Questions - OCR - A-Level Maths

OCR Further Additional Pure AS 2024 June Q7

12 marks Standard +0.3

7 In a long-running biochemical experiment, an initial amount of 1200 mg of an enzyme is placed into a mixture. The model for the amount of enzyme present in the mixture suggests that, at the end of each hour, one-eighth of the amount of enzyme that was present at the start of that hour is used up due to chemical reactions within the mixture. To compensate for this, at the end of each six-hour period of time, a further 500 mg of the enzyme is added to the mixture.

Let \(n\) be the number of six-hour periods that have elapsed since the experiment began. Explain how the amount of enzyme, \(\mathrm { E } _ { \mathrm { n } } \mathrm { mg }\), in the mixture is given by the recurrence system \(E _ { 0 } = 1200\) and \(E _ { n + 1 } = \left( \frac { 7 } { 8 } \right) ^ { 6 } E _ { n } + 500\) for \(n \geqslant 0\).
Solve the recurrence system given in part (a) to obtain an exact expression for \(\mathrm { E } _ { \mathrm { n } }\) in terms of \(n\).
Hence determine, in the long term, the amount of enzyme in the mixture. Give your answer correct to \(\mathbf { 3 }\) significant figures.
In this question you must show detailed reasoning. The long-running experiment is then repeated. This time a new requirement is added that the amount of enzyme present in the mixture must always be at least 500 mg . Show that the new requirement ceases to be satisfied before 12 hours have elapsed. \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series.
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OCR Further Additional Pure AS 2021 November Q1

5 marks Moderate -0.3

1 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } = \left( \begin{array} { l } 3 \\ 0 \\ 0 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 0 \\ 4 \\ 0 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) respectively, relative to the origin \(O\).

1. Calculate \(\mathbf { a } \times \mathbf { b }\), giving your answer as a multiple of \(\mathbf { c }\).
2. Explain, geometrically, why \(\mathbf { a } \times \mathbf { b }\) must be a multiple of \(\mathbf { c }\).
Use a vector product method to calculate the area of triangle \(A B C\).

OCR Further Additional Pure AS 2021 November Q2

4 marks Moderate -0.8

2 The surface \(S\) is given by \(z = x ^ { 2 } + 4 x y\) for \(- 6 \leqslant x \leqslant 6\) and \(- 2 \leqslant y \leqslant 2\).

1. Write down the equation of any one section of \(S\) which is parallel to the \(x\)-z plane
2. Sketch the section of (a)(i) on the axes provided in the Printed Answer Booklet.
Write down the equation of any one contour of \(S\) which does not include the origin.

OCR Further Additional Pure AS 2021 November Q3

6 marks Standard +0.8

3 For positive integers \(n\), the sequence of Fibonacci numbers, \(\left\{ \mathrm { F } _ { \mathrm { n } } \right\}\), starts with the terms \(F _ { 1 } = 1 , F _ { 2 } = 1 , F _ { 3 } = 2 , \ldots\) and is given by the recurrence relation \(\mathrm { F } _ { \mathrm { n } } = \mathrm { F } _ { \mathrm { n } - 1 } + \mathrm { F } _ { \mathrm { n } - 2 } ( \mathrm { n } \geqslant 3 )\).

Show that \(\mathrm { F } _ { 3 \mathrm { k } + 3 } = 2 \mathrm {~F} _ { 3 \mathrm { k } + 1 } + \mathrm { F } _ { 3 \mathrm { k } }\), where \(k\) is a positive integer.
Prove by induction that \(\mathrm { F } _ { 3 n }\) is even for all positive integers \(n\).

OCR Further Additional Pure AS 2021 November Q4

6 marks Standard +0.3

4

Let \(a = 1071\) and \(b = 67\).
1. Find the unique integers \(q\) and \(r\) such that \(\mathrm { a } = \mathrm { bq } + \mathrm { r }\), where \(q > 0\) and \(0 \leqslant r < b\).
2. Hence express the answer to (a)(i) in the form of a linear congruence modulo \(b\).
Use the fact that \(358 \times 715 - 239 \times 1071 = 1\) to prove that 715 and 1071 are co-prime.

