62226765 Power Point Buku Bengkel Matematik Tambahan Spm (1)

7/28/2019 62226765 Power Point Buku Bengkel Matematik Tambahan Spm (1)

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

then (x - ) (x - ) = 0 or x 2 - ( ) x () = 0. SOR POR 5. Quadratic equation ax 2 + bx + c = 0,

has 3 types of roots. a) two different real roots if b 2 - 4ac > 0 b) two equal roots or one root only if b 2 - 4ac = 0 c) no real roots if b 2 - 4ac < 0


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Quadratic Equations Question 1: SPM 2003 : Solve the quadratic

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


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Quadratic Equations Question 2: SPM 2005: Solve the quadratic

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


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Quadratic Equations Question 3: SPM 2004: From the quadratic

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


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Quadratic Equations Question 4: SPM 2006: A quadratic equation

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


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Quadratic Equations

Question 5: SPM 2003: The quadratic equation4x - k = 3x(x - 2) has two distinct roots. Find therange of values of k. (3 Marks)


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Quadratic Equations Question 6: SPM CLONE : Given and are

the roots of the quadratic equation2x2 + 4x - 7 = 0. Form the quadratic equationwhich roots 2 and 2. (4 Marks)


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Quadratic Equations

Question 7: SPM CLONE : The quadraticequation 3px - 5 = (qx) 2 - 1, has two equalroots. Find p: q. (3 Marks)


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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)


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Quadratic Functions Notes :

1. The general form : ax 2 + bx c = f(x), a 0, a , b anc c areconstant, 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 formf(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 minimumvalue is q.

b) when a < 0 , the maximum point is (-p, q) and the maximumvalue is q.


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Quadratic Functions4. A graph quadratic function ax 2 + 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 andreplace 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) orreplace x = 0 on to f(x),

d) Find the points of intersection at the x-axis by solvingthe f(x) = 0.

5. The axis of symmetry passes through the maximum or minimumpoint (turning point (h,k)) are parallel to y-axis, it is x = h.


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Quadratic Functions Question 1: Find the maximum or minimum

point for the following quadratic equation bythe completing the square method.a) h(x) = x2 + 4x - 8

b) g(x) = -4x2 + 12x 5


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

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


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

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


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Simultaneous Equations

Notes :Step to solve simultaneous equations :

a) Identiying linear equation, and express one unknownin terms of the other unknown.

b) Substitute the equation in a) into the nonlinearequation.

c) Solve the quadratic equation ax 2 + bx + c = 0 formed. d) Get the second unknown, substitude the unknown

found c) into linear equation.


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Simultaneous Equations Question 1: Solve the simultaneous equations

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


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Question 2: Solve the equationsx + y = 3x2 - y2/2 = 1.


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Question 3: Solve the simultaneousequations -x + y = 1, and x + 4 = y 2. Giveyour answers correct to three decimal places.


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Question 4: SPM 2004 : Solve thesimultaneous equations p - m = 2 andp + 2m = 8. Give your answer corret toThree decimal places. [5 marks]


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Simultaneous Equation Question 5: SPM 2005 : Solve the

simultaneous equations x + y = 1 andy2 - 10 = 2x. [5 marks]


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Question 6: SPM 2006 : Solve thesimultaneous equations 2x + y = 1 and

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


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Simultaneous Equation Question 7: SPM 2007 : Solve the

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

[5 marks]

INDICES AND LOGARITHMS


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Notes : 1. a 0 = 1 , a 0 2. a -n = 1/a n 3. a m/n = na m = (na) m 4. a m x an = am + n 5. a m an = am - n 6. (a m)n = am x n 7. a m bm = (a/b) m 8. If a = b x , then log b a = x .

Conversely if log b a = x , then a = b x .



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9. log a 1 = 0 10. log a a = 1 11. a loga b = b 12. log a (xy) = loga x + loga y. 13. log a (x/y) = log a x - loga y. 14. Loga x n = n log a n. 15. Loga b = log n b / log n a. 16. Solving the equation of logarithms. Change the equation of the logarithms in index form .


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Question 1: Solve the equation 8 . 16 x1 = 4 2x.

12 INDICES AND LOGARITHMS


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12. INDICES AND LOGARITHMS

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


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INDICES AND LOGARITHMS Question 3: Solve the equation 3 x = 2x + 1 ,

give your answer in three decimal places.


