Add Maths Year 11

8/14/2019 Add Maths Year 11

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Topic Learning Outcomes Resources/Activities Time

1 INTEGRATION

1.1 Indefinite Integrals

1.2 Definite Integrals

1.3 Integration ofTrigonometric

Functions

1.4 Integration ofExponential

Functions

Understand integration as the reverse process ofdifferentiation.

Find indefinite integrals.(Note: Stress on the need to write the arbitrary

constant c.)

Integrate axn and sum of terms in powers ofx,excluding n = 1.

Integrate functions of the form (ax + b)n where n -- 1.

Perform integration to get the equation of the curve

and determine the arbitrary constant in theequation of the curve.

Evaluate definite integrals of algebraic expressions.

Know the results of the following definite integrals:

(i) ( ) =a

a

dxxf 0

(ii) ( ) ( ) =b

a

a

b

dxxfdxxf

(iii) ( ) ( ) ( ) =+b

a

c

b

c

a

dxxfdxxfdxxf .

Integrate functions of the form ( )bax +sin , ( )bax +cos

and ( )bax +2sec , where a 0 .

Evaluate the definite integrals of trigonometricfunctions.

New AdditionalMathematics Chapter 20Additional MathematicsChapter 17

http://www.mathsnet.net/asa2/2004/c16int.html

http://www.mathsnet.net/asa2/2004/c27notation.html

5 weeks

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Integrate functions of the form baxe + , where a 0 .

Evaluate the definite integrals of exponentialfunctions.

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2 APPLICATION OFINTEGRATION

2.1 Plane Area Find the area enclosed by a curve (consider bothcases: abovex axis and belowx axis), thexaxis and the linesx= a andx= b.

Find the area enclosed by a curve, they axis andthe linesy= a andy= b (consider the positive and negative area).

Find the area enclosed by a curve and a line.

Find the area enclosed by two curves.

New AdditionalMathematicsChapter 21Additional MathematicsChapter 17 and 18

http://www.mathsnet.net/asa2/2004/c27area_2.html

3 weeks

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3 KINEMATICS

3.1 Displacement, Velocityand

Acceleration

Know that for a particle moving in a straight linewith displacementx, velocity vand acceleration a:

dt

dxv = and

dt

dva =

2

2

dt

xd=

= dtvx and = dtav . Know that when a particle is at instantaneous rest,

v= 0 and this will be followed by a change indirection of motion.

Use a number line to denote the positions of aparticle at the start of the motion, at the time t 1when v = 0 (if 0 < t1 < T) and at time T seconds, tofind the total distance travelled in the first Tseconds of the motion.

Use the formula Average speed =

takentimetotaltravelleddistancetotal .

Apply the differentiation and integration tokinematics problems that involve displacement,velocity and acceleration of a particle moving in astraight line with variable or constant acceleration.

(Suggestion: For motion with constant accelerationteachers may introduce the equation v = u + atbasedon the definition of acceleration


4 weeks

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3.2 Displacement-Time andVelocity-

Time Graphs

as rate of change of velocity with respect to time,

t

uva

= ).

Describe briefly the motion of the particle byobserving thex-tor the v-tgraph given.

Sketch thex-tor the v-tgraph for the motiondescribed.

Know that for anx-tgraph,(a) a straight line shows motion of uniform

(constant) velocity,(b) the velocity at any instant is given by the

gradient of the graph at that point.

Know that for a v-tgraph:(a) a straight line shows motion of uniform

acceleration,(b) the acceleration at any instant is given by the

gradient of the graph at that instant,(c) both the change in displacement and the

distance travelled may be found by consideringthe appropriate areas under the graph.

(Suggestion: For motion with constant acceleration,

derive the equation tvus )(2

1+= by using the area

under the v-tgraph which is represented by atrapezium with parallel sides u and vand width t).

Solve related problems.

4 SETS

4.1 Introduction to Sets New Additional 3 weeks

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Use set language and notation to describe setsand represent relationships between sets as follows:

A = {x : xis a natural number}B = { (x, y) : y = mx + c}

C = { x : a x b}D = { a, b, c, }.

Mathematics Chapter 1Additional MathematicsChapter 1

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4.2 Intersection and Unionof Sets

Define the terms finite and infinite sets,empty/null set, equal sets, subsets, universal setsand complement of a set.

Understand and use of the following notation:Number of elements in set A n(A)

is an element of

is not an element of Complement of Set A A

The empty set Universal set

A is a subset of B A B

A is a proper subset of B A B

A is not a subset of B A B

A is not a proper subset of B A B

Use Venn diagram to show the relationshipbetween sets.

Know the terms intersection of sets and unionof sets.

Use the following notation:

Union of A and B AB

Intersection of A and B AB

Shade the region defined by the set notations

and vice versa. Describe set notations in words.