OCR Further Additional Pure AS 2021 November Q5

11 marks Challenging +1.8

5 A trading company deals in two goods. The formula used to estimate \(z\), the total weekly cost to the company of trading the two goods, in tens of thousands of pounds, is \(z = 0.9 x + \frac { 0.096 y } { x } - x ^ { 2 } y ^ { 2 }\),
where \(x\) and \(y\) are the masses, in thousands of tonnes, of the two goods. You are given that \(x > 0\) and \(y > 0\).

In the first week of trading, it was found that the values of \(x\) and \(y\) corresponded to the stationary value of \(z\). Determine the total cost to the company for this week.
For the second week, the company intends to make a small change in either \(x\) or \(y\) in order to reduce the total weekly cost. Determine whether the company should change \(x\) or \(y\). (You are not expected to say by how much the company should reduce its costs.)

OCR Further Additional Pure AS 2021 November Q6

11 marks Challenging +1.8

6 The set \(S\) consists of the following four complex numbers. \(\begin{array} { l l l l } \sqrt { 3 } + \mathrm { i } & - \sqrt { 3 } - \mathrm { i } & 1 - \mathrm { i } \sqrt { 3 } & - 1 + \mathrm { i } \sqrt { 3 } \end{array}\) For \(z _ { 1 } , z _ { 2 } \in S\), the binary operation \(\bigcirc\) is defined by \(z _ { 1 } \bigcirc z _ { 2 } = \frac { 1 } { 4 } ( 1 + i \sqrt { 3 } ) z _ { 1 } z _ { 2 }\).

1. Complete the Cayley table for \(( S , \bigcirc )\) given in the Printed Answer Booklet.
2. Verify that ( \(S , \bigcirc\) ) is a group.
3. State the order of each element of \(( S , \bigcirc )\).
Write down the only proper subgroup of ( \(S , \bigcirc\) ).
1. Explain why ( \(S , \bigcirc\) ) is a cyclic group.
2. List all possible generators of \(( S , \bigcirc )\).

OCR Further Additional Pure AS 2021 November Q7

10 marks Challenging +1.2

7

Let \(f ( n ) = 2 ^ { 4 n + 3 } + 3 ^ { 3 n + 1 }\). Use arithmetic modulo 11 to prove that \(\mathrm { f } ( n ) \equiv 0 ( \bmod 11 )\) for all integers \(n \geqslant 0\).
Use the standard test for divisibility by 11 to prove the following statements.
1. \(10 ^ { 33 } + 1\) is divisible by 11
2. \(10 ^ { 33 } + 1\) is divisible by 121

OCR Further Additional Pure AS 2021 November Q8

7 marks Challenging +1.8

8 A sequence \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is defined by the recurrence system \(u _ { 1 } = 1\) and \(\mathrm { u } _ { \mathrm { n } + 1 } = \mathrm { a } - \frac { \mathrm { a } ^ { 2 } } { 2 \mathrm { u } _ { \mathrm { n } } }\) for \(n \geqslant 1\), where \(a\) is a positive constant.
Determine with justification the behaviour of the sequence for all possible values of \(a\). \section*{END OF QUESTION PAPER}

OCR FM1 AS 2021 June Q2

11 marks Standard +0.8

2 \includegraphics[max width=\textwidth, alt={}, center]{60f72141-4a99-4907-93b1-adb0cd66948e-2_211_1276_1427_365} Three particles \(A , B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are \(1.2 \mathrm {~kg} , 1.8 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively (see diagram). The coefficient of restitution in collisions between any two of them is \(\frac { 3 } { 4 }\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving with a velocity of \(4.0 \mathrm {~ms} ^ { - 1 }\) towards \(B\).
a) Show that immediately after the collision between \(A\) and \(B\) the speed of \(B\) is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
b) Find the velocity of \(A\) immediately after this collision. \(B\) subsequently collides with \(C\).
c) Find, in terms of \(m\), the velocity of \(B\) after its collision with \(C\).
d) Given that the direction of motion of \(B\) is reversed by the collision with \(C\), find the range of possible values of \(m\). The car is attached to a trailer of mass 200 kg by a light rigid horizontal tow bar. The greatest steady speed of the car and trailer on the road is now \(30 \mathrm {~ms} ^ { - 1 }\). The resistance to motion of the trailer may also be assumed constant.
(b) Find the magnitude of the resistance force on the trailer. The car and trailer again travel along the road. At one instant their speed is \(15 \mathrm {~ms} ^ { - 1 }\) and their acceleration is \(0.57 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
(c) (i) Find the power of the engine of the car at this instant.
(ii) Find the magnitude of the tension in the tow bar at this instant. In a refined model of the motion of the car and trailer the resistance to the motion of each is assumed to be zero until they reach a speed of \(10 \mathrm {~ms} ^ { - 1 }\). When the speed is \(10 \mathrm {~ms} ^ { - 1 }\) or above the same constant resistance forces as in the first model are assumed to apply to each. The car and trailer start at rest on the road and accelerate, using maximum power.
(d) Without carrying out any further calculations,

