Mrsm Maths P2 2004

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MAKTAB RENDAH SAINS MARA

PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 2004 MATEMATIK

Kertas 2 Dua jam tiga puluh minit JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

Calon dibenarkan menjawab keseluruhan atau sebahagian soalan sama ada dalam bahasa Melayu atau bahasa Inggeris. INFORMATION FOR CANDIDATES 1.

This question paper consistes of two sections: Section A and Section B. Answer all questions in Section A and four questions in Section B

2.

Write your answers clearly in the spaces provided in the question paper.

3.

Show your working. It may help you to get marks

4.

If you wish to change your answer, neatly cross out the work you have done. Then write down the new answer.

5.

The diagrams in the questions provided are not drawn to scale unless stated.

6.

The marks allocated for each question and sub-part of a question are shown in brackets.

7.

A list of formulae is provided.

8.

A booklet of four-figure matematical tables is provided.

9.

You may use a non-programmable scientific calculator

10.

This question paper must be handed in at the end of the examination.

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The followi ng formul ae may be helpful in answering the questions. The symbol s giv en are the ones commonly used. RELATIONS

1

am × an = am+n

2

am ÷ an = am-n

3

(am)n = amn

4

A–1 =

5

P(A') =

6

P(A') = 1 – P(A)

7

Distance =

8

Midpoint

1 ad − bc

 d − b   − c a 

n( A) n( S )

(x1 − x2 )2 + ( y1 − y 2 )2

(x, y ) =  x1 + x 2 , y1 + y 2  

2

2



distance travelled time taken

9

Average speed =

10

Mean =

sum of data number of data

11

Mean =

sum of (class mark × frequency) sum of frequencies

12

Pythagoras Theorem c2 = a2 + b2 y 2 − y1 x 2 − x1

13

m=

14

m=–

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y − intercept x − intercept

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SHA PE AND SPACE

1 × sum of parallel sides × height 2

1

Area of trapezium =

2

Circumference of circle = πd = 2πr

3

Area of circle = πr2

4

Curved surface area of cone

= πrs

5

Curved surface area of cylinder

= 2πrh

6

Surface area of sphere

= 4πr2

7

Volume of right prism

= cross sectional area × length

8

Volume of cylinder

= πr2h

9

Volume of cone

=

1 2 πr h 3

10

Volume of sphere

=

4 3 πr 3

11

Volume of right pyramid

=

1 × base area × height 3

12

Sum of interior angles of a polygon

= (n – 2) × 180o

13

arc length angle subtended at centre = circumference of circle 360 o

14

area of sector angle subtended at centre = area of circle 360 o

15

PA' Scale factor, k = PA

16

Area of image = k2 × area of object

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Section A [52 mark s] Answe r all questions in this section., 1

Solve the equation

x – 18 = 4( x + 2 )( x – 3 ).

[4 mark s]

Answe r:

2

Calcul ate the va lues of v a nd w that satisfy the si mult aneous linear equations : 2w − 3v = 4

and

w −

1 v = − 5 3 [4 mark s]

Answe r:

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On the graph provided, shade the region which sa tisfies the three inequalities x + y ≤ 6 , x ≥ 2 and y > 1 [3 mark s ] Answe r:

y 6 4 2 0

4

2

4

6

x

Figure 1 shows a prism with a horizontal re ctangular base ABCD .The uniform cross- section of the prism is an equilateral triangle. R and S are mi d points of AB and C D respectively.

Q P D

A

B

S

C

FIGURE 1

Given that AB = 8 cm and BC = 15 cm, calculate the angle betwee n plane ABQ and pl ane ABP. [4 mark s] Answe r:

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In Figure 2, O is the origin. PQR is a stra ight line and the area of the shaded re gi on is 25 unit 2 .

y

R Q( 4,5 ) P

O

x F IGU RE 2

Find the (a )

coordi nate of P,

(b)

equation of the st raight line PR,

(c )

l engt h OR.

