algebraic relations - · pdf file118 1 5. solve 6 ⋅ 5x = 750 without using a calculator....

1

Identities, Formulas and Simultaneous Equations

1

EquationsEquations are used to state the equality of two expressions containing one or more variables.For example, (x - 1)2 = 5

IdentitiesIdentities are a special kind of equations, which are always true regardless of the values of the variables.For example, (x - 1)2 ≡ x2 - 2x + 1

FormulasFormulas are used to express the general relation between quantities.

For example, V r h= 13

2π

1. Substitution2. Change of Subject

Algebraic Relations

Linear Equations in One UnknownStandard form: ax + c = 0For example, 3x - 1 = 0

HKEP

Linear Equations in Two UnknownsStandard form: ax + by + c = 0For example, 3x + y - 1 = 0

Simultaneous Linear Equations

For example, y xx y= −=

12

There may be one, no or infinitely many points of intersection for a pair of simultaneous linear equations.

1. One point of intersection 2. No intersection 3. Infinitely many points of intersection

4

Mathematics: Conventional Questions (Compulsory Part) Book 1 (Second Edition)

FormulasFormulas are used to express the relation between quantities.

For example, the volume V of a cone with base radius r and

height h is given by the formula = πV r h1

32 .

1. Substitution

In a formula, the value of a variable can be found by the method of

substitution if the values of all other variables are given.

In = πV r h1

32 , if r = 2 and V = 8π, then

2. Change of Subject

In a formula, the variable expressed in terms of other variables is called

the subject of the formula. It can be changed from one variable to

another.

For = πV r h1

32 , the subject is V.

After rewriting the formula as =π

hV

r

32 , the subject changes to h.

Recent HKDSE Questions:

2014 2013 2012 Practice Paper Sample Paper

HKDSE 5 2, 4 2, 5 2, 5 2, 5

Recent HKCEE Questions:

2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000

HKCEE 1, 6 5, 6 1, 6 3, 6 1, 7 1, 5 2, 7 1, 6 6 1

C

π = π

=

=

h

h

h

81

3(2)

84

36

2

In the formula y = 5x + 3, which

variable is the subject?

5

How many variables are there

in the formula A = πr2?

4

At first, V is expressed in terms

of r and h. Afterwards, h is

expressed in terms of V and r.


17

21. Solve the simultaneous equations 0 3 0 5 5

2 2. .x y

x y= +

+ =

.

22. If m + 2n + 13 = 2m - n = 11, find the value of n. Ref. DSE 2012 Paper 2, Q.5

23. If 3p + 2q = p - q = 5, find the value of q. Ref. CE 2010 Paper 2, Q.8

24. There are 25 coins in a bag. Some are $2 coins and the rest are $5 coins. If the total amount of the coins is $80, find the number of $2 coins. Ref. DSE 2013 Paper 1, Q.4

25. There are 192 students in a training camp consisting of 8 groups. Each group has the same number of students. In each group, there are 6 more female students than male students. Find the number of male students in the training camp. Ref. DSE 2012 Paper 1, Q.5

26. In a company, the ratio of the number of senior staff members to that of junior staff members is 5 : 8. If 20 senior staff members and 6 junior staff members leave the company, then the number of junior staff members will be 2 times that of the senior staff members. Find the original number of junior staff members in the company. Ref. CE 2011 Paper 1, Q.6

27. Let a, b, m and n be constants. If ax3 + 3x2 + bx - 1 ≡ x3 + mx2 - 7x + n, find the values of a, b, m and n.

28. If A, B and C are constants such that (x - 2)2 + 9 ≡ Ax2 + Bx + C, find the values of A, B and C.

29. If a, b and c are constants such that (2x + 1)2 + 3 ≡ ax2 + bx + c, find the values of a, b, and c.

30. Let a and b be constants. Find the values of a and b if (x + 1)2 + x2 ≡ ax2 + bx + 1.

31. Let a, b and c be constants. Find the values of a, b and c if x3 + ax2 + bx + c ≡ x(x + 2)(x - 3). Ref. CE 1999 Paper 2, Q.6

32. If m and n are constants such that mx - (x - 4)2 ≡ n - 7x - x2 , find the values of m and n. Ref. CE 2010 Paper 2, Q.5

33. If h and k are constants such that h(x2 + x) - k(x2 - x) ≡ 3x2 + 7x , find the values of h and k. Ref. CE 2009 Paper 2, Q.5

34. Let a, b and m be constants. If a(x + 1)(x + 2) + b ≡ 2x2 + mx + 4, find the values of a, b and m.

35. If A, B and C are constants such that x2 + 5x + 6 ≡ A(x + 1)2 + Bx + C, find the values of A, B and C. Ref. CE 2000 Paper 2, Q.10

36. If a, b and c are constants such that (ax + b)(x - 1) - c ≡ 2x2 - 3x + 4, find the values of a, b and c.


21

Foundation Level 1 Sum of the six numbers = 5 × average of the first five numbers + the last number

Rewrite A = B = C as ==

A C

B C

.................. (1)

.................. (2).