explain whether the time taken to attain a speed of \(20 \mathrm {~m} ^ { - 1 }\) would be predicted to be lower, the same or higher using the refined model compared with the original model,
explain whether the greatest steady speed of the system would be predicted to be lower, the same or higher using the refined model compared with the original model.

OCR FM1 AS 2021 June Q2

8 marks Standard +0.3

2 A particle moves in a straight line with constant acceleration. Its initial and final velocities are \(u\) and \(v\) respectively and at time \(t\) its displacement from its starting position is \(s\). An equation connecting these quantities is \(s = k \left( u ^ { \alpha } + v ^ { \beta } \right) t ^ { \gamma }\), where \(k\) is a dimensionless constant.

Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\).
By considering the case where the acceleration is zero, determine the value of \(k\).

OCR FM1 AS 2021 June Q3

10 marks Standard +0.8

3
Two particles \(A\) and \(B\) are connected by a light inextensible string. Particle \(A\) has mass 1.2 kg and moves on a smooth horizontal table in a circular path of radius 0.6 m and centre \(O\). The string passes through a small smooth hole at \(O\). Particle \(B\) moves in a horizontal circle in such a way that it is always vertically below \(A\). The angle that the portion of the string below the table makes with the downwards vertical through \(O\) is \(\theta\), where \(\cos \theta = \frac { 4 } { 5 }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{75f629e7-969d-43ae-8222-031875ae54ae-02_453_696_1571_552}

Find the time taken for the particles to perform a complete revolution.
Find the mass of \(B\).

OCR FM1 AS 2021 June Q1

6 marks Standard +0.3

1 \includegraphics[max width=\textwidth, alt={}, center]{d6a0d7a6-4166-4c26-a461-39b2414c0412-02_494_390_251_255} A smooth wire is shaped into a circle of radius 2.5 m which is fixed in a vertical plane with its centre at a point \(O\). A small bead \(B\) is threaded onto the wire. \(B\) is held with \(O B\) vertical and is then projected horizontally with an initial speed of \(8.4 \mathrm {~ms} ^ { - 1 }\) (see diagram).

Find the speed of \(B\) at the instant when \(O B\) makes an angle of 0.8 radians with the downward vertical through \(O\).
Determine whether \(B\) has sufficient energy to reach the point on the wire vertically above \(O\).

OCR FM1 AS 2021 June Q2

11 marks Moderate -0.3

2 A student is studying the speed of sound, \(u\), in a gas under different conditions.
He assumes that \(u\) depends on the pressure, \(p\), of the gas, the density, \(\rho\), of the gas and the wavelength, \(\lambda\), of the sound in the relationship \(u = k p ^ { \alpha } \rho ^ { \beta } \lambda ^ { \gamma }\), where \(k\) is a dimensionless constant. (The wavelength of a sound is the distance between successive peaks in the sound wave.)

Use the fact that density is mass per unit volume to find \([ \rho ]\).
Given that the units of \(p\) are \(\mathrm { Nm } ^ { - 2 }\), determine the values of \(\alpha , \beta\) and \(\gamma\).
Comment on what the value of \(\gamma\) means about how fast sounds of different wavelengths travel through the gas. The student carries out two experiments, \(A\) and \(B\), to measure \(u\). Only the density of the gas varies between the experiments, all other conditions being unchanged. He finds that the value of \(u\) in experiment \(B\) is double the value in experiment \(A\).
By what factor has the density of the gas in experiment \(A\) been multiplied to give the density of the gas in experiment \(B\) ? Particles \(A\) of mass \(2 m\) and \(B\) of mass \(m\) are on a smooth horizontal floor. \(A\) is moving with speed \(u\) directly towards a vertical wall, and \(B\) is at rest between \(A\) and the wall (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d6a0d7a6-4166-4c26-a461-39b2414c0412-03_211_795_285_244} \(A\) collides directly with \(B\). The coefficient of restitution in this collision is \(\frac { 1 } { 2 }\). \(B\) then collides with the wall, rebounds, and collides with \(A\) for a second time.
1. Show that the speed of \(B\) after its second collision with \(A\) is \(\frac { 1 } { 2 } u\). The first collision between \(A\) and \(B\) occurs at a distance \(d\) from the wall. The second collision between \(A\) and \(B\) occurs at a distance \(\frac { 1 } { 5 } d\) from the wall.
2. Find the coefficient of restitution for the collision between \(B\) and the wall.

