Created by T.
Madas
IYGB GCE
Mathematics MMS
Advanced Level
Practice Paper D
Difficulty Rating: 3.4333/0.7992
Time: 3 hours
Candidates may use any calculator allowed by the
regulations of this examination.
Information for Candidates
This practice paper follows closely the Pearson Edexcel Syllabus, suitable for first
assessment Summer 2018.
The standard booklet “Mathematical Formulae and Statistical Tables” may be used.
Full marks may be obtained for answers to ALL questions.
The marks for the parts of questions are shown in round brackets, e.g. (2).
There are 17 questions in this question paper.
The total mark for this paper is 150.
Advice to Candidates
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit.
Non exact answers should be given to an appropriate degree of accuracy.
The examiner may refuse to mark any parts of questions if deemed not to be legible.
Created by T. Madas
Created by T. Madas
SECTION 1 – STATISTICS
Question 1 (***+)
In a histogram the weights of peaches, correct to the nearest gram, are plotted on the
x axis.
In this histogram the class 146 − 150 has a frequency of 75 and is represented by a
rectangle of base 2.8 cm and height 7.5 cm .
In the same histogram a different class is represented by a rectangle of base 5.6 cm
and height 10.5 cm .
Determine the frequency of this class. ( 4)
Question 2 (****)
The continuous random variable Y is Normally distributed with a mean of 122 and a
standard deviation of 14 .
a) Find P (125 < Y < 139 ) . ( 4)
b) Determine the value of a such that P (101 < Y < a ) = 0.8276 ( 6)
Question 3 (***)
The events A and B satisfy
P ( A ) = 0.4 and P ( A ∪ B ) = 0.79 .
Determine P ( B ) in each of the two following cases.
a) If A and B are mutually exclusive. ( 2)
b) If A and B are independent. ( 3)
Created by T. Madas
Created by T. Madas
Question 4 (***)
The ages of the residents of Arnold Street are denoted by x the ages of the residents
of Benedict Street are denoted by y .
These are summarized in the following back to back stem and leaf diagram.
x y
50
5,5,3,3 1
9, 9,1 2 5
9,8, 6, 5,5, 4,3, 2, 2, 2,1 3 6, 7,8
6, 4,1, 0, 0, 0, 0 4 1, 2, 2, 3, 4,8
9 5 1, 4, 4, 4, 4,5,8,8
6 1, 3, 4, 4,5, 9,9
7 2, 6,9
where 2 3 9 = 32 in Arnold Street and 39 in Benedict Street .
a) Find separately for the residents of Arnold Street and Benedict Street, ...
i. ... the mode.
ii. ... the lower quartile, the median and the upper quartile.
iii. ... the mean and the standard deviation. (10 )
You may assume x = 866 , x 2 = 31514 , y = 1516 , y 2 = 86880 .
A coefficient of skewness is defined as
mean − mode
.
standard deviation
b) Evaluate this coefficient for the ages in each street. (1)
c) Compare the distribution of the ages between the two streets. ( 3)
Created by T. Madas
Created by T. Madas
Question 5 (**)
The percentage test exam marks, of a random sample of 8 students, in Physics and
Chemistry are recorded in the table below.
Student A B C D E F G H
Physics 70 36 56 56 58 45 67 72
Chemistry 78 49 55 50 75 50 60 57
Test, at the 5% level of significance, whether there is evidence of positive correlation
between the percentage test marks in Physics and Chemistry. ( 5)
Question 6 (***+)
Taxis in Pajan have to pass an additional safety test consisting of three parts, one for
the brakes, one for the tyres and one for the lights. A taxi must pass all three parts.
The individual probabilities that a taxi will fail the “brake part”, the “tyre part” and
the “light part” are 1 , 1 and 1 , respectively.
6 4 5
These probabilities are independent of one another.
A taxi from Pajan is tested at random.
a) Find the probability that it will fail exactly one of the three parts of the test. ( 4)
Safety regulations change so that the test has to be performed in the order “brake part”
first, “tyre part” next and “lights part” last.
If the taxi fails one of the three parts the test results in failure, without any of
remaining parts of the test having to be carried out.
A taxi from Pajan is tested at random under these new regulations.
b) Find the probability that it will fail the test. ( 4)
c) Given a taxi failed the test, determine the probability it failed the “lights part”.
( 3)
Created by T. Madas
Created by T. Madas
Question 7 (***+)
The discrete random variable X has the following probability distribution
x 0 1 3
P ( X = x) 1 1 1
6 3 2
Three independent observations of X are made, denoted by X1 , X 2 and X 3 .
Calculate P ( X 1 + X 2 + X 3 ≥ 4 ) . (7)
Question 8 (****)
It has been established over a long period of time that in a particular variety of rose
bushes, 0.2 produce pink flowers.
A selection of 10 rose bushes of this variety, are bought.
a) Find probability that more than 4 of these bushes will produce red flowers.
( 2)
b) Calculate the least number of rose bushes that need to be bought so that the
probability of producing at least 1 plant with pink flowers exceeds 0.975 .
( 4)
Another selection of 125 rose bushes of this variety are bought.
c) Use a Normal distribution approximation to determine the probability
that the number of bushes which will produce pink flowers will be more
than 21 but no more than 30 . (7)
Finally a selection of 25 rose bushes of the same variety are considered. When
these rose bushes flowered they produced 10 plants with pink flowers.
d) Stating your hypotheses clearly, test at the 1% level of significance,
whether this constitutes evidence that this variety of rose bushes have a
higher probability than 0.2 in producing pink flowers.