(Caution students on correct use of terms and thenecessity to write statements in detail especially incases involving the and symbols e.g. If M ={set of students studying mathematics} and P ={set of students studying physics},

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(i) M P is the set of students studyingmathematics or physics or both mathematics andphysics,(ii) P M means all students studying physics alsostudy mathematics.)

Solve related Set problems including the

maximum and minimum possible value.

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5 PERMUTATIONSAND COMBINATIONS

5.1 The Basic CountingPrinciple

5.2 Notation ofn Factorial,(n!)

5.3 Permutation ofnDifferent

Objects

5.4 Permutation ofrObjects From n

Different Objects

5.5 Permutation withRestrictions

5.6 Combinations

5.7 Combination ofrObjects From

Determine the number of ways of performingseveral tasks in succession by using the basiccounting principle.

Use the notation ofn! = n (n-1) (n-2) x . . . x 3 x2 x 1 or n (n-1)!

[Note: 0! =1].

Recognise a permutation as an arrangement ofobjects in a definite order.

Determine the number of permutations ofn

different objects as n!.

Evaluate)!(

!

rn

nPrn

= and apply the rule that

the number of permutations of r objects from n

different objects is)!(

!

rn

nPrn

= .

Solve simple problems onarrangements/permutations with restrictions suchas the arrangement of letters in a word beginning

with a vowel, the number of 4-digit odd numbers,etc.

Recognise a combination as any selection ofobjects where the order of the objects is of noconcern.


http://www.themathpage.com/aPreCalc/permutations-combinations.htm#perm

http://www.themathpage.com/aPreCalc/permutations-combinations-

2.htm#Cfactorial

Relate to the use of rnC

in the BinomialExpansion.

3 weeks

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n Different Objects Evaluate

!)!(

!

rrn

nCr

n

= and apply the rule

that the number of combinations ofrobjects from n

different objectsis rnC .

Know the relation,!r

PC r

n

rn = .

Solve miscellaneous problems.

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(Note: Stress on the need to distinguish betweenpermutations and combinations. Cases withrepetition of objects or with objects arranged in a

circle or involving both permutations andcombinations are excluded).

6 VECTORS IN TWODIMENSIONS

6.1 Basic Concepts

6.2 Operations on Vectors

(Revision on vectors). Define a vector as adirected line segment which has magnitude anddirection. Give examples of vector quantities:displacement, velocity and acceleration.

Know vector notation:

=

3

2AB , a or

=

3

2a .

Define equal vectors and zero vector.

Define a negative vector as a vector having thesame magnitude but opposite in direction i.e.

=BA AB .

Perform addition of vectors by using the TriangleLaw and Parallelogram Law of Addition.

Use vector diagram to show ACBCAB =+ .

Perform subtraction of vectors a b as (b a)

using vector diagrams. Perform scalar multiplication of a vector.

(Show students that vector ka has a magnitude k

times that of vector a, by using examples thatvector 3a is actually a + a + a, etc.)

Know that vector ka is parallel to vector a and is inthe same direction as a if k is positive but is


Remark: Students have

prerequisite knowledgeon Vectors fromMathematics Syllabus D.

3 weeks

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opposite in direction if k is negative.

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6.3 Magnitude of Vectors,Unit

Vectors and Parallel

Vectors

6.4 Position Vectors

Use Pythagoras Theorem to find the magnitude of avector.

(If

= y

x

AB then22

yxAB+=

.)

Define unit vector as a vector with a magnitude of 1unit.

(IfOP is a unit vector, then 1=OP . )

Find the unit vector in the same direction as a givenvector a.

(Required unit vector, a aa.

1= .) Define parallel vectors as two non-zero vectors a

and b having the same or opposite direction and

that a = kb. Know that ifa = kb , then bk=a .

Know that for two non-parallel vectors a and b: sqandrpsrqp ==+=+ baba .

Solve related problems.

Know position vectors as vectors which aredescribed relative to the origin O.(e.g. the position of a point P with respect to anorigin O is indicated by the directed line segment

OP . Thus the vector OP or p is called the position

vector ofP relative to O and =OP xi +yj where i

and j are unit vectors in the positive direction alongthex axis and they axis respectively. In column vector form,

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=

y

xOP .)

Find the magnitude ofOP using the formula22 yxOP += .


6.4 Position Vectors

6.5 MiscellaneousProblems on

Vectors

Find the unit vector in the direction ofOP using the

formula 22 yx

yx

OP

OP

+

+=

jior

+ y

x

yx 22

1.

Know that ifa and b are the position vectors of

points A and B with respect to the origin, then =AB

b a .

Express given vectors in terms of given positionvectors and solve problems including finding(i) the position vector of the midpoint M of the line

segmentAB as2

1=OM (a + b),

(ii) the position vector of point R given e.g.

RCAR3

2= ,

(iii) the unit vector in the direction ofAB .