OCR FM1 AS 2021 June Q1

7 marks Standard +0.3

1 A particle \(A\) of mass 3.6 kg is attached by a light inextensible string to a particle \(B\) of mass 2.4 kg . \(A\) and \(B\) are initially at rest, with the string slack, on a smooth horizontal surface. \(A\) is projected directly away from \(B\) with a speed of \(7.2 \mathrm {~ms} ^ { - 1 }\).

Calculate the speed of \(A\) after the string becomes taut.
Find the impulse exerted on \(A\) at the instant that the string becomes taut.
Find the loss in kinetic energy as a result of the string becoming taut.

OCR FM1 AS 2021 June Q2

11 marks Standard +0.3

2 A car of mass 1500 kg has an engine with maximum power 60 kW . When the car is travelling at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road using maximum power, its acceleration is \(3.3 \mathrm {~ms} ^ { - 2 }\). In an initial model of the motion of the car it is assumed that the resistance to motion is constant.

Using this initial model, find the greatest possible steady speed of the car along the road. In a refined model the resistance to motion is assumed to be proportional to the speed of the car.
Using this refined model, find the greatest possible steady speed of the car along the road. The greatest possible steady speed of the car on the road is measured and found to be \(21.6 \mathrm {~ms} ^ { - 1 }\).
Explain what this value means about the models used in parts (a) and (b). \includegraphics[max width=\textwidth, alt={}, center]{aa25b8a6-9a5a-4de2-9534-18db8a175c34-03_583_378_169_255} As shown in the diagram, \(A B\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length 1 m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m _ { 1 } \mathrm {~kg}\). One end of another light inextensible string of length 1 m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m _ { 2 } \mathrm {~kg}\), which is free to move on \(A B\). Initially, \(P\) moves in a horizontal circle of radius 0.6 m with constant angular velocity \(\omega \mathrm { rads } ^ { - 1 }\). The magnitude of the tension in string \(A P\) is denoted by \(T _ { 1 } \mathrm {~N}\) while that in string \(P R\) is denoted by \(T _ { 2 } \mathrm {~N}\).
1. By considering forces on \(R\), express \(T _ { 2 }\) in terms of \(m _ { 2 }\).
2. Show that
  1. \(T _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right)\),
  2. \(\omega ^ { 2 } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } }\).
  3. Deduce that, in the case where \(m _ { 1 }\) is much bigger than \(m _ { 2 } , \omega \approx 3.5\). In a different case, where \(m _ { 1 } = 2.5\) and \(m _ { 2 } = 2.8 , P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.
3. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega \mathrm { rads } ^ { - 1 }\) to zero.

OCR FM1 AS 2021 June Q4

14 marks Standard +0.3

4
2.4
&

B1 for each of two correct statements about the models.

If commenting on the accuracy of (a), must emphasise that (a) is very inaccurate or at least quite inaccurate

Do not allow e.g.

- model (a) is not very effective

- Neither model is accurate

- (a) and (b) are not very accurate

Clear comparison between the accuracy of the two models (must emphasise that (b) is fairly accurate or considerably more accurate than (a)), or other suitable distinct second comment

Do not allow e.g.

