MATHEMATICS at ‘A’ LEVEL
MOCK EXAMINATION 2
NTU/NUS ENTRANCE EXAMINATION PREP
COPYRIGHT : Agus Leonardi ([email protected])
TIME ALLOTED:
2 HOURS
ANSWER ALL QUESTIONS IN THE SPACE
PROVIDED
NAME : ________________________________________
FOR EXAMINER’S USE:
1 / 25
2 / 25
3 / 25
4 / 25
TOTAL / 100
MARKS %
GOOD LUCK! GOD BLESS.
Created by: Agus L.
Mathematics – Mock Exam 2
2
Question 1
(a) Show that the Maclaurin series for ( )xy secln= , as far as the term in 4x , is
42
12
1
2
1xx +
Hence, show that ( ) 086.0secln4
0≈∫ dxx
π
. [12 marks]
(b) Using de Moivre’s theorem, show that θθ
θθθ
42
3
tantan61
tan4tan44tan
+−
−=
Hence find, correct to 3 significant figures, the two positive roots of the equation
016 234=+−−+ xxxx . [13 marks]
Question 2
(a) Solve the differential equation xydx
dyx sectan =+ , given that π=y when x = 0.
[10 marks]
(b) Given that x and y are positive numbers such that 122=++ yxyx .
Show that 2342
1
2
1xxy −+−= .
R is the region enclosed by the curve and the axes.
Find the exact area of R.
Find also the exact volume of the solid of revolution obtained when R is rotated
completely about the x-axis. [15 marks]
Question 3
(a) Prove that ( )32
13
2
1
1
+
−=
+∑
=
nr
nn
rr
[9 marks]
(b) Given a quadratic equation 02=++ baxx . The coefficients a and b of the quadratic
equation are to be taken randomly from the set of integers { }9,...,2,1,0 . If ba ≠ ,
what is the probability that the equation does not have any real roots?
[7 marks]
(c) Given that x and y are related by ( ) ( )1lnsin 22+=+ xyyx . If π<≤ y0 , what is the
value of dx
dy and
2
2
dx
yd at 0=x ? [9 marks]
COPYRIGHT:
Agus Leonardi S. ([email protected])
Created by: Agus L.
Mathematics – Mock Exam 2
3
Question 4
(a) Four rods with lengths a, b, c, d respectively are connected to form a quadrilateral.
Prove that the area enclosed by the four rods will be maximum when the
quadrilateral is cyclic.
Hint: A quadrilateral is cyclic if and only if the sum of the opposite angles is 1800.
You are not required to show that the value you have obtained is the maximum.
[9 marks]
(b) Find an equation of the plane passing through the point (2, 3, 3) and containing the
line of intersection of the two planes 7432 =++ zyx and 1024 =++ zyx
[7 marks]
(c) Evaluate dxx∫6
0
3sec
π
. [9 marks]
COPYRIGHT:
Agus Leonardi S. ([email protected])
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