Created by T.
Madas
IYGB GCE
Mathematics MP2
Advanced Level
Practice Paper T
Difficulty Rating: 5.000/1.7143
Time: 3 hours 30 minutes
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 175.
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
Question 1 (*****)
C D
A B
E
4 cm
The figure above is constructed as follows.
A semicircle with diameter AB of 4 cm is first drawn.
Then another semicircle is drawn, with its diameter CD parallel to AB .
The semicircle with CD as its diameter is circumscribed by the semicircle with AB
as its diameter, as shown in the figure.
Show that the area of the shaded region is ( 2π − 2 ) cm 2 . (6)
Question 2 (*****)
Show by a suitable algebraic method that
602 − 592 + 582 − 57 2 + ... + 222 − 212 = 1620 . (5)
Created by T. Madas
Created by T. Madas
Question 3 (*****)
1
f ( x) ≡ , x <1.
2 5
(1 − 5 x )
It is given that the equation
3
f ( x ) − ( 8 x + 3) = −37 x3 − 475 x 2 − 157 x + 27
has a solution α , which is numerically small.
Find an approximate value for α . (10 )
Question 4 (*****)
A factory gets permission to dispose, at the start of every day, 600 kg of waste into a
stream of water.
The running stream removes 40% of the any waste present, by the end of the day.
Determine a simplified expression for the amount of waste present in the stream at the
end of the n th day. (8)
Question 5 (*****)
A quartic curve C has the following equation.
y = x ( x − 4 )( x + 2 )( x − 6 ) , x ∈ ! .
By considering suitable transformations, show that C is even about the straight line
with equation x = 2 . (7)
Created by T. Madas
Created by T. Madas
Question 6 (*****)
T y
P
C
x
O
The figure above shows the curve C with parametric equations
x = 4cos θ , y = 3sin θ , 0 ≤ θ < 2π .
The point P lies on C where θ = α , where 0 < α < π .
2
The line T is a tangent to C at P .
The tangent T meets the coordinate axes at the points A and B .
The area of the triangle OAB , where O is the origin, is less than 24 square units.
Find the range of the possible values of α . (12 )
Question 7 (*****)
It is given that x , a and b are positive real numbers, with a > b and x 2 > ab .
Use proof by contradiction to show that
x+a x+b
− > 0. (8)
x2 + a2 x 2 + b2
Created by T. Madas
Created by T. Madas
Question 8 (*****)
Use an appropriate substitution followed by integration by parts to find a simplified
expression for
ln x 2 + 1 − 2 ln x x 2 + 1
( )
4
dx . (12 )
x
Question 9 (*****)
The piecewise continuous function f is given below.
2 x − 2 x ≤ 5
f ( x) ≡
x+3 x >5
a) Determine an expression, in similar form to that of f ( x ) above, for the
inverse function, f −1 ( x ) . (5)
b) Sketch a detailed graph for the composition ff ( x ) . (7)
Question 10 (*****)
With respect to a fixed origin, the points A and B have position vectors 10i + 9 j − 6k
and 6i − 3 j + 10k , respectively.
The position vector of the point C has i component equal to 2 .
The distance of C from both A and B is 12 units.
Show that one of the two possible position vectors of C is 2i + 5 j + 2k and determine
the other. (12 )
Created by T. Madas
Created by T. Madas
Question 11 (*****)
dy y4 − y2
= , x > 0 , y > 0.
dx x4 − x2
Find the solution of the above differential equation subject to the boundary condition
2
y= at x = 2 .
3
2x
Give the answer in the form y = , where f ( x ) is a function to be found. (15 )
f ( x)
Question 12 (*****)
A curve C is defined in the largest real domain by the equation
y = log x 2 .
a) Sketch a detailed graph of C . ( 2)
The point P , where x = 2 lies on C .
The normal to C at P meets C again at the point Q .
b) Show that the x coordinate of Q is a solution of the equation
[1 + x ln 4 − ln16] ln x = ln 2 . (8)
c) Use an iterative formula of the form xn+1 = e ( n ) , with a suitable starting
f x
value, to find the coordinates of Q , correct to 3 decimal places.
(4)
Created by T. Madas
Created by T. Madas
Question 13 (*****)
The acute angles x and y , satisfy the following relationships.
7
2 tan x = 1 and sin ( x + y ) = .
50
Determine the possible values of tan y . (12 )
Question 14 (*****)
A container is in the shape of a hollow inverted right circular cone, whose ratio of its
base radius to its height is π : 1 .
The container is initially empty when water is begins to flow in at the constant rate k .
At time t , the area of the circular surface of the water in the cone is A .
Show that at time t = T , the rate at which A is changing is
2π 3 f ( k ,T ) ,
where f ( k , T ) is an expression to be found. (10 )
Created by T. Madas
Created by T. Madas
Question 15 (*****)
O
x
2 x 2 + 2 xy + y 2 = 50
The figure above shows the curve with equation
2 x 2 + 2 xy + y 2 = 50 .
Determine the area of the finite region bounded by the x axis and the part of the
curve for which y ≥ 0 .
(16 )
Question 16 (*****)
A curve C has equation
x2
y2 = , x ∈ ! , x > 1.
x −1
Show that there exist exactly two tangents to C which pass through the point (1,2 ) ,
and find their equations. (16 )
Created by T. Madas