Answe r:

[5 mark s]

(a)

(b)

(c )

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Figure 3 shows a solid cy linder after the shaded cone has been removed. The radi us and height of the cy linder are 7 cm and 18 cm respectively . The ratio of the height of cone and the height of cy linder is 2 : 3.

FIGURE 3 By using π =

22 , calcul ate the total surface area of the remaining solid. 7 [4 mark s]

Answ er:

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In F igure 4, ABZ D and CBWD are quadrants of circles centred at A and C re spectively. The triangle XY Z touches the arc BZ D at Z where Z is the midpoint of arc BZ D .

A

B W

Z

D

Y

X C

FIGURE 4 Given DC = 7 cm, calcul ate the (a ) (b) (c )

length of XZ , peri met er of t he shade d regi on, area of the shaded region.

[Use π =

22 ] 7 [6 mark s]

Answe r:

(a)

(b)

(c )

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9

( a)

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Write the true statement using the quantifiers “all” or “some ” using the object and feature below Object

: proper fraction

Fe ature : value less than 1. (b)

State the converse of the implication be low: If the va lue of θ lies be tween 180° and 270°, then cos θ = – cos (θ - 180°)

(c )

“88, 82, 72, 58. ……… .” is a list of number pattern and can be written as 88 82 72 58

= = = =

90 90 90 90

– – – –

2 2 2 2

× × × ×

1 4 9 16

Make a genera l conclusion by induction regarding the list of number pattern ba sed on the information above. [4 mark s] Answe r: (a)…………… ……………………… ………………………… ………….. …… ……………………… ………………………… …………………… (b)…………… ……………………… ………………………… ………….. ……………………… ………………………… ……………………… .. (c )………………………… ……………………… ………………………. ….………… ………………………… ……………………… …………

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Ta ble 1 shows the number of plates according to colours and letters X and Y in a box. Three plates are drawn from the box at random, one at a time and without re pl acement. Letters on plates Colours Blue Green Red

X

Y

2 3 5 TAB LE 1

4 9 7

Ca l c ul a te t he proba bi l i t y that (a )

the second plate drawn is marked Y ,

(b)

all the three plates are of different colours. [5 mark s]

Answe r::

(a)

(b)

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Figure 5 shows the speed-time graph of a particle in motion for 18s. The distance traveled in the fi rst (t + 4) s is 183 m .

Speed (ms–1) 27

14 7

0

5

t

t+4

18

Time (s)

F IGU RE 5 Calcul ate (a )

the va lue of t ,

(b)

the di stance traveled by the particle in the second ha lf of its mot i o n,

(c)

the rate of change of speed in the last 3 se conds. [6 mark s]

Answe r::

(a)

(b)

(c )

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 1

11

− (a ) The inverse of matrix  2   m

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 1  n

is

 4 − 2   3 −1

.

Find the values of m a nd n . (b) By using matrices , calculate the values of x a nd y that satisfy this simultaneous equation . 4x − 2y = 10 3x − y = 6 [6 mark s] Answe r:

(a)

(b)

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Section B [48 mark s] Answe r fou r questions from this section. (a ) Complete Table 2 below for the values of y for y = 3 + 10x – 2x 3 .

12

Answe r: x y

-3 27

-2 -1

-1

-0.5 -1.75

0 3

0.5

1 11

2 7

3

TABLE 2 [3 mark s] (b)

For this part of the question,use the graph paper provided on page 27 . You may use a flexible curve. By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of y = 3 + 10x – 2x 3 for –3 ≤ x ≤ 3 . [3 mark s]

(c ) Draw a suitabl e straight line on y our graph to determine the values of x that satisfy the equation 2x 3 – 15x = 8 for –3 ≤ x ≤ 3. State the values of x. [4 mark s] Answe r:

…………………… ……………..… . …………… ……………………… ...