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INDICES AND LOGARITHMS Question 4: Solve the equation

2 log 2 x - 2 = log2 (x - 1).


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12. INDICES AND LOGARITHMS

Question 5: Given log 4 P - 2 = log2 T , express P in terms of T.


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Question 6: Given log 10 (x2 y) = 3 and log10 (x/y2 ) = 4. Find log 10 x and log 10 y.


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13. TRIGONOMETRIC FUNCTIONS

Learning Focus : SPM - General Pattern Of Transformations.- Sketch the trigonometricGraph.-

Paper 2 part A Compulsory Questions. -6 - 7 Marks.


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Question 1: Tranformation type 1 VerticalTranslation.

On the same axis, sketch the graph of y = sinx + 2 and y = sin x - 1 for O 0 x 180 0 .


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Question 2: Tranformation type 1 VerticalTranslation.

On the same axis, sketch the graph of y = 2cos x and y = cos x for O 0 x 2 .


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13.1 TRIGONOMETRIC FUNCTIONS Question 3 : Sketching the trigonometric graph

(Clone SPM exam questions) a) Sketch the graph of y = cos 2x for O 0 x 180 0 . b) Hence, by drawing a suitable straight line on the

same axes , find the number of solution satisfying theequation 2 cos x = 2 - x/360 for O x 180 .


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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 calculatorthe value of ,

a) tan 2x b) Cos x/2.


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13.1 TRIGONOMETRIC FUNCTIONS

Question 5: Strategy to provetrigonometric identity.

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

14 LINEAR LAW


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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


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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 theequation y = pk x , where p and k are constants.

Table 1.

a) Plot log 10 y against x by using a scale of 2 cm to 2 units on the x-axisand 2 cm to 0.2 unit on the log 10 y-axis. Hence, draw the line of bestfit. [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

14 LINEAR LAW


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

a) Plot xy against x, 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 5.5 y 5.5 4.7 5.0 6.5 7.7 8.4

14 LINEAR LAW


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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 theequation y = pk x + 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


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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 theequation y = 2kx + 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)


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

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

Solution :

14 2 LINEAR LAW (P 1)


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Question 2: y (8, 16)

( 1, 6) O The diagram shows part of the straight line graph obtained by

plotting y against x 2 . Express y in terms of x . [3 marks] Solution:

14.2 LINEAR LAW (Paper 1)

14 2 LINEAR LAW (P 1)


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Question 2:

x 2

y (k, 44) ( 1, h) O x 2 The diagram shows part of the straight line graph of x 2y against x 2 .

Given that y = 3x + 4/x 2 .Calculate the value of h and k. [3marks]

Solution:

14.2 LINEAR LAW (Paper 1)

15 : INDEX NUMBER


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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 year2004based on the year 2002

Year 2002 Year 2004

A 120 130 x

B 40 y 110

C z 70 125

15 : INDEX NUMBER


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15 : INDEX NUMBER Question 2 : SPM 2005 : Table shows the number of cars sold in theyear

1990 and 1995 for the four different models. Calculatea) 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 25H 25 30 T 40 50M 60 90

15 : INDEX NUMBER


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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 forthe year 2004

based on theyear 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


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16 : Differentiation Question 1 : SPM 2004 : Differentiate 2x(4 - x 2)4 with respect to x .

[3 marks]



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16 : Differentiation Question 2 : SPM 2005 :

Given that f(x) = 1/(2 - 3x) 4 , evaluate f (1) . *4 marks+



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16 : Differentiation Question 3 : SPM 2006 : Given that y = 5v 4/6 ,where v = 4 - 3x. Find dy/dx in terms of x. [3 marks]



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16 : Differentiation Question 4 : SPM 2003 : Given that y = 5x - x 2 ,find the small change in y using the differentiationwhen x increases from to 2.01. [3 marks]



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16 : Differentiation Question 5 : SPM 2006 : Given that y = 5v 4/6 ,

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



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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 :



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

function px 2 - 4x, where p is a constant. The tangent tothe curve at the point (1 , 3) is parallel to the straight liney + x - 5 = 0. Find

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



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16 : Differentiation Question 8: SPM 2007 : A curve with gradient function

2x - 2 /x 2 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


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62226765 Power Point Buku Bengkel Matematik Tambahan Spm (1)

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