- model (b) is more accurate than model (a)

- (b) is not accurate

Do not allow statement claiming that resistance is proportional to speed, or to speed \({ } ^ { 2 }\)

Suitable comments for (a):

- is very inaccurate

- predicted speed is nearly three times the actual value

- constant resistance is not a suitable model

- both models underestimate the resistance (as top speed is lower than expected)

For the linear model (b)

- is fairly accurate (but probably underestimates the resistance at higher speeds)

- resistance is not proportional to speed but is a much better model than constant resistance

3

(a)

\(T _ { 2 } \cos \theta = m _ { 2 } g\) \(T _ { 2 } = \frac { m _ { 2 } \times 9.8 } { 0.8 } = 12.25 m _ { 2 }\)

M1

A1

[2]

1.1a

1.1

Resolving \(T _ { 2 }\) vertically and balancing forces on \(R\)

Do not allow extra forces present

Allow use of g, e.g. \(\frac { 5 } { 4 } g m _ { 2 }\)

In this solution \(\theta\) is the angle between \(R P\) and \(R A\) Sin may be seen instead if \(\theta\) is measured horizontally.

Do not allow incomplete expressions e.g. \(\frac { m _ { 2 } g } { \sin 53.13 }\)

3

(b)

(i)

\(\begin{aligned}

T _ { 2 } \cos \theta + m _ { 1 } g = T _ { 1 } \cos \theta

T _ { 1 } = T _ { 2 } + \frac { 9.8 m _ { 1 } } { 0.8 } =

\qquad 12.25 m _ { 2 } + 12.25 m _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right) \end{aligned}\)

M1

A1

[2]

3.1b

2.1

Vertical forces on \(P\); 3 terms including resolving of \(T _ { 1 }\); allow sign error

AG Dividing by \(\cos \theta ( = 0.8 )\), substituting their \(T _ { 2 }\) and rearranging

Allow 12.25 instead of \(\frac { 49 } { 4 }\)

Or \(T _ { 1 } \cos \theta = m _ { 1 } g + m _ { 2 } g\) (equation for the system as a whole)

At least one intermediate step must be seen

3

(b)

(ii)

\(\begin{aligned}

T _ { 1 } \sin \theta + T _ { 2 } \sin \theta = m _ { 1 } a

12.25 \left( m _ { 1 } + m _ { 2 } \right) \times 0.6 + 12.25 m _ { 2 } \times 0.6 = m _ { 1 } \times 0.6 \omega ^ { 2 }

\omega ^ { 2 } = \frac { 7.35 m _ { 1 } + 14.7 m _ { 2 } } { 0.6 m _ { 1 } } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } } \end{aligned}\)

M1

A1

[3]

3.1b

1.1

2.1

NII horizontally for \(P ; 3\) terms including resolving of tensions; allow sign error

Substituting for \(T _ { 1 }\), their \(T _ { 2 } , \sin \theta\) and \(\alpha\)

AG Must see an intermediate step

Could see \(a\) or \(0.6 \omega ^ { 2 }\) or \(\frac { v ^ { 2 } } { 0.6 }\) or \(\omega ^ { 2 } r\) or \(\frac { v ^ { 2 } } { r } \sin \theta = 0.6\)

must be \(a = 0.6 \omega ^ { 2 }\)

3

(c)

\(\begin{aligned}

\text { E.g } m _ { 1 } \gg m _ { 2 } \Rightarrow \frac { 2 m _ { 2 } } { m _ { 1 } } \approx 0 \text { or } \frac { 49 m _ { 2 } } { 4 m _ { 1 } } \approx 0

\omega \approx \sqrt { \frac { 49 m } { 4 m } } = 3.5 \end{aligned}\)

M1 A1

[2]

1.1

Allow argument such as if \(m _ { 1 } \gg m _ { 2 }\) then \(m _ { 1 } + 2 m _ { 2 } \approx m _ { 1 }\)

AG \(m\) may be missing

SC1 for result following argument that \(m _ { 2 }\) is negligible (by comparison with \(m _ { 1 }\) ) without justification, or using trial values of \(m _ { 1 }\) and \(m _ { 2 }\) with \(m _ { 1 } \gg m _ { 2 }\).

Do not allow the assumption that \(m _ { 2 } = 0\)

If using trial values, \(m _ { 1 }\) must be at least \(70 \times m _ { 2 }\) to give \(\omega = 3.5\) to 1 dp .

