2015 H2 Mathematics Prelim Paper 2 Solutions
Qn
1.
(a)
Solutions
Locus of R is a straight line passing through the origin and parallel to AB .
1.
(b)
a and b are parallel
a = ( 2aib ) b
2
a = 2 a b cos
b=
1.
(c)
1
2
Using ratio theorem,
1
2
1
ON = a + b
FS
O
PR
O
PA
G
E
3
U
N
C
O
R
R
EC
TE
D
Composite
functions,
transformations
and inverses
3.1 Kick off with CAS
3.2 Composite functions and functional equations
3.3 Transformations
3.4 Transformations using matrices
3.5 Inverse graphs and relations
3.6 Inv
Skill Sets
Chapter 5 Functions
No.
1.
Skills
Sketch the graph of the
function according to the
domain.
Examples of questions involving the skills
(Lecture Notes Example 2 (b)
g : x x2 2 x 1, x , x 2
- Students tend to sketch
the graph without referring
to
MSA 1 Revision Package 2013 Solutions
Pre-requisites
1
(a)
3x 1
3x 1
2
2 x x 1 (2 x 1)( x 1)
=
A
B
2x 1 x 1
1
3( ) 1
1
3(1) 1 4
2
Using Cover-Up Rule, A
, B
1
3
2(1)
1
3
1
2
.
(b)
3x 1
1 1
4
2
2x x 1 3 2x 1 x 1
2 5 x 15 x 2
A
Bx C
2
(2 x)(1 2 x ) 2
Functions Summary
1. Definitions
Domain ( D f ) : set of x values for which the function f (x) is defined.
Range ( R f ) : set of y values for which the function f ( x) is defined.
For completeness of description, any function definition must include a
re
A Level Mathematics Tuition
1 to 1 or small group
http:/alevelmathtuition.sg
94385929 / 81502027
H2 Mathematics 2016 Paper 2 Revision
1
The curve C has equation
f(x)
ax 2 bx 1
, where a, b and c are real constants.
xc
Given that the line y 2 x 1 is an as
Figuring out asymptotes in "exp" and "ln" functions
Objective: To find the horizontal asymptotes of functions such as f(x) = x2 e-x
or g(x) = (ln x) / x. Recall, this involves taking limits as x
.
* Observe: (1) ex increases much faster than x.
(2) x inc
VECTORS
UNIT VECTOR
1 What is a unit vector?
3
2 Avectora=3ij+4k =[_1)
4
What is the unit vector 21 ?
1
3 What is the unit vector in the direction of [ 2 j ?
1
4 What is the formula for a vector?
1
5 A force F; is 6N and acts in the direction of vector v=
VECTORS
ANGLE BETWEEN TWO VECTORS
1 What is the formula for the angle between two vectors?
2 From analysis of the forces in a new bridge it is calculated that the following forces are
imposed at one ofthe bridge supports (in Newtons).
F1 = (1,1,1)
F2 is 6
VECTORS
RELATIVE VELOCITY
1 A bus B is travelling past a tree at 45 km/h. To a passenger on bus B,
tree T appears to be moving at 45km/h backwards, i.e. in the opposite direction
of the motion of the bus. Find the velocity ofT relative to B denoted by VTB
VECTORS
VECTOR EQUATION OF A STRAIGHT LINE
(VECTOR FORM)
1 at is the position vector of a fixed pointA on the line I . The line is parallel to AB
Find a vector equation of line I.
2 Let r be a variable position vector of any point R on a line I.
Find an e
VECTORS
LENGTH OF PROJECTION (i.e. ADJACENT)
1 What is the length of ON?
2 What is the length of ON in terms of vectors only?
3 Leta=() andb:(_;,).
Find the length of the projection of vector b onto vector 3.
4 Given the points A(1,2,-1), B(3,0,1) and C
VECTORS
CROSS PRODUCT (VECTOR PRODUCT)
5 What are the laws for cross product?
6 How do we evaluate a><b ?
7 Given the three vectors a = ( i) , b = (
Find axb VECTORS
CROSS PRODUCT (VECTOR PRODUCT)
8 What is the geometrical meaning of la-h ?
and laxh
9
VECTORS
EQUATION OF A STRAIGHT LINE
(CARTESIAN FORM)
4
1 A straight line 1 passes through the point Q(2, 1, 2) and is parallel to AB> = 7 ]
Find a equation of line I in cartesian form.
2 The equation of a line given below:
2 l
r: 0 +1] Where XER
2 1
Find
VECTORS
FOOT (POSITION VECTOR OR LENGTH)
finding the foot of the perpendicular and distance
from a point to a line '
Referred to the origin 0, the position vectors of points A and B are
4i11j+4k and 7i+j+7k
respectively.
(i) Find a vector equation for th
VECTORS
FOOT PLANE
finding the foot of the perpendicular from a point to a plane
The line 1 passes through the points A and B with coordinates (1, 2, 4) and (2, 3, 1) respectively.
