Created by T.
Madas
IYGB GCE
Mathematics SYN
Advanced Level
Synoptic Paper M
Difficulty Rating: 4.02/0.7071
Time: 3 hours
Candidates may use any calculator allowed by the
regulations of this examination.
Information for Candidates
This synoptic practice paper follows closely the Advanced Level Pure Mathematics
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 24 questions in this question paper.
The total mark for this paper is 200.
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 (**)
Solve the following inequality.
2
<x (7)
x +1
Question 2 (***)
A circle C has equation
x 2 + y 2 − 6 x − 10 y + k = 0 ,
where k is a constant.
a) Determine the coordinates of the centre of C . ( 3)
The x axis is a tangent to C at the point P .
b) State the coordinates of P and find the value of k . ( 2)
Question 3 (***+)
3x − 1 1
f ( x) = 2
, x<
(1 − 2 x ) 2
Show that if x is small, then
f ( x ) ≈ −1 − x + 4 x 3 . (6)
Created by T. Madas
Created by T. Madas
Question 4 (***+)
A ( 2,3)
B
D C ( 3,0 )
The figure above shows a rhombus ABCD , where the vertices A and C have
coordinates ( 2,3) and ( 3,0 ) , respectively.
a) Show that an equation of the diagonal BD is
x − 3y + 2 = 0 . ( 5)
b) Given that an equation of the line through A and D is
3x − 4 y + 6 = 0 ,
find the coordinates of D . ( 3)
c) State the coordinates of B . ( 2)
Question 5 (****+)
Determine the value of k .
2399 − 2395
= 32k .
15
You must show full workings. ( 5)
Created by T. Madas
Created by T. Madas
Question 6 (****)
A polynomial p ( x ) is given by
p ( x ) = 4 x3 − 2 x 2 + x + 5 .
a) Find the remainder and the quotient when p ( x ) is divided by x 2 + 2 x − 5 . ( 3)
A different polynomial q ( x ) is defined as
q ( x ) = 4 x3 − 2 x 2 + ax + b .
b) Find the value of the constants a and b so that when q ( x ) is divided by
x 2 + 2 x − 5 there is no remainder. ( 6)
Question 7 (***+)
D
A
6cm
2θ
C 6cm B
The figure above shows a sector CADB , of radius 6 cm and angle 2θ radians.
Given that the area of the triangle ABC and the area of segment ABD are in the ratio
4 :1 , show that
8θ − 5sin 2θ = 0 . ( 6)
Created by T. Madas
Created by T. Madas
Question 8 (****)
y C
y=6
R
P
x
O
The figure above shows the curve C , with parametric equations
π
x = 6t sin t , y = 3sec t , 0 ≤ t < .
2
The curve meets the coordinate axes at the point A .
The line y = 6 meets C at the point P .
a) Show that the area under the arc of the curve between A and P , and the x
axis is given by the integral
3
18 t + tan t dt . ( 5)
0
The shaded region R is bounded by C , the line y = 6 and the y axis.
b) Show that the area of R is approximately 10.3 square units. ( 5)
Created by T. Madas
Created by T. Madas
Question 9 (****)
Relative to a fixed origin O , the position vectors of three points A , B and C are
OA = i − 2k , AB = 2i + 10 j + 2k and BC = 6i − 12 j .
a) Show that AC is perpendicular to AB . ( 4)
b) Show further that the area of the triangle ABC is 18 6 . ( 2)
c) Hence, or otherwise, determine the shortest distance of A from the straight
line through B and C . ( 2)
Question 10 (****)
Use an appropriate substitution, followed by partial fractions, to show that
e5
5
2 x ( ln x )
2
+ ln x − 6
( )
dx = ln 3 .
2
e3
You may assume that the integral converges. (10 )
Question 11 (****+)
f ( x ) = ln(1 + sin x) , sin x ≠ ±1 .
Show clearly that
f ( x ) − f ( − x ) = 2ln ( sec x + tan x ) . (6)
Created by T. Madas
Created by T. Madas
Question 12 (***+)
Find, in exact simplified surd form, the roots of the following equation.
6
3 x + = 9 , x ≠ 0 .
x
Detailed workings must be shown in this question. ( 6)
Question 13 (****)
y
y = 2 x3 + 3x 2 − 11x − 6
O
x
P Q R
The figure above shows the curve with equation
y = 2 x3 + 3x 2 − 11x − 6 .