3

\multirow{3}{*}{(d)}

\(v = r \omega = 0.6 \sqrt { \frac { 49 \times 2.5 + 98 \times 2.8 } { 4 \times 2.5 } }\)

Final energy \(= 2.5 \times g \times 1\) \(\text { Initial } \mathrm { KE } = \frac { 1 } { 2 } \times 2.5 \times 0.6 ^ { 2 } \times \frac { 49 \times 2.5 + 98 \times 2.8 } { 4 \times 2.5 }\)

Initial PE \(= 2.5 \times g \times 1.2 + 2.8 \times g \times 0.4\)

Energy loss \(= 17.8605 + 40.376 - 24.5 = 33.7365\)

M1

B1

M1

A1

1.2

1.1

3.2a

Use of \(v = r \omega\) with values for \(m _ { 1 }\) and \(m _ { 2 }\)

(Assuming zero PE level at 2 m below \(A\); other values possible)

Do not allow use of \(\omega = 3.5\)

oe with different zero PE level awrt 33.7

( \(v = 3.78 , v ^ { 2 } = 14.2884\) )

NB \(\omega = 6.3\) (24.5)

(17.8605)

(40.376)

Alternate method \(v = r \omega = 0.6 \sqrt { \frac { 49 \times 2.5 + 98 \times 2.8 } { 4 \times 2.5 } }\)

Initial KE \(= \frac { 1 } { 2 } \times 2.5 \times 0.6 ^ { 2 } \times \frac { 49 \times 2.5 + 98 \times 2.8 } { 4 \times 2.5 }\)

\(\triangle P E\) for \(m _ { 1 } = \pm 2.5 \times 9.8 \times ( 0.8 - 1 )\)

\(\triangle P E\) for \(m _ { 2 } = \pm 2.8 \times 9.8 ( 1.6 - 2 )\)

Energy loss \(= 17.8605 + 4.9 + 10.976\)

M1

A1

Use of \(v = r \omega\) with values for \(m _ { 1 }\) and \(m _ { 2 }\)

Or \(- \triangle P E\) \(= 2.5 \times 9.8 \times 0.2 + 2.8 \times 9.8 \times 0.4\)

awrt 33.7

( \(v = 3.78 , v ^ { 2 } = 14.2884\) ) \(\mathrm { NB } \omega = 6.3\)

(17.8605)

\(( \pm 4.9 )\)

\(( \pm 10.976 )\)

\(( \pm 15.876 )\)

Or 15.876 + 17.8605

[5]

OCR FM1 AS 2021 June Q1

7 marks Moderate -0.8

1 A particle \(P\) of mass 4.5 kg is moving in a straight line on a smooth horizontal surface at a speed of \(2.4 \mathrm {~ms} ^ { - 1 }\) when it strikes a vertical wall directly. It rebounds at a speed of \(1.6 \mathrm {~ms} ^ { - 1 }\).

Find the coefficient of restitution between \(P\) and the wall.
Determine the impulse applied to \(P\) by the wall, stating its direction.
Find the loss of kinetic energy of \(P\) as a result of the collision.
State, with a reason, whether the collision is perfectly elastic.

OCR FM1 AS 2021 June Q2

14 marks Moderate -0.3

2 A particle \(P\) of mass 2.4 kg is moving in a straight line \(O A\) on a horizontal plane. \(P\) is acted on by a force of magnitude 30 N in the direction of motion. The distance \(O A\) is 10 m . \begin{enumerate}[label=(\alph*)] \item Find the work done by this force as \(P\) moves from \(O\) to \(A\). The motion of \(P\) is resisted by a constant force of magnitude \(R \mathrm {~N}\). The velocity of \(P\) increases from \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(O\) to \(18 \mathrm {~ms} ^ { - 1 }\) at \(A\). \item Find the value of \(R\). \item Find the average power used in overcoming the resistance force on \(P\) as it moves from \(O\) to \(A\). When \(P\) reaches \(A\) it collides directly with a particle \(Q\) of mass 1.6 kg which was at rest at \(A\) before the collision. The impulse exerted on \(Q\) by \(P\) as a result of the collision is 17.28 Ns . \item

Find the speed of \(Q\) after the collision.
Hence show that the collision is inelastic. It is required to model the motion of a car of mass \(m \mathrm {~kg}\) travelling at a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) around a circular portion of banked track. The track is banked at \(30 ^ { \circ }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b9741472-f230-4e2d-9c8b-47f7e168e938-03_355_565_269_274} In a model, the following modelling assumptions are made.
For a particular portion of banked track, \(r = 24\).
(b) Find the value of \(v\) as predicted by the model. A car is being driven on this portion of the track at the constant speed calculated in part (b). The driver finds that in fact he can drive a little slower or a little faster than this while still moving in the same horizontal circle.
(c) Explain

OCR FP1 AS 2021 June Q2

12 marks Standard +0.3

2
The position vector of point \(A\) is \(\mathbf { a } = - 9 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k }\).
The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf { a }\).