Foundation Level 2 Express P and R in terms of Q.

Use the two ratios to set up a pair of simultaneous equations.

Write x2 + y2 as x2 + y2 - 2xy + 2xy.

Non-foundation Level 1 Consider (h + k)2 = h2 + 2hk + k2.

Consider h3 + k3 = (h + k)(h2 - hk + k2) and use the result of (a).

Make x the subject of the formula.

75

Quadratic Equations in One Unknown

11. Let k be a constant. If 9x2 + kx + 16 = 0 has one double positive real root, find the value(s) of k. (5 marks)

Suggested Solution

D = (k)2 - 4(9)(16) 1M

= k2 - 576Since the equation has one double real root, D = 0.

k2 - 576 = 0 1M k = ±24 1A

When k = 24,

9x2 + 24x + 16 = 0

(3x + 4)2 = 0

x = - 43

(double root)

When k = -24,

9x2 - 24x + 16 = 0

(3x - 4)2 = 0

x = 43

(double root) 1M

\ k = -24 1A

12. It is given that 2x3 - x2 + kx - 8 ≡ (x - 2)(ax2 + bx + c), where k, a, b and c are constants.

(a) Find the values of a, b and c. (4 marks)

(b) Someone claims that all roots of the equation 2x3 - x2 + kx - 8 = 0 are real numbers. Do you agree? Explain your answer. (3 marks)

Suggested Solution

(a) R.H.S. = (x - 2)(ax2 + bx + c)

= ax3 + bx2 + cx - 2ax2 - 2bx - 2c

= ax3 + (b - 2a)x2 + (c - 2b)x - 2c

By comparing the coefficients of like terms,

a = 2 1A

b - 2a = -1 ..........(1) 1M

-2c = -8 ..........(2)

Substituting a = 2 into (1),

b - 2(2) = -1

b = 3 1A

From (2), -2c = -8

c = 4 1A

Ref. CE 2004 Paper 2, Q.6


Cross

Ref. DSE 2013 Paper 1, Q.12

N o t e t h a t t h e re a re t w o

conditions:

1. Double root

2. Positive root

Students may forget to justify

the answer with respect to the

condition that the equation has

a positive root.

Expand the R.H.S. of the

identity and then comparing the

coefficients of like terms.

118


5. Solve 6 ⋅ 5x = 750 without using a calculator. (3 marks)

Suggested Solution

6 ⋅ 5x = 750

5x = 125 1M

5x = 53 1M

∴ x = 3 1A

6. Solve 49x - 2 = 343. (3 marks)

Suggested Solution

49 343

(7 ) 7

7 72( 2) 3

23272

2

2 2 3

2( 2) 3

x

x

x

x

x

x

=

=

=− =

− =

=

−

−

−

∴

Foundation Level 2

7. Solve 4x + 2 - 6(4x + 1) + 2(4x) = -1536. (5 marks)

Suggested Solution

4x + 2 - 6(4x + 1) + 2(4x) = -1536

4x(42) - 6(4x ⋅ 4) + 2(4x) = -1536

16(4x) - 24(4x) + 2(4x) = -1536 1M + 1A

-6(4x) = -1536

4x = 256 1M

4x = 44 1M

∴ x = 4 1A

Ref. CE 2003 Paper 1, Q.4

1M

1M

1A

Try to express both sides of

the equation in the form 5x and

5y respectively. Then use the

property: if ax = ay, then x = y.

Express 125 as an exponent of

5.

Express 49 and 343 as

exponents of 7.

If ax = ay, then x = y.

am + n = am × an

Express 256 as an exponent of

4.


154

Does (3a + 3b)(3c - 3d) equal

3(a + b)(c - d)?

4

Polynomials in One Variable

A polynomial in one variable is a sum of terms. Each term is a product of the

variable with a non-negative integral exponent and a real number called the

coefficient. The degree of the polynomial is the degree of the term with the

highest degree.

