power point buku bengkel matematik tambahan spm

 4. 10 perkara kenapa saya tak patut 

gagal Additional Mathematics.

1. SASARAN  PENCAPAIAN  SPM  TAHUN :

“Mata Pelajaran  Gred

1. BAHASA MELAYU

2. BAHASA INGGERIS

3. MATEMATIK

4. SEJARAH

5. PENDIDIKAN ISLAM / MORAL

6. FIZIK

7. KIMIA

8. BIOLOGI

9. MATEMATIK TAMBAHAN

• 2. SAYA PERLU BERJAYA CEMERLANG KERANA (Berikan tiga sebab utama):• • 1. ………………………………………………………………………………………………………...• ……………………………………………...……• • 2. ……………………………………………………...………………………………………………...• ………………………………………………...…• • 3. ……………………………………………...………………………………………………………….• ……………………………………………...……• ………………………………………………...…• Dengan ini saya : (nama anda) • ……………………………...……………………...……………………………………………………….• anak kepada (nama ibu) • ……………………………………………………...……………………………………………………….• • Demimu ibu, saya berjanji akan berusaha dengan bersungguh-sungguh secara berterusan mulai sekarang supaya

sasaran saya dalam SPM tahun ………. tercapai.• • Tanda tangan : …………………………………..

• Saksi…………………………………………:

3. Pesan ibu beribu-ribu… 10  pesan  ibu  kepada  saya ...

3. Pesan ibu beribu-ribu… 10  pesan  ibu  kepada  saya ...

3. Pesan ibu beribu-ribu… 10  pesan  ibu  kepada  saya ...

4. 10 perkara kenapa saya tak patut gagal Additional Mathematics

4. 10 perkara kenapa saya tak patut gagal Additional Mathematics

4. 10 perkara kenapa saya tak patut gagal Additional Mathematics

5. PAGE 4 Anda  letak  diri                        anda  di  mana?

7. Analisis   Topik  Dalam SPM

CHAPTER TOPIK   TINGKATAN  4 PAPER  1 PAPER  2

1. Functions / /

2. Quadratic Equations /  

3. Quadratic Functions /  

4. Simaultaneous Equations   /

5. Indices and Logarithms /  

6. Coordinate Geometry / /

7. Statistics / /

8. Circular Measure / /

9. Differentiation / /

10. Solution Of Triangles   /

11. Index Number   /

7. Analisis   Topik  Dalam SPM

CHAPTER TOPIK   TINGKATAN  5 PAPER  1 PAPER  2

1. Progressions / /

2. Linear Law / /

3. Integration / /

4. Vectors / /

5. Trigonometric Functions / /

6. Permutations and Combinations

/  

7. Probability /  

8. Probability Distrubutions / /

9. Motion Along a Straight Line   /

10. Linear Programming   /

6. Saya  boleh dapat 90 markah Kertas          Mat. Tambahan

KERTAS   1 25  SOALAN (SEMUA WAJIB  JAWAB)

MASA : 2  JAM

R S T6 3 1

KERTAS  2R S T4 3 3

8: FUNCTIONS • Notes: • 1. 4 types of relation.• 2. Notation of function : f : x —> y or f (x) = y.• 3. Composite Functions.• 4. Inverse Functions, if f -¹(x) = y then x = f (y)

8: FUNCTIONS Question 1: Paper 1 : Given that f –1 : x —> (x + 1)/4 , find f (x) . [ 2 marks ]

SOLUTION : f –1 (x) =y , f(y) = x

y = (x + 1)/4 jadikan x perkara rumus4y = x + 14y – 1 = x f(y) = x f(y) = 4y – 1 f(x) = 4x – 1

functions

• Question 2: Paper 1 : Given that f (x) = (1 + 5x ), and g(x) = (x/2) + 3 , find f g(x) . [2 marks]• Solution: f (x) = (1 + 5x ), g(x) = (x/2) + 3 f g(x) = f ((x/2) + 3) = 1 + 5((x/2) + 3) = 1 + 5x/2 + 15 f g(x) = 16 + 5x/2

functions• Question 3: SPM 2004 : Given that f : x —> p - 3x and f-1 : x —> (qx/2) + 2/3, where

p and q are constant. Find the value of p and q : (3 Marks)• Solution : f–1(x) = y , f(y) = x f(y) = x f–1(x) = y P – 3y = x or p – x = 3y = x/-3 + p/3y = (x – p)/- 3 or (p – x)/3 = y compare : q/2 = -1/3 p = 2 , q = -2/3

functions• Question 4: 1. SPM 2004 : Given that the

function f(x) = 4x - 1 and fg(x) = 5x, find: • a) g (x) ,• b) the value of x when gf(x) = 9. (4 marks)• Solution : a) fg(x) = 5x b) gf(x) = 9.