d)

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Shade the region defined by the following three inequa lities y ≤ –5x – 5 , y > 3x and y + 2x 3 ≥ 3 + 10x and labe l the re gion “ R ”. [2 mark s]

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13. (a) In Figure 6, ∆ AB′ C′ is the image of ∆ ABC under transformation V and ∆ A′ ′B′′ C′ ′ is the image of ∆ AB′ C′ under transformation W.

y

B′′

9

B′ C B

C′ A

8 7 6 5 4 3 2 1

-8 -7 -6 -5 -4 -3 -2 -1 0

C′′ A′′

1 2 3 4

x

FIGURE 6 Fully (i) (ii) (iii) (b)

descri be transformation V, transformation W . a single transformation which is equivalent to transformation WV

[8 mark s] PQRS is a quadrilateral . (i) In the space provided on page 31, draw the image of PQRS under enlargement with scale factor 2 about the origin. (ii)

If the area of the image obtained in (b) i) is 16 cm 2 , calculate the area of PQRS. [4 marks]

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15 (a)

i)

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-----------------------------------------------------------------------------------------------------------------------------

i i ) ----------------------------------------------------------------------------------------------------------------------------iii) ----------------------------------------------------------------------------------------------------------------------------(b)

i)

y

P O R

S

x

Q

ii)

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Table 3, shows percentage distribution of carbon content in a chemic al substance for 72 samples that are tested. % Carbon 3.1 3.6 4.1 4.6 5.1 5.6 6.1 -

(a)

Cont ent Frequency 3.5 4 4.0 10 4.5 22 5.0 16 5.5 10 6.0 7 6.5 3 TAB LE 3 Calculate the estimate of the mean. Answe r:

(b )

Construct a cumulative frequency table from the data above. Answe r: Class 3.1 3.6 4.1 4.6 5.1 5.6 6.1

(c )

-

Upper class boundary

Cumulative Frequenc y

3.5 4.0 4.5 5.0 5.5 6.0 6.5

For this part of the question , use the graph paper prov ided on page 35. By using a scale of 2 cm to 0.5% on the x-axis and 2 cm to 10 samples on the y-axis, draw an ogive base d on the data.

(d)

F rom the ogive: (i) find the Median (ii) chemical substance exceeding the third quartile has to be tested again. State the minimum perc entage of carbon content for the chemical substance that has to be tested again. [12 mark s ] Answe r: (i)

……………… ……………………… ……..

(ii) … ……………………… …………………..

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Figure 7 shows a solid joined with a half cylinde r at the plane BC K J. ADH E,ABFE,BC GF and DCGH are vertical planes. EFH is an inclined plane. The re ctangular base ABCD and the semi circular base BC M, lie on a horizontal ta ble. H

G

7 cm

F

K

2 cm

L J

5 cm

D

E

6 cm

2 cm

A

C M

Y

B

FIGURE 7 X Draw a full scale: (a)

plan of this solid,

(b)

elevation of the combined solid on a verticle pl ane parallel to AB as viewed from X,

(c)

elevation of the combined solid on a ve rticle plane parallel t o BC as viewed from Y . [12 mark s] Answer: (a)

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Answe r: (b) and (c)

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P (56 0 N , 45 0 W), Q, R and S are four points on the surface of the earth, such that PQ is a diameter of the eart h. S is located to the north of Q. (a )

State the location of Q.

(b)

Calculate the latitude of R which is located 3900 nautical miles to t he s ou t h o f P.

(c )

The plane X took off at P and flew e ast towards S with a speed of 300 knots. Two hours later , anot her plane Y took off at P and fl ew t ow a rds S using the shortest di stance on the earth’ s surface. C al cul a te t he (i)

distance travelled by plane Y ,

(ii)

speed of plane Y. [12 marks]

Answe r:

(a)

(b)

(c ) (i )

(ii)

END OF QUESTION PAPER

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