Arithmetic Operations of Polynomials

1. Addition

For example,

(6x3 + 2x2 - x + 5) + (3x - 2) = 6x3 + 2x2 - x + 5 + 3x - 2

= 6x3 + 2x2 + 2x + 3

2. Subtraction

For example,

(6x3 + 2x2 - x + 5) - (3x - 2) = 6x3 + 2x2 - x + 5 - 3x + 2

= 6x3 + 2x2 - 4x + 7

3. Multiplication

For example,

(6x3 + 2x2 - x + 5)(3x - 2)

= 3x(6x3 + 2x2 - x + 5) - 2(6x3 + 2x2 - x + 5)

= 18x4 + 6x3 - 3x2 + 15x - 12x3 - 4x2 + 2x - 10

= 18x4 - 6x3 - 7x2 + 17x - 10

Factorization of Polynomials

Some polynomials can be factorized into a product of two or more factors by

the following methods.

1. By taking out common factors

For example, ab + ac = a(b + c)

abc - abd = ab(c - d)

2. By grouping terms

For example, ax + ay + bx + by = a(x + y) + b(x + y) = (x + y)(a + b)

ax + ay - bx - by = a(x + y) - b(x + y) = (x + y)(a - b)

A

B

C

Write down the degree of each

of the following polynomials.

(a) x3 - x2 + 5x + 1

(b) x4 + x3y2 + 8

1

Consider two polynomials P(x)

and Q(x). Determine whether

the following must be true.

(a) P(x) - Q(x) = Q(x) - P(x)

(b) P(x) × Q(x) = Q(x) × P(x)

2

Simplify (2x2 + 3x) - (4x2 - 5x).

3

We can only add or subtract

like terms.

C h a n g e t h e s i g n w h e n

removing the brackets of 3x - 2.

Introduction to Coordinates

317

• Unlessotherwisespecified,numericalanswers shouldbeeitherexactorcorrect to3significantfigures.

• Thediagramsarenotnecessarilydrawntoscale.

Foundation Level 1

1. Writedownthecoordinatesofthepoints A,BandCasshowninthefigure.

2. Markthefollowingpointsonthepolarcoordinateplane. A(4,60°),B(3,120°),C(1,150°)

3. Accordingtothefigure,find∠POQ.

4. ThepolarcoordinatesofCandDare(4,120°)and(9,300°)respectively.FindthelengthofCD. Ref. CE 2006 Paper 2, Q.27

5. FindthelengthofthelinesegmentjoiningA(4,9)andB(-1,2).(Leavetheanswerinsurdform.)


321

26.ConsidertwopointsA(-6,5)andB(2,-7).

(a) Writedown thecoordinatesof the imagewhenpointA is reflectedabout the following linesrespectively.

(i) x=3 (ii) y=-6

(b) Writedownthecoordinatesof the imagewhenpointB is reflectedabout the following linesrespectively.

(i) x=y (ii) x+y=0 Ref. CE 2010 Paper 2, Q.29

27.A(-1,8),B(6,10),C(4,3)andD(-5,1)are theverticesofaquadrilateral. It is rotatedabout theoriginO through90° inaclockwisedirection to formanotherquadrilateralA′B′C′D′.Draw thequadrilateralsABCDandA′B′C′D′onthesamecoordinateplane.

28.FindthenewcoordinatesofX(a,b)ifitisfirstreflectedaboutthex-axisandthenrotated270°anti-clockwiseabouttheoriginO.

29.PointY(6,-6) isrotated90°anti-clockwiseaboutO topointZ.Describethetransformation,otherthanrotation,totransformpointZbacktopointY.

30.WhataretherectangularcoordinatesofapointAifthepolarcoordinatesofAare( 2,135°)? Ref. CE 2010 Paper 2, Q.30

31.IftherectangularcoordinatesofapointBare( 3 3- ,-3),findthepolarcoordinatesofB. Ref. CE 2007 Paper 2, Q.30

32.FindthedistancebetweenthepointsCandD if thepolarcoordinatesofCandDare(6,50°)and(8,140°)respectively. Ref. CE 2006 Paper 2, Q.27

Foundation Level 2

33.ConsidertheverticesD(4,4),E(6,4),F(6,-5)andG(4,-5)ofaquadrilateral.

(a) Mark the fourverticesand thequadrilateralon therectangularcoordinateplaneandstate the typeof thequadrilateralformed.

(b) D andEare translated3units leftwardswhileFandG are translated 1 unit rightwards. Draw the newquadrilateralonthesamerectangularcoordinateplaneandstatethetypeofthenewquadrilateralformed.