Determine the shortest distance between the origin, \(O\), and \(l\). \(l\) is also perpendicular to the vector \(\mathbf { b }\) where \(\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }\).
Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).
Write down an equation of \(l\) in vector form. \(P\) is a point on \(l\) such that \(P A = 2 O A\).
Find angle \(P O A\) giving your answer to 3 significant figures. \(C\) is a point whose position vector, \(\mathbf { c }\), is given by \(\mathbf { c } = p \mathbf { a }\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf { r } = \mathbf { c } + \mu \mathbf { b }\). The point with position vector \(9 \mathbf { i } + 8 \mathbf { j } - 12 \mathbf { k }\) lies on \(m\).
Find the value of \(p\). \section*{In this question you must show detailed reasoning.} You are given that \(\alpha , \beta\) and \(\gamma\) are the roots of the equation \(5 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0\).
1. Find the value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
2. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\) giving your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) where \(a , b , c\) and \(d\) are integers.

OCR FP1 AS 2021 June Q4

6 marks Standard +0.3

4 In this question you must show detailed reasoning. \(\mathbf { M }\) is the matrix \(\left( \begin{array} { l l } 1 & 6 \\ 0 & 2 \end{array} \right)\).
Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & 3 \left( 2 ^ { n + 1 } - 2 \right) \\ 0 & 2 ^ { n } \end{array} \right)\), for any positive integer \(n\).

OCR FP1 AS 2021 June Q1

4 marks Easy -1.3

1 Matrices \(\mathbf { P }\) and \(\mathbf { Q }\) are given by \(\mathbf { P } = \left( \begin{array} { c c c } 1 & k & 0 \\ - 2 & 1 & 3 \end{array} \right)\) and \(\mathbf { Q } = ( ( 1 + k ) - 1 )\) where \(k\) is a constant.
Exactly one of statements A and B is true.
Statement A: \(\quad \mathbf { P }\) and \(\mathbf { Q }\) (in that order) are conformable for multiplication.
Statement B: \(\quad \mathbf { Q }\) and \(\mathbf { P }\) (in that order) are conformable for multiplication.

State, with a reason, which one of A and B is true.
Find either \(\mathbf { P Q }\) or \(\mathbf { Q P }\) in terms of \(k\).

OCR FP1 AS 2021 June Q2

14 marks Standard +0.3

2 In this question you must show detailed reasoning. You are given that \(\mathrm { f } ( z ) = 4 z ^ { 4 } - 12 z ^ { 3 } + 41 z ^ { 2 } - 128 z + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\).

Express \(\mathrm { f } ( z )\) as the product of two quadratic factors with integer coefficients.
Solve \(\mathrm { f } ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( z ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } \cdot R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
Find the exact area of \(R\).
\(\omega\) is the sum of all the roots of \(\mathrm { f } ( z ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).

OCR FP1 AS 2021 June Q3

5 marks Standard +0.8

3 A transformation T is represented by the matrix \(\mathbf { T }\) where \(\mathbf { T } = \left( \begin{array} { c c } x ^ { 2 } + 1 & - 4 \\ 3 - 2 x ^ { 2 } & x ^ { 2 } + 5 \end{array} \right)\). A quadrilateral \(Q\), whose area is 12 units, is transformed by T to \(Q ^ { \prime }\). Find the smallest possible value of the area of \(Q ^ { \prime }\).

OCR FP1 AS 2021 June Q4

7 marks Standard +0.3

4 A transformation A is represented by the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { c c c } - 1 & x & 2 \\ 7 - x & - 6 & 1 \\ 5 & - 5 x & 2 x \end{array} \right)\).
The tetrahedron \(H\) has vertices at \(O , P , Q\) and \(R\). The volume of \(H\) is 6 units. \(P ^ { \prime } , Q ^ { \prime } , R ^ { \prime }\) and \(H ^ { \prime }\) are the images of \(P , Q , R\) and \(H\) under A .

In the case where \(x = 5\)

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