Know the term collinear points: IfA, B and C arethree points lying on a straight line, thenA, B and C

are collinear points. Use one of the equations BCkAB = or ACkAB = or

BCkAC = to show that pointsA, B and C are

collinear.

Solve miscellaneous problems related to vectors.

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7 RELATIVE VELOCITY

7.1 Composition ofVelocities

Understand that velocity is a vector quantity and sothe composition of two velocities results in a

resultant velocity that has the same effect as thetwo velocities combined and the resultant velocitycan be found by using the Parallelogram or TriangleLaw of Addition.


5 weeks

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7.2 Resolution of Velocities

7.3 Introduction to RelativeVelocity

Find the resultant velocity , Rv of two given

velocities 1v and 2v where

(i) 1v is parallel to 2v by using the i orj direction,

(ii) 1v is perpendicular to 2v e.g. 3i and 4j by using

Pythagoras Theorem to find the magnitude andtangent ratio to find the direction/bearing of theresultant velocity,

(iii) 1v and 2v are in any direction by using

trigonometry e.g. 1v is 6 ms-1

on a bearing of 060and 2v is 8 ms-1 due north-

east.

Understand that the reverse process of combining

two velocities(vectors) is resolving a velocity i.e.splitting it into two perpendicular components,usually in the i j directions.

Show that a velocity of magnitude V making anangle ofwith the vertical can be resolved intothe components V sini and V cosj.

Resolve given velocities into the i j components.

Obtain the resultant of the two velocities e.g. given

in (iii) above, 1v is 6 ms-1 on a bearing of 060and

2v is 8 ms-1 due north-east by

- resolving each velocity as components1

v = 6 sin

60i + 6 cos 60

j and 2v = 8 sin 45i + 8 cos 45

j ,

- adding the components giving Rv = 10.853i +

8.657j ,

- then finding the magnitude of Rv and the

direction.(Suggestion: The method of resolution of velocities

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into the i j components to find the resultantvelocity is an alternative method for students whohave difficulty dealing with the direction of arrowsusing trigonometry.)

Understand the concepts involving relative motion

of a moving object and a stationary object e.g. aboy in a moving car looking at a tree, and relativemotion of two moving objects e.g. two carsapproaching each other, etc.

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7.4 Relative Motion

7.5 Relative Velocity -Motion in a Current andMotion in the Air

Know the term true (actual) velocity, e.g. v A, thetrue velocity of a moving objectA relative to theEarth.

Know the term relative (apparent) velocity, e.g. v A/B,the velocity of a moving objectA relative to amoving object B (observer).

Use relative velocity equation,

BABAvvv =

/ to solve problems involving motion

along a straight line and non parallel motion.

Use the alternative equation, BBAA vvv += / .

Know the terms actual path, track, actual speed,

ground speed, course, speed in still water, speed instill air, air speed and speed of current.(Teachers may use the examples of a man rowing aboat on a river and an aircraft flying in the wind.Show, using vector diagrams, the composition of thevelocities and use the parallelogram law to get theresultant velocity.)

Use the relative velocity equation:

v WA / + v w = v A

where v WA / is the velocity of aircraft relative to the

wind or the velocity of boat relative to water

current, v w is the velocity of the wind or watercurrent, and v A is the true velocity of the aircraft or

boat.

Solve problems involving motion in the water suchas river crossing problems, like finding:- the actual velocity across the river,

http://webphysics.davidson.edu/physlet_resources/bu_semester1/c4_relv1D

.html

http://www.saburchill.com/physics/chapters/0083.html

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- the time taken to cross the river,- the angle of motion,- the resultant speed of the boat on return journey,

etc.

Solve problems involving motion of aircraft in theair:- find the true velocity and direction (ground speed

and track),- find the course and time taken for the journey,

- find the course of outward and return journey,by drawing vector diagrams and using the

trigonometry method.


7.5 Relative Velocity -Motion in a Current and

Motion in the Air

7.6 Relative Motion of TwoMoving

Objects

(Caution: Students must be aware that the windvelocity is usually given from the direction it

is blowing from, e.g. if the wind is blowing fromeast, then the direction of the vector is towardswest.)

Define the apparent path of A relative to B as thepath that would be taken by A as observed by B if Bis assumed to be stationary.(Suggestion: Plot actual positions at regular timeintervals to show the apparent path of A relative toB.)

Know that for interception or collision to

occur, the apparent path of A relative to B must beparallel to AoBo where Ao and Bo are the given initial

positions of A and B i.e. B/Av is parallel to AoBo.

Solve problems on relative motion of two movingobjects.

Solve problems involving interception (collision) oftwo moving objects (but not closest approach).

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Text books1 New Additional Mathematics (Ho Soo Thong & Khor Nyak Hiong)2 Additional Mathematics (H H Heng, JF Talbert)

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Add Maths Year 11