4g(x) – 1 = 5x (5f(x) + 1)/4 = 9

4g(x) = 5x + 1 5f(x) = 35 4x = 8

g(x) = 5x/4 + ¼ f(x) = 7 x = 2

4x – 1 = 7

functions• 5. SPM 2005: f:x —>3 - 4x and g:x —>1 + 2x

are two functions, find f-1 g(x). (3 marks)• Solution : f:x —>3 - 4x f–1(x) = (3 – x)/4 f –1 (x) =y , f(y) = x, f–1 g(x) = (3 – (1 + 2x))/4 3 – 4y = x = (2 - 2x)/43 – x = 4y f–1 g(x) = (1 – x)/2(3 – x)/4 = yf –1 (x) =y f –1 (x) = (3 – x)/4

functions• 6. SPM 2003 : Given that f:x —> 1 + 2x and

g : x —> x2 + 4x - 3, find :• a) f-1 (4), • b) gf(x). (4 Marks)•

functions• 7. SPM 2006 : Given that f : x —> 4 + 5x and

g : x —> (x/2) - 1, find : a) f-1 (x), (1 Mark ) b) gf -1 (x). (2 Marks) c) h(x) such that hg(x) = 6 - 5x. (3 Marks)

functions• 8. SPM 2006 : Given that f : x —> 3x - 2

and g : x —> (x/5) + 1, find :• a) f-1 (x) [ 1 mark ]• b) f-1 g(x) [ 2 marks]• c) h(x) such that hg(x) = 2x + 6. [3marks].

Quadratic Equations• Notes : • 1. The general form : ax2 + bx + c = 0, a = 0,

a , b anc c are constant, x is a variable.• 2. If k is the root of the equation ax2 + bx + c =

0, then ak2 + bk + c = 0.• 3. Quadratic equation can be solved by:• a) factorization,• b) using the formula,• c) completing the square,•

Quadratic Equations• 4. If α and β are the roots of the quaratic equations,

then• (x - α) (x - β) = 0 or x2 - (α + β) x + (αβ) = 0.• SOR POR• • 5. Quadratic equation ax2 + bx + c = 0, has 3 types of roots.• a) two different real roots if b2 - 4ac > 0• b) two equal roots or one root only if b2 - 4ac = 0• c) no real roots if b2 - 4ac < 0

•

Quadratic Equations• Question 1: SPM 2003 : Solve the quadratic

equation (2 - x) (x + 1) = x (x – 5)/4. Give your answers correct to four significant figures. (3 Marks)

•

Quadratic Equations• Question 2: SPM 2005: Solve the quadratic

equation (3x - 5)/(1 - 2x) = 4x . Give your answers correct to three decimal places. (3 Marks)

•

Quadratic Equations• Question 3: SPM 2004: From the quadratic

equation which has the roots –5 and 2/3 . Give your answers in the general form. (2 Marks)

•

Quadratic Equations• Question 4: SPM 2006: A quadratic equation

x2 - kx + 4 = 8x has two equal roots. Find the possible values of k. (3 Marks)

•

Quadratic Equations• Question 5: SPM 2003: The quadratic equation

4x - k = 3x(x - 2) has two distinct roots. Find the range of values of k. (3 Marks)

•

Quadratic Equations• Question 6: SPM CLONE : Given α and β are

the roots of the quadratic equation 2x2 + 4x - 7 = 0. Form the quadratic equation

which roots 2α and 2β. (4 Marks)•

Quadratic Equations

• Question 7: SPM CLONE : The quadratic equation 3px - 5 = (qx)2 - 1, has two equal roots. Find p: q. (3 Marks)

•

Quadaratic Equations• Question 8: SPM CLONE : Find the range of

values of p if the quadratic equation (p - 1)x2 - 8x = 4, has no roots. (3 Marks) •

Quadratic Functions• Notes : 1. The general form : ax2 + bx + c = f(x), a ≠ 0, a , b anc c are constant, x is a variable.2. The shape of the graphs is known as a parabola, if : a) when a > 0 , the graph is a parabola with a minimum point, b) when a < 0 , the graph is a parabola with a maximum point,3. A quadratic function can be expressed in the form f(x) = a (x + p)2 + q, where a, p and q are constants. a) when a > 0 , the minimum point is (-p, q) and the minimum value is q. b) when a < 0 , the maximum point is (-p, q) and the maximum value is q.