The curve crosses the x axis at the points P , Q and R ( 2,0 ) .
The tangent to the curve at R is the straight line L1 .
a) Find an equation of L1 . (5)
The normal to the curve at P is the straight line L2 .
The point S is the point of intersection between L1 and L2 .
b) Show that PSR = 90° . (7)
Created by T. Madas
Created by T. Madas
Question 14 (****)
The sum of the first n terms of a geometric series is denoted by S n .
The common ratio of the series, r , is greater than 1 .
a) If S 4 = 5S 2 find the value of r . ( 6)
b) Given further that S3 = 21 determine the value of S10 . (4)
Question 15 (***+)
A function f is defined by
1
f ( x) = 2 + , x ∈ , x ≥ 0 .
x +1
a) Find an expression for f −1 ( x ) , as a simplified fraction. ( 4)
b) Find the domain and range of f −1 ( x ) . ( 3)
Question 16 (****)
Show that the following simultaneous equations
e2 y + 4 = x
ln ( x + 1) = 2 y − 1
are satisfied by the solution pair
4+e 1 5e
x= , y = ln
1− e 2 1− e
and hence explain why the equations have no real solutions. (7)
Created by T. Madas
Created by T. Madas
Question 17 (****)
The curve C has equation
2x + 3
y= , x∈, x ≠ 2.
x−2
a) Show clearly that
2x + 3 7
≡ 2+ . ( 2)
x−2 x−2
b) Find the coordinates of the points where C meets the coordinate axes. ( 2)
c) Sketch the graph of C showing clearly the equations of any asymptotes. (4)
d) Determine the coordinates of the points of intersection of C and the straight
line with equation
y = 7 x − 12 . ( 4)
Question 18 (****)
Find the range of values that the constant k can take so that
2x2 + ( k + 2) x + k = 0
has two distinct real roots. (6)
Question 19 (***+)
Find the set of values of x for which
x 2 − 4 > 3x . ( 6)
Created by T. Madas
Created by T. Madas
Question 20 (****+)
y
M
y = − x2 + 8x − 7
A ( 6,5 )
x
O
The figure above shows the quadratic curve with equation
y = − x2 + 8x − 7 .
The point M is the maximum point of the curve and A is another point on the curve
whose coordinates are ( 6,5) .
Find the exact area of the shaded region, bounded by the curve, the x axis and the
straight line segment from A to M . (10 )
Question 21 (****)
A grass lawn has an area of 225 m 2 and has become host to a parasitic weed.
Let A m 2 be the area covered by the parasitic weed, t days after it was first noticed.
The rate at which A is growing is proportional to the square root of the area of the
lawn already covered by the weed.
Initially the parasitic weed has spread to an area of 1 m 2 , and at that instant the
parasitic weed is growing at the rate of 0.25 m 2 per day.
By forming and solving a suitable differential equation, calculate after how many
days, the weed will have spread to the entire lawn. (10 )
Created by T. Madas
Created by T. Madas
Question 22 (*****)
6x
A solid machine component, made of metal, is in the shape of a right circular cylinder,
with radius x cm and length 6 x cm .
The component is heated so that it is expanding at the constant rate of 6 π cm3 s −1 .
7
Given that the initial volume of the component was 36π cm3 , find the rate at which
the surface area of the component is increasing 14 s after the heating started.
You may assume that the shape of the component is mathematically similar to its
original shape at all times. (10 )
Question 23 (*****)
It is given that
1
x = t2 , t > 0.
Given further that y is a function of x , show clearly that
d2y dy d2y
=2 + 4t 2 . (7)
dx 2 dt dt
Created by T. Madas
Created by T. Madas
Question 24 (*****)
f ( x ) = 3sin x − cos x + 3 , x ∈ .
g ( x ) = sin x + cos x , x ∈ .
a) Express f ( x ) in the form
A × g ( x ) + B × g′ ( x ) + 3 , ( 5)
where A and B are constants.
b) Express g ( x ) in the form
R cos ( x − ϕ ) ,
where R and ϕ are positive constants. ( 3)
c) Hence find a simplified expression for
f ( x)
dx . (7)
g ( x)
Created by T. Madas
