$ with the horizontal. The block can move on the horizontal surface, which is also rough. The system is released from rest, and in the subsequent motion $P$ moves down the plane and $B$ does not reach the pulley. It is given that the coefficient of friction between $P$ and the inclined plane is twice the coefficient of friction between $B$ and the horizontal surface.
Determine, in terms of $g$, the tension in the string. When $P$ is moving at $2 \mathrm {~ms} ^ { - 1 }$ the string breaks. In the 0.5 seconds after the string breaks $P$ moves 1.9 m down the plane.
Determine the deceleration of $B$ after the string breaks. Give your answer correct to 3 significant figures.

OCR H240/03 2021 November Q1

1 Show in a sketch the region of the $x - y$ plane within which all three of the following inequalities hold.
$y \geqslant x ^ { 2 } , \quad x + y \leqslant 2 , \quad x \geqslant 0$. You should indicate the region for which the inequalities hold by labelling the region $R$.

OCR H240/03 2021 November Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{699c5e1e-1476-42cb-b3c4-ca08c4d81cb6-04_492_422_982_246} The diagram shows triangle $A B C$ in which angle $A$ is $60 ^ { \circ }$ and the lengths of $A B$ and $A C$ are $( 4 + h ) \mathrm { cm }$ and $( 4 - h ) \mathrm { cm }$ respectively.

Show that the length of $B C$ is $p \mathrm {~cm}$ where $$p ^ { 2 } = 16 + 3 h ^ { 2 } .$$
Hence show that, when $h$ is small, $p \approx 4 + \lambda h ^ { 2 } + \mu h ^ { 4 }$, where $\lambda$ and $\mu$ are rational numbers whose values are to be determined.

OCR H240/03 2021 November Q3

3 An arithmetic progression has first term 2 and common difference $d$, where $d \neq 0$. The first, third and thirteenth terms of this progression are also the first, second and third terms, respectively, of a geometric progression. By determining $d$, show that the arithmetic progression is an increasing sequence.

OCR H240/03 2021 November Q4

4

Sketch, on a single diagram, the following graphs.
- $y = | x - 1 |$
- $y = \frac { k } { x }$, where $k$ is a negative constant
- Hence explain why the equation $x | x - 1 | = k$ has exactly one real root for any negative value of $k$.
- Determine the real root of the equation $x | x - 1 | = - 6$.

OCR H240/03 2021 November Q5

5 A particle $P$ moves along a straight line in such a way that at time $t$ seconds $P$ has velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $$v = 12 \cos t + 5 \sin t .$$

Express $v$ in the form $R \cos ( t - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$. Give the value of $\alpha$ correct to $\mathbf { 4 }$ significant figures.
Hence find the two smallest positive values of $t$ for which $P$ is moving, in either direction, with a speed of $3 \mathrm {~ms} ^ { - 1 }$.

OCR H240/03 2021 November Q6

6 The equation $6 \arcsin ( 2 x - 1 ) - x ^ { 2 } = 0$ has exactly one real root.

Show by calculation that the root lies between 0.5 and 0.6 . In order to find the root, the iterative formula $$x _ { n + 1 } = p + q \sin \left( r x _ { n } ^ { 2 } \right)$$ with initial value $x _ { 0 } = 0.5$, is to be used.
Determine the values of the constants $p , q$ and $r$.
Hence find the root correct to $\mathbf { 4 }$ significant figures. Show the result of each step of the iteration process.

OCR H240/03 2021 November Q7

7 A curve $C$ in the $x - y$ plane has the property that the gradient of the tangent at the point $P ( x , y )$ is three times the gradient of the line joining the point $( 3,2 )$ to $P$.

Express this property in the form of a differential equation. It is given that $C$ passes through the point $( 4,3 )$ and that $x > 3$ and $y > 2$ at all points on $C$.
Determine the equation of $C$ giving your answer in the form $y = \mathrm { f } ( x )$. The curve $C$ may be obtained by a transformation of part of the curve $y = x ^ { 3 }$.
Describe fully this transformation.

OCR H240/03 2021 November Q8

8
\includegraphics[max width=\textwidth, alt={}, center]{699c5e1e-1476-42cb-b3c4-ca08c4d81cb6-06_485_912_1046_242} The diagram shows the curve $M$ with equation $y = x \mathrm { e } ^ { - 2 x }$.

Show that $M$ has a point of inflection at the point $P$ where $x = 1$. The line $L$ passes through the origin $O$ and the point $P$. The shaded region $R$ is enclosed by the curve $M$ and the line $L$.
Show that the area of $R$ is given by
$\frac { 1 } { 4 } \left( a + b \mathrm { e } ^ { - 2 } \right)$,
where $a$ and $b$ are integers to be determined.

OCR H240/03 2021 November Q9

9 There are three checkpoints, $A , B$ and $C$, in that order, on a straight horizontal road. A car travels along the road, in the direction from $A$ to $C$, with constant acceleration. The car takes 20 s to travel from $B$ to $C$. The speed of the car at $B$ is $14 \mathrm {~ms} ^ { - 1 }$ and the speed of the car at $C$ is $18 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the acceleration of the car. It is given that the distance between $A$ and $B$ is 330 m .
Determine the speed of the car at $A$.

OCR H240/03 2021 November Q10

10
\includegraphics[max width=\textwidth, alt={}, center]{699c5e1e-1476-42cb-b3c4-ca08c4d81cb6-07_362_754_1123_242} A block $D$ of weight 50 N lies at rest in equilibrium on a fixed rough horizontal surface. A force of magnitude 15 N is applied to $D$ at an angle $\theta$ to the horizontal (see diagram).

