Created by T.
Madas
IYGB GCE
Mathematics MMS
Advanced Level
Practice Paper S
Difficulty Rating: 4.8067/1.6760
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 16 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 (***)
The discrete random variable X has binomial distribution B ( n, p ) .
Given further that n × p = np (1 − p ) = 0.95 determine the value of n . (4)
Question 2 (****+)
It is given that for a sample of data x1 , x2 , x3 , x4 , x5 , … xn the mean x and standard
deviation σ are
2
n n n
x=
1
n x = 2
r =1
r and σ =
1
n
r =1
2
n
1
( xr ) − 2
r =1
xr = 3 .
Determine, in terms of n , the value of
( x + 1) .
r =1
r
2
(8)
Question 3 (****)
The continuous random variables X and Y are independent of one another and have
Normal distributions
(
X ~ N 300, 252 ) and (
Y ~ N 210,502 . )
Given further that P ( X < k ) = P (Y > k ) , determine the value of k .
Detailed workings must be presented in this question. (8)
Created by T. Madas
Created by T. Madas
Question 4 (*****)
Two candidates, A and B , are about to be interviewed by a company.
The probability that both candidates will be unsuccessful is the same as the
probability that candidate B will be unsuccessful.
Show that the probability that both candidates will be successful is the same as the
probability that candidate A will be successful.
You may not assume independence in this question. (4)
Question 5 (*****)
The probability distribution of a discrete random variable X is given by
1 x = 1, 2, 3, ... 7
P ( X = x) = 7
0 otherwise
The probability distribution of another discrete random variable Y is given by
1 x = 2, 3, 6
P (Y = y ) = y
0 otherwise
Two observations of X are made, denoted by X1 and X 2 , and one observation of Y ,
denoted by Y1 are considered.
Assuming these three observations are independent, calculate P ( X1 + X 2 ≥ 9 + Y1 ) . (10 )
Created by T. Madas
Created by T. Madas
Question 6 (*****)
It is given that the discrete random variable X satisfies
X ~ B ( n, p ) .
Given further that P ( X = 2 ) = P ( X = 3) , show that
E( X ) = 3− p .
You may assume that E ( X ) = mean of the distribution , defined as n × p . (8)
Question 7 (*****)
The events A and B are such so that
P ( A ∪ B′ ) = 0.92 , P ( A′ ∪ B ) = 0.5 and P ( A′ ∪ B′ ) = 0.88 .
Determine the value of P ( A′ ∩ B′ ) . (10 )
Created by T. Madas
Created by T. Madas
Question 8 (*****)
1 1 2 3 4 5 6 6
The figure below shows a contraption where balls are rolling down a slope and fall
into one of 8 vertical tubes. There is equal chance of a ball falling into any of these
8 vertical tubes. The ball scores according to which tube it falls in, and these scores
are marked clearly in the figure below.
Two balls are rolled in succession.
Determine the probability that …
a) … their sum is even. ( 3)
b) … the second ball will score higher than the first. (4)
c) … their sum is even and the second ball will score higher than the first. (5)
[continues overleaf]
Created by T. Madas
Created by T. Madas
[continued from overleaf]
Three balls are rolled in succession.
Determine the probability that …
d) … they will fall into different tubes. ( 3)
e) … their scores will be different. (8)
Created by T. Madas
Created by T. Madas
SECTION 2 - MECHANICS
Question 9 (****+)
A particle P is moving on the x axis and its velocity v ms −1 in the positive x
direction, t seconds after a given instant, is given by
v = 1 t 2 − 3t + 4 , t ≥ 0 .
2
The particle is passing through the origin when t = 0
Determine the displacement of the particle from the origin, when it has covered a
total distance of 13 m . (9)
Question 10 (****+)
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 from point A to point B , with constant acceleration of
( pi + qj) ms−2 , where p and q are constants. The velocity of P at A is
(10i − 13j) ms−1 and the velocity of P at B is ( 22i + 22 j) ms −1 .
Given that the magnitude of the acceleration is 3.7 ms −2 , find the value of p and the
value of q . (9)
Created by T. Madas
Created by T. Madas
Question 11 (*****)
A child is sitting over the shoulders of his father whose mass is 100 kg .
The father, with the child over his shoulders, is standing in a lift of mass 900 kg .
When the tension in the cable of the lift is 10200 N , the lift and its occupants is
accelerating, with the child exerting a force of 200 N on the shoulders of his father.
Find the magnitude of the force exerted by floor of the lift to the feet of the father. (9)
Question 12 (*****)
Two particles A and B have masses 4 kg and 1 kg , respectively.
A small, smooth light fixed pulley P , is located 1.6 m above a horizontal floor.
The two particles are connected by a light inextensible string, of length L m , which
passes over P .
The particles are held at rest, with B level with the floor and A hanging above the
floor, with the string taut and the hanging parts of the string vertical.
The system is released from rest and A hits the floor, from which it does not rebound.
B continues moving upwards and comes to instantaneous rest as it reaches P .
Determine the value of L . (10 )
Created by T. Madas
Created by T. Madas
Question 13 (*****)
A B
particle
Two identical small rings, A and B , are threaded on a fixed, rough horizontal wire.
A light inextensible string of length 78 cm connects the two rings and a particle is
attached to the midpoint of the string. The particle is hanging in equilibrium as shown
in the figure above.
It is further given that the mass of each ring is 1 of the mass of the particle, and the
3
coefficient of friction between each ring and the wire is 1 .
4
Show that the length of AB is at most 30 cm . (10 )
Question 14 (*****)
Two cars, A and B , are travelling on a straight horizontal road.
At time t = 0 , A drives past a fixed point O on the road with speed 4.95 ms −1 ,
accelerating uniformly at 2 ms −2 .
After passing through O , A continues accelerating for 4 s , reaching a speed which it
maintains thereafter.
At time t = 2 , B starts from rest from O , accelerating uniformly at 1.5 ms −2 , until it
reaches the point P , where it draws level with A .
Determine the distance OP . (10 )
Created by T. Madas
Created by T. Madas
Question 15 (*****)
A θ
The figure above shows a uniform rigid rod AB resting on a rough peg. The rod has
weight W N and length 6 m and rests on the peg at the point C , where AC is 4 m .
The coefficient of friction between the rod and the peg is 0.5 .
The end A of the rod rests in on rough ground, where the coefficient of friction
between the rod and the ground is µ .
The rod is inclined at angle θ to the horizontal and the points A , B and C lie in a
vertical plane which is perpendicular to the ground.
Given the equilibrium is limiting both at A and at C , show clearly that
6 tan θ − 3
µ= . (10 )
8 tan 2 θ − 3 tan θ + 2
Created by T. Madas
Created by T. Madas
Question 16 (*****)
185 N
60 N
B D
63 N
84 N
The figure above shows a light rigid framework ABCD , where BDA = 90° and
BCD = 90° .
It is also given that AB = 0.37 m , BC = 0.28 m , CD = 0.21 m and AD = 0.12 m .
Forces of magnitude 185 N , 84 N , 63 N and 60 N are acting along AB , BC , CD
and AD , in the directions indicated by the arrows in the figure.
The 4 forces reduce to a single force acting at some point P on AD , and at right
angles to AD .
Determine the distance AP . (8)
Created by T. Madas
