Created by T.
Madas
IYGB GCE
Mathematics MP2
Advanced Level
Practice Paper C
Difficulty Rating: 3.565/1.1499
Time: 2 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 11 questions in this question paper.
The total mark for this paper is 100.
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 (***)
f ( x ) = 225 + 15 x , x < 15 .
a) Expand f ( x ) as an infinite series, up and including the term in x 2 . (4)
b) By substituting x = 1 in the expansion of f ( x ) , show that
1859
15 ≈ . ( 3)
480
Question 2 (***+)
Three consecutive terms in geometric progression are given in sequential order as
(1 − 5 p ) , 1 and ( 4 p − 2) ,
2
where p is a constant.
Show that one possible value of p is 1 and find the other. (5)
4
Question 3 (***+)
The curve with equation y = 2 x intersects the straight line with equation y = 3 − 2 x
at the point P , whose x coordinate is α .
a) Show that 0 < α < 1 . ( 3)
b) Starting with x = 0.5 , use the Newton Raphson method to find the value of α ,
correct to 3 decimal places. (6)
Created by T. Madas
Created by T. Madas
Question 4 (****)
12cm R
3π
10
O C B
The figure above shows a circular arc OAB of radius 12 cm , subtending an angle of
3π
radians at O .
10
Find to three significant figures …
a) … the length of the arc AB . ( 2)
b) … the area of the sector OAB . ( 2)
The point C lies on OB so that OC = AC .
The region R , shown shaded in the figure, is bounded by the arc AB and the straight
lines AC and BC .
c) Determine, to three significant figures, the perimeter and area of R . (5)
Question 5 (***)
Use the method of proof by contradiction to show that if x then
1
x+ ≥ 2. (5)
x
Created by T. Madas
Created by T. Madas
Question 6 (***)
The variable points A ( 2t , t , 2 ) and B ( t , 4,1) , where t is a scalar variable, are referred
relative to a fixed origin O .
a) Show that
AB = 2t 2 − 8t + 17 . ( 3)
b) Hence find the shortest distance between A and B , as t varies. (5)
Question 7 (***+)
A curve C is given by the parametric equations
3t − 2 t 2 − 2t + 2
x= , y= , t ∈ , t ≠ 1.
t −1 t −1
a) Show clearly that
dy
= 2t − t 2 . (7)
dx
( )
The point P 1, − 5 lies on C .
2
b) Show that the equation of the tangent to C at the point P is
3 x − 4 y − 13 = 0 . (4)
Created by T. Madas
Created by T. Madas
Question 8 (****)
e2 x
y= , x∈
ex + 1
a) Calculate the missing values of x and y in the following table.
x ln 2 x2 x3 x4 ln 8
y 1.333 y2 y3 y4 7.111
( 3)
b) Use the trapezium rule with all the values from the completed table of part (a)
to find an estimate for
ln 8
e2 x
dx . ( 3)
ln 2 ex +1
c) Use the substitution u = e x + 1 to find an exact simplified value for
ln 8
e2 x
dx . (6)
ln 2 ex +1
Created by T. Madas
Created by T. Madas
Question 9 (***)
The shape of a weather balloon remains spherical at all times.
It is filled with a special type of gas and is floating at very high altitude.
The rate at which the volume of the balloon is decreasing is directly proportional to
the square of the surface area of the balloon at that instant.
Let r m be the radius of the balloon, t hours since r = 5 .
a) By relating the volume, the surface area and the radius of the weather balloon
show that
dr
= − kr 2 ,
dt
where k is a positive constant. (4)
volume of a sphere of radius r is given by 4 π r 3
3
surface area of a sphere of radius r is given by 4π r 2
When t = 10 , r = 4.8 .
b) Determine the value of t when r = 4 . (7)
Question 10 (****)
y = cot x , 0 < x < π2 .
Show, with detailed workings, that
dy
a) = − cosec 2 x . ( 3)
dx
d2y
b)
dx 2 ( )
= 2 y y2 +1 . (5)
Created by T. Madas
Created by T. Madas
Question 11 (****)
f ( x ) = 3 sin x + cos x , x ∈ .
a) Express f ( x ) in the form R cos ( x − α ) , R > 0 , 0 < α < 90° . ( 3)
b) State the maximum value of f ( x ) and find the smallest positive value of x
for which this maximum occurs. ( 3)
The temperature of the water T °C in a tropical fish tank is modelled by the equation
T = 32 + 3 sin (15t ) ° + cos (15t ) ° , 0 ≤ t < 24 ,
where t is the time in hours measured since midnight.
c) State the maximum temperature of the water in the tank and the time when this
maximum temperature occurs. ( 3)
d) Show that the temperature of the water in the tank reaches 30.5 °C at 13 :14
and at 18 : 46 .
[You may not verify the answers in this part] (6)
Created by T. Madas
