form 5 midyear paper 2 2013

7/27/2019 Form 5 Midyear Paper 2 2013

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1

Rumus-rumus berikut boleh membantu anda menjawab soalan. Simbol-simbol yang diberi adalah yang biasa

digunakan.

ALGEBRA

2 41. 5. log log log 9. 1

2

2. 6. log = log log 10. 2 12

3.

a a a n

m n m n

a a a n

m n m n

b b acx mn m n T a n d

a

m na a a m n S a n d n

a a a

1

1 17. log log 11. , 1

1 1

log4. 8. log 12.

log

n n

n

a a n

nm mn nc

a

c

a r a r m n m S r

r r

ba a b Tn ar

a

13. , 11

aS r

r

KALKULUS (CALCULUS)

1. , 4. Luas di bawah lengkung (Area under a curve)

= atau (or)

dy dv duy uv u v

dx dx dx

ydx xdy

b

a

2

2 2

2. , 5. Isipadu janaan (Volume generated)

3. = atau (or)

b

a

b

a

du dvv u

u dy dx dxyv dx v

dy dy duy dx x dy

dx du dx

b

a

STATISTIK (STATISTICS)

n11. x 6. 100 11. ( ) , 1

2. x 7.

r n r

r

o

i i

i

x QI P X r C p q p q

N Q

fx W II

f W

2 2

2 n

2 2

2 n

12. Min (Mean),

x !3. x 8. 13.

!

x !4. x 9. 14.

! !

1

25.

r

r

m

np

x x nP npq

N N n r

f x fx n XC Z

f f n r r

N F

m Lf

10. ( ) ( ) ( )C P A B P A P B P A B

SULIT 3472/2


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2

GEOMETRI (GEOMETRY)

2 2

2 1 2 1

1 2 2 3 3 1 2 1 3 2 1 3

1. Jarak (Distance) 4. Luas segitiga (Area of triangle)

12. Titik tengah (Midpoint)

2

x x y y

x y x y x y x y x y x y

2 21 2 1 2

2 2

, , 5.2 2

3. Titik yang membahagi suatu tembereng garis 6.

(A point dividing a segment of a

x x y yx y r x y

xi yjr

x y

1 2 1 2

line)

, ,nx mx ny my

x ym n m n

TRIGONOMETRI (TRIGONOMETRY)

2

1. Panjang lengkok, 8. sin sin sin

Arc length, sin sin cos cos sin

12. Luas sektor,

2

s j A B AkosB kosA B

s r A B A B A B

L j

2

2 2

9. kos sin sin

1Area of sector, cos cos cos sin sin

2

3. sin 1

A B kosAkosB A B

A r A B A B A B

A kos A

2 2

2

2 2

2 2

tan tan10. tan

1 tan tan

sin cos 12tan

11. tan21 tan

4. sek 1 tan

sec 1 tan

A BA B

A B

A AA

AA

A A

A A

2 2 2 2 2

2 2 2 2

12.sin sin sin

5. kosek 1 13. 2

cosec 1 cot

a b c

A B C

A kot A a b c bc kosA

A A a b c

2

2 2

2 cos

6. sin2 2sin 14. Luas segitiga (Area of triangle)

1sin2 2sin cos = sin

2

7. kos2 sin

bc A

A AkosA

A A A ab C

A kos A A

2

2

2 2

2

2

2 1

1 2sin

cos2 cos sin

2cos 1

1 2sin

kos A

A

A A A

A

A


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3

Section A

[40 marks]

Answer all questions

1. Solve the following simultaneous equations:2 2 7

2 5

x y xy

x y

[5 marks]

2. a) The quadratic equation 2 6 7 2 3x x m x has two equal roots. Find the possible valuesofm. [3 marks]

b) Hence, determine the stationary point and determine the axis of symmetry for the above

equation. [3 marks]

3. A closed rectangular box is made of very thin sheet metal, and its length is three times its width. If

the volume of the box is 288 cm3, show that its surface area is equal to 2

7686x

x cm2, where x cm is

the width of the box. [3 marks]Find by differentiation the dimension of the box of least surface area. [3 marks]

4. A set of data which consists of 15 numbers has a mean of 12 and a standard deviation of 3.

a) For the set of data, find

i) the sum of the numbers,

ii) the sum of squares of the numbers [3 marks]

b) Another set of data which consists of 5 numbers with a mean of 11 and a variance of 8 is

added to the original set of data. For the combined set of data, find

i) the new mean

ii) the new standard deviation [5 marks]

5. Diagram 1 shows a triangle OXY. The straight line AYintersects the straight line XB at C. It is given

that1

, , and3

OX x OY y OA OX OB BY

a) Express each of the following vectors in terms of andx y

i) AB

ii) BX

iii) AY

[5 marks]

b) Given that andBC hBX AC k AY

, find the

value ofh and ofk. [4 marks]

6. The histogram below shows the marks obtained by a Form 5 class

of students in an Additional Mathematics test.

a) Without drawing an ogive, calculate the median mark [3 marks]

b) Calculate the standard deviation of the marks distribution. [3 marks]

X

O

A

BC

Y

Diagram 1

5

7

12

9

21


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4

O SQ

B

AP

R

1 rad

Diagram 3

Section B

[40 marks]

Answer any four questions from this section.