The plane p has equation 3x y + 22 = 17. Find
(i) the coordinates of the p
H2 Mathematics 2016 Paper 2 Revision Solutions
f(x)
ax 2 bx 1
1 bc ac 2
(ax b ac)
xc
xc
1
ax b ac 2 x 1
a 2 (ans) and b ac 1 b 2c 1 (shown)
Given c 1, f(x) 2 x 1
(i)
2
x 1
2
x 1
( x 1) y (2 x 1)( x 1) 2
Let y 2 x 1
2 x 2 (1 y ) x (1 y ) 0
For all re
Summary: FUNCTIONS
INVERSE functions
Key Concepts:
1)
For inverse function to exist, the function f(x) must be one-to-one for the given
domain. When asked to determine if a function is one-to-one, students must
apply the horizontal line test (HLT). Refer
H2 MATH
GRAPHS AND TRANSFORMATION
Include:
use of a graphing calculator to graph 'a given
function
important characteristics of graphs such as
symmetry, intersections with the axes, turning
points and asymptotes of the following:
27+:2-1
_1 31-,
a2 b2 ,
Victoria Junior College
Preliminary Examinations 2014
H2 Mathematics H2 (9740) Paper 1
Solutions
1
Common Mistakes
3
dy
=m
m
dx
dy
dy
k
y 2 = kx 2 y
=k
=
dx
dx 2 y
Since the line is a tangent to the parabola for all values of m,
k
=m
2y
y = mx +
2
k
k
H2 MATHS (9740) JC2 PRELIMINARY EXAM 2010
PAPER 1 SUGGESTED SOLUTIONS
Qn
1
Solution
Method of Differences
N
N
r 1
r
2
f (r )
r 1 ! r 2 !
r 2
r 2 r !
1 2 2
2! 1! 0!
2 3 2
3! 2! 1!
3 4 2
4! 3! 2!
4 5 2
5! 4! 3!
M
N 3
N 2
2
N 2 ! N 3 ! N 4 !
N
TPJC 2014 JC2 Preliminary Examination
H2 Mathematics Paper 2 Solutions
Section A: Pure Mathematics
1 (i)
(8m)
Im (z)
z (9 4i) = 5
1
14
4
0
P
Re(z)
(9,4)
5
9
(ii)(a)
The smallest value of z is the length of OP =
=
(b)
9 2 + 42 5
97 5
Using similar triangle
2010 TPJC H2 Mathematics (9740) Prelim Exam paper 1 Mark Scheme
1
Let x be the cost of each ride for the timid.
Let y be the cost of each ride for the adventurous.
Let z be the cost of each ride for the thrill seeker.
5x + 3 y + 2z =13.5
5x + 5y
= 12.5
2x
2009 GCE A Level Solution Paper 1 (Contributed by Hwa Chong Institution)
1i)
Let un = an2 + bn + c.
u1 = a + b + c = 10
u2 = 4a + 2b + c = 6
u3 = 9a + 3b + c = 5
Using GC, a = 1.5, b = 8.5, c = 17.
un = 1.5n2 8.5n + 17.
(ii)
Teaching Point:
The set of si
TEMASEK JUNIOR COLLEGE, SINGAPORE
Preliminary Examination
Higher 2
MATHEMATICS
9740/01
Paper 1
15 September 2010
Additional Materials:
Answer paper
List of Formula (MF15)
3 hours
Solutions
This document consists of 16 printed pages and 0 blank page.
[ Tur
HCI 2008 Prelim Paper 1 Solution
Qn
Solution
Let
x,
y
and
z
be
the
no
of
10
cent,
20
cents
and
50
cents coins respectively.
1
The 3 equations are 10 x 20 y 50 z 1500 , x y z
x yz
50 1500
2
10 20 50 1500
0 rref A =
Aug matrix A = 1 1 1
1 1 1
60
2
1 0
VECTORS
PLANE EQUATION
7 (i) The plane Hl passes through the points A, B and C whose position vectors are given by
l 0 2
(I) , <2) and (0) respectively. Find a
O 3 3
(a) parametric, (b) scalar product, (c) cartesian equation of H1. VECTORS
PLANE EQUATION
GRAPHS
Simple Hyperbola
X2 (y4)2
8 A curve has equation I 9 : 1.
(i) Sketch the curve, stating the equations of the asymptotes and the coordinates
of the vertices. [3] The curve C has equation
32(X+1)2 7122? :1
where a and b are positive constants.
Give
A LEVEL QUESTIONS 2000 - 2009
l The functions f and g are dened as follows:
f:xl>x2+2x, xZl,
gszx-HL xER.
(1) A Find the value ofx such that gf(x) = 7. -
_r(ii) V Find an expression for f1(x).
~2, The function f is dened for x 2 1 by
f(x) = x + L .
x
(i)
Pioneer Junior College
Mathematics Department H2 J C2 -2013
Revision Lecture 2 : Function (Solution)
Q1
6)
Every horizontal line y=k intersects the curve y=f(x) at most once, f is a oneone function,
hence f1 exists.
R <oo,oo)
f1_
(ii)
y: eZXl, XER
1n(y+
CMM Subject Support Strand: GRAPHS Unit 3 Functions: Text
STRAND: GRAPHS
Unit 3 Functions
TEXT
Contents
Section
3.1
Functions, Mappings and Domains
3.2
Composite Functions
3.3
Inverse Functions
3.4
Transformations of Graphs of Functions
CMM Subject Suppor