•

Quadratic Functions4. A graph quadratic function ax2 + bx + c = f(x) can be sketched by the following steps, a) identifying the value of a. (Notes 2) b) determine the maximum or minimum point by completing the square method (Notes 3), or use x = - b/2a and replace the value of x on to f(x) to the find f(x), c) Find the points of intersection at the y-axis by finding f(0) or replace x = 0 on to f(x), d) Find the points of intersection at the x-axis by solving the f(x) = 0.5. The axis of symmetry passes through the maximum or minimum point (turning point (h,k)) are parallel to y-axis, it is x = h.

•

Quadratic Functions• Question 1: Find the maximum or

minimum point for the following quadratic equation by the completing the square method.

a) h(x) = x2 + 4x - 8 b) g(x) = -4x2 + 12x –5

Quadratic Functions• Question 2: Express the equation 3x2 - 6x - 1 =

0 in the form a (x + p)2 + q = 0. Hence state the values of a, p and q. (3 Marks)

•

Quadratik Functions• Question 3: Sketch the graph for y = (x - 3)2 -

4 for -1 ≤ x ≤ 6. Hence, state the range for the corresponding given domain. (4 Marks)

•

Simultaneous Equations

Notes : Step to solve simultaneous equations :• a) Identiying linear equation, and express one unknown in terms of the other

unknown. • b) Substitute the equation in a) into the nonlinear equation.• c) Solve the quadratic equation ax2 + bx + c = 0 formed.• d) Get the second unknown, substitude the unknown found c) into linear

equation.

• •

Simultaneous Equations• Question 1: Solve the simultaneous

equations x - 1 - 2y = 0 and y/x + 5x/y = 6.•

• Question 2: Solve the equations x + y = 3x2 - y2/2 = 1.•

• Question 3: Solve the simultaneous equations -x + y = 1, and x + 4 = y2. Give your answers correct to three decimal places.

•

• Question 4: SPM 2004 : Solve the simultaneous equations p - m = 2 and

p + 2m = 8. Give your answer corret to Three decimal places. [5 marks]

•

Simultaneous Equation• Question 5: SPM 2005 : Solve the

simultaneous equations x + ½ y = 1 and y2 - 10 = 2x. [5 marks]•

• Question 6: SPM 2006 : Solve the simultaneous equations 2x + y = 1 and

• 2x2 + y2 + xy = 5. Give your answers correct to three decimal places. [5 marks]

•

Simultaneous Equation• Question 7: SPM 2007 : Solve the

following simultaneous equations • 2x - y - 3 = 0 , and 2x2 - 10x + y + 9 = 0.

[5 marks]•

INDICES  AND  LOGARITHMS

• Notes : • 1. a0 = 1 , a ≠ 0• 2. a-n = 1/an

• 3. a m/n = n√am = (n√a)m

• 4. am x an = am + n • 5. am ÷ an = am - n • 6. (am)n = am x n • 7. am ÷ bm = (a/b)m • 8. If a = bx , then logb a = x .

Conversely if logb a = x , then a = bx .•

INDICES  AND  LOGARITHMS

• 9. log a 1 = 0

• 10. log a a = 1

• 11. a loga b = b

• 12. loga (xy) = loga x + loga y.

• 13. loga (x/y) = loga x - loga y.

• 14. Loga x n = n loga n.

• 15. Loga b = logn b / logn a.

• 16. Solving the equation of logarithms. • Change the equation of the logarithms in index form• .

•

INDICES  AND  LOGARITHMS

• Question 1: Solve the equation 8 . 16x—1 = √42x .

•

12.                   INDICES  AND  LOGARITHMS

• Question 2: Solve the equation • 3 . 2x + 1 + 2x = 14•

INDICES  AND  LOGARITHMS• Question 3: Solve the equation 3x = 2x + 1 , give your answer in three decimal places.•

INDICES  AND  LOGARITHMS• Question 4: Solve the equation 2 log2 x - 2 = log2 (x - 1).

•

12.  INDICES  AND  LOGARITHMS

• Question 5: Given log4 P - 2 = log2 T ,

• express P in terms of T.

INDICES  AND  LOGARITHMS

• Question 6: Given log10 (x2 y) = 3 and

• log10 (x/y2 ) = 4. Find log10 x and log10 y.