Complete the diagram in the Printed Answer Booklet showing all the forces acting on $D$. It is given that $D$ remains at rest and the coefficient of friction between $D$ and the surface is 0.2 .
Show that $$15 \cos \theta - 3 \sin \theta \leqslant 10 .$$ \begin{figure}[h]
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\end{figure} A golfer hits a ball from a point $A$ with a speed of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $15 ^ { \circ }$ above the horizontal. While the ball is in the air, it is modelled as a particle moving under the influence of gravity. Take the acceleration due to gravity to be $10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The ball first lands at a point $B$ which is 4 m below the level of $A$ (see diagram).
Determine the time taken for the ball to travel from $A$ to $B$.
Determine the horizontal distance of $B$ from $A$.
Determine the direction of motion of the ball 1.5 seconds after the golfer hits the ball. The horizontal distance from $A$ to $B$ is found to be greater than the answer to part (b).
State one factor that could account for this difference.

OCR H240/03 2021 November Q12

12
\includegraphics[max width=\textwidth, alt={}, center]{699c5e1e-1476-42cb-b3c4-ca08c4d81cb6-08_405_1227_1583_242} A beam, $A B$, has length 4 m and mass 20 kg . The beam is suspended horizontally by two vertical ropes. One rope is attached to the beam at $C$, where $A C = 0.5 \mathrm {~m}$. The other rope is attached to the beam at $D$, where $D B = 0.7 \mathrm {~m}$ (see diagram). The beam is modelled as a non-uniform rod and the ropes as light inextensible strings. It is given that the tension in the rope at $C$ is three times the tension in the rope at $D$.

Determine the distance of the centre of mass of the beam from $A$. A particle of mass $m \mathrm {~kg}$ is now placed on the beam at a point where the magnitude of the moment of the particle's weight about $C$ is 3.5 mg N m . The beam remains horizontal and in equilibrium.
Determine the largest possible value of $m$.

OCR H240/03 2021 November Q13

13 In this question the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are in the directions east and north respectively.
At time $t$ seconds, where $t \geqslant 0$, a particle $P$ of mass 2 kg is moving on a smooth horizontal surface under the action of a constant horizontal force $( - 8 \mathbf { i } - 54 \mathbf { j } ) \mathrm { N }$ and a variable horizontal force $\left( 4 t \mathbf { i } + 6 ( 2 t - 1 ) ^ { 2 } \mathbf { j } \right) \mathrm { N }$.

Determine the value of $t$ when the forces acting on $P$ are in equilibrium. It is given that $P$ is at rest when $t = 0$.
Determine the speed of $P$ at the instant when $P$ is moving due north.
Determine the distance between the positions of $P$ when $t = 0$ and $t = 3$.

OCR H240/03 Q1

1

If $| x | = 3$, find the possible values of $| 2 x - 1 |$.
Find the set of values of $x$ for which $| 2 x - 1 | > x + 1$. Give your answer in set notation.

OCR H240/03 Q2

2

Use the trapezium rule, with four strips each of width 0.25 , to find an approximate value for $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 + x ^ { 2 } } } \mathrm {~d} x$.
Explain how the trapezium rule might be used to give a better approximation to the integral given in part (a).

OCR H240/03 Q3

3 In this question you must show detailed reasoning. Given that $5 \sin 2 x = 3 \cos x$, where $0 ^ { \circ } < x < 90 ^ { \circ }$, find the exact value of $\sin x$.

OCR H240/03 Q4

4 For a small angle $\theta$, where $\theta$ is in radians, show that $1 + \cos \theta - 3 \cos ^ { 2 } \theta \approx - 1 + \frac { 5 } { 2 } \theta ^ { 2 }$.

OCR H240/03 Q5

5

Find the first three terms in the expansion of $( 1 + p x ) ^ { \frac { 1 } { 3 } }$ in ascending powers of $x$.
The expansion of $( 1 + q x ) ( 1 + p x ) ^ { \frac { 1 } { 3 } }$ is $1 + x - \frac { 2 } { 9 } x ^ { 2 } + \ldots$. Find the possible values of the constants $p$ and $q$.

OCR H240/03 Q6

6 A curve has equation $y = x ^ { 2 } + k x - 4 x ^ { - 1 }$ where $k$ is a constant. Given that the curve has a minimum point when $x = - 2$

find the value of $k$
show that the curve has a point of inflection which is not a stationary point.

OCR H240/03 Q7

7

Find $\int 5 x ^ { 3 } \sqrt { x ^ { 2 } + 1 } \mathrm {~d} x$.
Find $\int \theta \tan ^ { 2 } \theta \mathrm {~d} \theta$. You may use the result $\int \tan \theta \mathrm { d } \theta = \ln | \sec \theta | + c$.

OCR H240/03 Q8

8 In this question you must show detailed reasoning. The diagram shows triangle $A B C$.
\includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-06_737_1383_456_342} The angles $C A B$ and $A B C$ are each $45 ^ { \circ }$, and angle $A C B = 90 ^ { \circ }$.
The points $D$ and $E$ lie on $A C$ and $A B$ respectively. $A E = D E = 1 , D B = 2$. Angle $B E D = 90 ^ { \circ }$, angle $E B D = 30 ^ { \circ }$ and angle $D B C = 15 ^ { \circ }$.

Show that $B C = \frac { \sqrt { 2 } + \sqrt { 6 } } { 2 }$.
By considering triangle $B C D$, show that $\sin 15 ^ { \circ } = \frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$.

OCR H240/03 Q9

9 Two forces, of magnitudes 2 N and 5 N , act on a particle in the directions shown in the diagram below.
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Calculate the magnitude of the resultant force on the particle.
Calculate the angle between this resultant force and the force of magnitude 5 N .

Questions H240/03 (80 questions)

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