7. Use the graph paper to answer this question

Table 1 shows the values of two variables, x and y, obtained from an experiment

The variables x and y are related by the equation2

1pxy

qx

, where p and q are constants.

a) Based ontable 1, construct a suitable table for the values ofx2y [1 mark]

b) Plot x2y against x, using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 0.5 unit on the x

2y - axis

Hence,draw the line of best fit. [3 marks]

c) Use the graph drawn to give the best estimated value of

i) y when x = 2.5ii) p

iii) q [6 marks]

8. Diagram 2 shows an equation ( 4)y x x , the x-axis, the straight line y = 5 and the straight line AB

Find

a) The turning point of the curve ( 4)y x x [2 marks]

b) Determine the axis of symmetry of the curve. [1 mark]

c) The equations of normal at point A and point B [3 marks]

d) Hence or otherwise, determine the point of intersection, D of the normal at point A and normal at the

point B [3 marks]

e) What can you say about the position of point D, the midpoint ofAB and the turning point of the

curve ( 4)y x x ? [1 mark]

9. Diagram 3 shows two arcs, PQ and RS, of two concentric circles, with the same centre O. RQ is

perpendicular to OS.

Given that OP = OQ = 5 cm and 1 radianPOQ ,

find

a) the perimeter of the shaded region A, [7 marks]

b) the area of the shaded region B. [3 marks]

x 1 2 3 4 5 6

y 2.601 0.551 0.194 0.089 0.040 0.017 Table 1

( 4)y x x

5y

Diagram 2

A B


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10. Solutions by scale drawing will not be accepted for this question.

Diagram 4 shows that the straight lines AB and BCare perpendicular to each other. The equation of

the straight line BCis 2 6x y

a) Find

i) the equation of the straight line AB

giving your answer in the general form,

ii) the coordinates of point B. [4 marks]

b) The straight line AB is extended to point D such

that AB : BD = 2 : 3. Calculate the area of triangle

ADO. [3 marks]

c) A point Mmoves such that the angle AMB is always

a right-angle. Find the equation of the locus ofM. [3 marks]

11. a) Diagram 5 shows a shaded region bounded by the curve 2y x k , the y-axis and the

straight lines 3 and 3y y . If the area of the shaded region is 30 units2, find the value of

k. [4 marks]

b) Find the ratio of the volumes if the shaded region is rotated about the y-axis for 180o

to that

of the shaded region if it is rotated about the x-axis for 180o. [6 marks]

Section C

[20 marks]

Answer any two questions from this section.

12. The cost to produce a tin of paint depends on the cost of the raw materials, the production cost and

the packaging cost. The table shows the price indices and weightages of those costs.

a) Given that the composite index of the cost to produce a tin of paint for the year 2005 based on the

year 2002 is 117.7, find the value ofx. [3 marks]

b) Given that the price of a tin of paint in the year 2002 was RM 30, calculate its corresponding price in

the year 2005. [1 mark]

c) Find the values of

i) h,ii) k. [6 marks]

Diagram 4

( 9, 4)A

B

C

y

x

O

2 6x y

y

xO

3

3

2y x k

Diagram 5

Cost 2002I (based on

the year 1999)

2005I (based on the

year 1999)

2005I (based on the

year 2002)Weightages, w

Raw materials 175 182 104 x

Production h 200 125 5

Packaging 145 k 120 2


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13. a) If 6 6log 3 and log 5,m n express 2log 45in terms ofm and n. [5 marks]

b) Given log 6, log 8 and log 3,a b abcx x x find the value of logc x [5 marks]

14. Diagram 6shows, AECand BED are straight lines and BE= ED. It is given that AB = 9 cm,

AD = 14 cm, CD = 15 cm, o o115 and 35AED ACD .

Calculate

a) the length of BD, [4 marks]

b) ,BAD [2 marks]

c) the area of the whole diagram. [4 marks]

15. Diagram 7shows a circular cylinder of height h and radius rsurmounted by a hemisphere of the

same radius.

Express the total surface area Sof the object and the total volume Vin terms ofh and r.

[2 marks]

If the total surface area Sis 20, express h in terms ofrand hence show that35

106

rV r

.

[2 marks]

Find the value ofrwhich makes Va maximum and calculate the maximum value ofV, giving your

answer in terms of [6 marks]

End of Questions

Diagram 6

r

h

Diagram 7

form 5 midyear paper 2 2013

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