•

13.       TRIGONOMETRIC  FUNCTIONS • Learning Focus : SPM - General Pattern Of

Transformations.- Sketch the trigonometric Graph.- • Paper 2 part A Compulsory Questions. - 6 - 7 Marks.• •

13.            TRIGONOMETRIC  FUNCTIONS

• Question 1: Tranformation type 1—Vertical Translation.

• On the same axis, sketch the graph of y = sin x + 2 and y = sin x - 1 for O0 ≤ x ≤ 1800 .

•

13.       TRIGONOMETRIC  FUNCTIONS

• Question 2: Tranformation type 1—

Vertical Translation.• On the same axis, sketch the graph of y = 2

cos x and y = ½ cos x for O0 ≤ x ≤ 2Π .•

13.1  TRIGONOMETRIC FUNCTIONS

• Question 3 : Sketching the trigonometric graph (Clone SPM exam questions)

• a) Sketch the graph of y = cos 2x for O0 ≤ x ≤ 1800 .

• b) Hence, by drawing a suitable straight line on the same axes , find the number of solution satisfying the equation 2 cos² x = 2 - x/360º for Oº ≤ x ≤ 180º .

•

13.1 TRIGONOMETRIC FUNCTIONS  • Question 4: SPM  2001  Paper 1

• Given that sin x = 5/13 , and that x is obtuse. Calculate without using a calculator the value of ,

• a) tan 2x• b) Cos x/2.•

13.1     TRIGONOMETRIC  FUNCTIONS

• Question 5: Strategy to prove trigonometric identity.

• Prove cos x cosec x + sin x sec x = cosec x sec x.

•

14 : LINEAR  LAW • Learning Focus : SPM - 1) Change the non linear

equation to linear form Y = m X + c.• 2) Complete the table for Y axis and X axis.• 3) Plot graph Y against X , use pair from table.• 4) Plot straight line of best fit. • 5) Determine the gradient, m and the Y

intercept, c from your graph (best fit line).• 6) State the value of m and c.•

14 : LINEAR  LAW• Question 1: SPM 2004 : Table 1 shows the values of two variables,

x and y, obtained from an experiment. Variables x and y are related by the equation y = pkx , where p and k are constants.

Table 1.

a) Plot log10 y against x by using a scale of 2 cm to 2 units on the x-axis and 2 cm to 0.2 unit on the log10 y-axis. Hence, draw the line of best fit. [4 marks]

b) Use your graph from (a) to find the value of : i) p . ii) k . [6 marks]

x 2 4 6 8 10 12

y 3.16 5.50 9.12 16.22 28.84 46.77

• Question 2 : SPM 2005 : Table 1 shows the values of two variables, x and y, obtained from an experiment. Variables x and y are related by the equation y = px + r/(px) , where p and r are constants.

a) Plot xy against x², by using a scale of 2 cm to 5 units on the both axes. Hence, draw the line of best fit. [5 marks]b) Use your graph from (a) to find the value of : i) p . ii) r . [5 marks]

14 : LINEAR  LAW

x 1.0 2.0 3.0 4.0 5.0 5.5y 5.5 4.7 5.0 6.5 7.7 8.4

14 : LINEAR  LAW• Question 3: SPM 2006 : Table 2 shows the values of two variables, x

and y, obtained from an experiment. Variables x and y are related by the equation y = pkx + 1 , where p and r are constants.

a) Plot log y against ( x + 1), using a scale of 2 cm to 1 unit on the ( x + 1 )-axis and 2 cm to 0.2 unit on the log y-axis. Hence, draw the line of best fit. [5 marks]b) Use your graph from (a) to find the value of : i) p . ii) k . [5 marks]

x 1 2 3 4 5 6

y 4.0 5.7 8.7 13.2 20.0 28.8

14 : LINEAR  LAW• Querstion 4 : SPM 2007 : Table 3 shows the values of two variables, x

and y, obtained from an experiment. Variables x and y are related by the equation y = 2kx² + px/k , where p and k are constants.

a) Plot y/x against x using a scale of 2 cm to 1 unit on both axes. The x -axis and 2 cm to 0.2 unit .Hence, draw the line of best fit. [4 marks]b) Use your graph from (a) to find the value of : i) p . ii) k . iii) y when x = 1.2 [6 marks]

x 2 3 4 5 6 7

y 8.0 13.2 20.0 27.5 36.6 45.5

14.2        LINEAR  LAW   (Paper  1) • Question 1: Reduce non-linear equation,

y = pxk - 1 , where p and k are constant, to linear equation. State the gradient and vertical intercept for the linear equation obtained. [3 marks]