324


44.Inapolarcoordinatesystem,Oisthepole.ThepolarcoordinatesofAandBare(20,190°)and(20,310°)respectively.

(a) Markthepointsonthegivenpolarcoordinateplane.

(b) FindthepolarcoordinatesofthepointofintersectionofLandABifListheaxisofreflectionalsymmetryof∆OAB.

Ref. DSE 2013 Paper 1, Q.6

45.Inapolarcoordinatesystem,O is thepole.ThepolarcoordinatesofCandDare (12,205°)and(e,295°)respectively,whereeisapositiveconstant.

(a) Is∆CODaright-angledtriangle?Explainyouranswer.

(b) FindthevalueofeifthelengthofCDis13.

(c) Findtheperimeterof∆COD. Ref. CE 2009 Paper 1, Q.8

46.E(5,-2)isrotatedclockwiseabouttheoriginOthrough90°toF.HisthereflectionimageofG(3,-4)withrespecttothey-axis.

(a) WritedownthecoordinatesofFandH.

(b) AreO,FandHcollinear?Explainyouranswer. Ref. CE 2008 Paper 1, Q.12

47.ConsiderfourpointsW(3,-9),X(-2,-5),Y(-7,-1)andZ(-9,7).

(a) Whichthreepointsarecollinear?Explainyouranswer.

(b) Ifoneof thesefourpoints is translatedpunitsdownwardssuchthat theyarecollinear, findthevalueofp.

Non-foundation Level 1

48.IfapointP(x,y)dividesthelinesegmentjoiningI(-2,1)andJ(3,-5)internallyintheratio4:3,findthecoordinatesofP.

49.Apoint Q12325

625

, −

dividesthelinesegmentKLinternallyintheratior:1wherethecoordinates

ofKandLare(5,-2)and 4910

15

,

respectively.Findthevalueofr.


325

50.ApointR lyingon thex-axisdivides the line segmentPQ internally such thatQRRP

s=1

. If the

coordinatesofPandQare(6,2.5)and(7,-7.5)respectively,findthecoordinatesof Randthevalueofs. Ref. CE 2000 Paper 2, Q.50

51.A(17,13)andB(3,-1)aretwopoints.A,BandCarecollinearsuchthatAB :BC=7:2.FindthecoordinatesofC. Ref. CE 1998 Paper 2, Q.54

52.Thefigureshowsapolarcoordinateplane.

(a) WritedownthecoordinatesofAandB.

(b) FindthelengthofAB.

Non-foundation Level 2

53.ConsidertwopointsA(-5,11)andD(8,-2).ThelinesegmentADcutsthey-axisandthex-axisatBandCrespectively.

(a) FindAB:BDandthecoordinatesofB.

(b) FindAC:CDandthecoordinatesofC.

(c) FindAB:BC:CD.

54.In the figure, thecoordinatesofpointD are (-3,-6)and they-coordinatesofEis3.PdividesDEinternallyintheratio1:s.

(a) FindthecoordinatesofPintermsofs.

(b) MingclaimsthattheslopeofOPis2 1

ss -

fors≠ 12

.Doyou

agree?Explainyouranswer.

Ref. CE A. Maths 2000 Paper 2, Q.5

55.LetObetheorigin.IfthecoordinatesofthepointsFandGare(21,20)and(-21,20)respectively,findthey-coordinateoftheorthocentreof∆FGO. Ref. DSE PP Paper 2, Q.42

346


Distribution of Related TopicsFoundation Level 1 Foundation Level 2 Non-foundation Level 1 Non-foundation Level 2

Question

Chapter1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

1 Identities, Formulas and Simultaneous equations ✓ ✓ ✓ ✓ ✓ ✓ ✓

2 Percentages ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

3 Transformation and Symmetry

4 Quadratic Equations in One Unknown ✓ ✓ ✓ ✓ ✓

5 Functions and Graphs ✓ ✓ ✓ ✓

6A Indices and Surds ✓ ✓ ✓

6B Exponential and Logarithmic Functions ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

7A Polynomials and Factorization ✓

7B More about Polynomials ✓

8 More about Equations ✓

9 More about Graphs of Functions ✓

10 More about Trigonometry ✓ ✓ ✓ ✓ ✓ ✓

11 Applications of Trigonometry ✓ ✓

12A Introduction to Coordinates ✓ ✓ ✓ ✓ ✓ ✓ ✓

12B Equations of Straight Lines ✓ ✓ ✓ ✓ ✓

algebraic relations - · pdf file118 1 5. solve 6 ⋅ 5x = 750 without using a calculator....

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