( 6)
Created by T. Madas
Created by T. Madas
SECTION 2 - MECHANICS
Question 9 (**)
3m
A B
C
8m
A non uniform plank of wood AB has length 8 m and mass 100 kg .
The plank is smoothly supported at its two ends A and B . A boy of mass 60 kg
stands on the plank at the point C , where AC = 3 m , as shown in the figure above.
The plank with the boy standing on the plank, remains in equilibrium with AB
horizontal. The plank is modelled as a non uniform rod and the boy as a particle.
a) Given that the reactions at the two supports are equal, determine the distance
of the centre of mass of the plank from A . ( 4)
b) Explain in the context of this problem the model of
i. … the plank is a rod (1)
ii. … the boy is a particle. (1)
Created by T. Madas
Created by T. Madas
Question 10 (***)
A lift is moving upwards.
The lift accelerates from rest with uniform acceleration 0.36 ms −2 until it reaches a
speed of 1.62 ms −1 .
It then travels at constant speed for 15 s before decelerating uniformly to rest in 6.5 s .
a) Sketch a speed time graph for the lift’s journey. ( 2)
b) Determine the distance covered by the lift during the journey. ( 4)
A man of mass 80 kg is standing in the lift during this journey.
c) Calculate the greatest value of the reaction exerted by the floor of the lift on
the man during the journey. ( 2)
Question 11 (****)
Relative to a fixed origin O , the horizontal unit vectors i and j are pointing due east
and due north, respectively.
A particle P is moving with constant acceleration of (101 i − 15 j) ms −2
.
(
It is initially observed passing through the point with position vector −20i − 15 j m
2 )
with velocity of ( 4i + 2 j) ms −1 .
a) Find an expression for the position vector of P , t s after it was first observed.
( 2)
b) Calculate the times when P is due east of the origin O . ( 4)
c) Determine the speed of P when it is travelling in a south-eastern direction. ( 6 )
Created by T. Madas
Created by T. Madas
Question 12 (***)
80 N
30°
30°
A box of mass 10 kg is pulled by a rope on a fixed rough inclined plane. The rope is
modelled as a light inextensible string and the box is modelled as a particle. The plane
is at an angle of 30° to the horizontal, as shown in the figure above.
The rope lies in a vertical plane containing a line of greatest slope of the incline plane
and is inclined at 30° to the plane. When the tension in the rope is 80 N the box is
travelling up the plane, at constant speed.
The normal reaction between the box and the plane is R N .
Given that the magnitude of the friction between the box and the plane is µ R , where
µ is a positive constant, determine the value of µ . (7)
Question 13 (****)
A particle is projected from a point O with speed 36 ms −1 at an angle of elevation β .
It reaches a point P which is at the same vertical level as O and at a horizontal
distance of 60 m from O .
The particle is subject to no other external forces except its weight.
a) Find the two possible values of β . ( 8)
b) Determine the shortest possible flight time for the journey. ( 2)
Created by T. Madas
Created by T. Madas
Question 14 (***+)
A θ
B
2a
The figure above shows a uniform rod AB of length 2a and of mass m smoothly
hinged at the point A , which lies on a vertical wall.
The rod is kept in a horizontal position by a light inextensible string BC , where C
lies on the same wall vertically above A .
The plane ABC is perpendicular to the wall and the angle ABC is denoted by θ .
Given that tan θ = 1 , show that …
2
a) … the tension in the string is 1 5mg . ( 3)
2
b) … the magnitude of the reaction at the hinge has the same magnitude as the
tension in the string. ( 5)
Created by T. Madas
Created by T. Madas
Question 15 (***)
A car is travelling along a straight horizontal road with constant acceleration a ms −2 .
The points A , B and C lie in that order on this road.
The car is passing through A with speed 11 ms −1 , through B with speed 17 ms −1 ,
and through C with speed 29 ms −1 .
The distance AB = 28 m .
By modelling the car as a particle calculate in any order …
a) … the distance AC ( 4)
b) … the time it takes the car to travel from A to C . ( 2)
Question 16 (****)
P
F N
150°
30 N
Two forces, act on a particle P so that the angle between the two forces is 150° .
The magnitude of one of these forces is 30 N and the magnitude of the other force is
F N , as shown in the figure above.
The resultant of these two forces has magnitude R N , and acts at 60° to the force
with magnitude F N .
Calculate in any order the value of R and the value of F . ( 6)
Created by T. Madas
Created by T. Madas
Question 17 (****)
B P
A particle A of mass 5 kg is connected to small box B of mass 7.5 kg by a light
inextensible string. The string passes over a light smooth pulley P , which is located
at the end of a rough horizontal house roof. The box is held on the roof with the
particle hanging vertically at the end of the roof, as shown in the figure above.
The system is released from rest with the string taut.
The string, A , P and B lie in a vertical plane at right angles to the end of the roof.
a) Given that the coefficient of friction between B and the roof is 0.2 , find in
any order…
i. … the acceleration of the system.
ii. … the tension in the string. ( 5)
On release B is at a distance d m from P . When A has moved a distance of 2.8 m
the string breaks. In the subsequent motion B comes to rest as it reaches P .
b) Calculate the value of d . (7)
Created by T. Madas