• Solution :

•

• Question 2: • √y • (8, 16)• • • ( 1, 6) • • O• The diagram shows part of the straight line graph obtained by

plotting √y against x2 . • Express y in terms of x . [3 marks]• Solution:

14.2        LINEAR  LAW   (Paper  1)

• Question 2: • x2y• (k, 44)• • • ( 1, h) • • O x2

• The diagram shows part of the straight line graph of x2y against x2 . Given that y = 3x + 4/x2 .Calculate the value of h and k. [3 marks]

• Solution:

14.2        LINEAR  LAW   (Paper  1)

15 :          INDEX  NUMBER

• Question 1 : SPM 2007 : Table shows the price and price indeces of three item A, B, and C, for the years 2002 and 2004. Calculate

• • • • • • •

• a) the value of x,• b) the value of y,• c) the value of z,

ITEM Price per unit (RM)

Price Index for the year 2004 based on  the  year 2002

Year 2002 Year 2004

A 120 130 x

B 40 y 110

C z 70 125

15 :          INDEX  NUMBER Question 2 : SPM 2005 : Table shows the number of cars sold in theyear 1990 and 1995 for the four

different models. Calculate a) The index number of model T in the year 1995 based on year 1990. b) The index number of model H in the year 1990 based on year 1995. c) The index number in the year 1997 based on the year 1990 for model P if the sale is forecast to increase by 20% from 1995 to 1997. d) The composite index for the year 1995 based on the year 1990. • Solution :•

• • • • •

Model of cars

Number of car sold (‘000)

Year 1990 Year 1995

P 20 25

H 25 30

T 40 50

M 60 90

• Question 3 : SPM 2005 : Table a) shows the price indeces for the four ingredients P, Q, R and S, used in • making biscuits of a participant kind. Diagram b) is a pie chart which represents the relative amount of the • Gradients P, Q, R and S , used in making this biscuits.• • • • • • • • • • • •

• a) Find the value of x, y and z . [3 marks]• b) (i) Calculate the composite index for the cost of making these biscuits in the year 2004 base on the• Year 2001.• [ii] Hence, calculate the corresponding cost of making these biscuits in the year 2001 if the cost• In the year 2004 was RM2985. [5 marks]• c) The cost of making these biscuits is expected to increase by 50% from the year 2004 to the year 2007.• Find the expected composite index for the year 2007 based on the year 2001. [2 marks].

Ingredients Price per kg (RM) Price Index for the year 2004

based on the year 2001

Year 2001 Year 2004

P 0.80 1.00 x

Q 2.00 y 140

R 0.40 0.60 150

S z 0.40 80

15 :          INDEX  NUMBER

16 :                     Differentiation • Question 1 : SPM 2004 :

• Differentiate 2x(4 - x2)4 with respect to x . [3 marks]

•

16 :                     Differentiation • Question 2 : SPM 2005 :

Given that f(x) = 1/(2 - 3x)4 , evaluate f’’ (1) . [4 marks]

• •

16 :                     Differentiation • Question 3 : SPM 2006 : Given that y = 5v4/6 ,

where v = 4 - 3x. Find dy/dx in terms of x. [3 marks]• •

16 :                     Differentiation • Question 4 : SPM 2003 : Given that y = 5x - x2 ,

find the small change in y using the differentiation when x increases from to 2.01. [3 marks]•

16 :                     Differentiation • Question 5 : SPM 2006 : Given that y = 5v4/6 ,

where v = 4 - 3x. Find dy/dx in terms of x . [3 marks]

•

16 :                     Differentiation Question 6: SPM 2004 : The gradient of function of a curve

which passes through A (1, -12) is 3x2 _ 6x . Finda) the equation of the curve. [3 marks]b) the coordinates of the turning points of the curve and determine the whether each of the turning points is a maximum

or minimum. [5 marks]• Solution :

16 :                     Differentiation • Question 7: SPM 2006 : A curve has a gradient function

px2 - 4x, where p is a constant. The tangent to the curve at the point (1 , 3) is parallel to the straight line y + x - 5 = 0. Find

• a) the value of p, [3 marks] • b) the equation of the curve. [3 marks] • Solution:

•

16 :                     Differentiation Question 8: SPM 2007 : A curve with gradient function 2x - 2 /x2 has a

turning point at (k, 8).a) find the value of k, (3 marks)b) determine whether the turning points is a maximum or minimum point. [2 marks]c) Find the equation of the curve. [3 marks]• Solution