modul peningkatan prestasi matematik tambahan …sulit 1 3472/2 3472/2 (c) 2016 hak cipta sm zon b...

SULIT 1 3472/2

3472/2 (C) 2016 Hak Cipta SM Zon B Kuching Sarawak [ Lihat halaman sebelah

SULIT

Modul Peningkatan Prestasi Matematik Tambahan (Kertas 2)

SPM 2016 Zon B Kuching Sarawak

Question Solution and Answer Scheme Sub-

Mk

Full

Mk

1( )w f x

( )f w x

1 2w x

1 2x w

12

xw

1 1( ) 1

2f x x

Let

(Shown)

Then

K1

N1

2 (a)

4 2y x 4

2

yx

or

Substitute (1) into (2)

Note:

1. If the solutions of x and y are

matched wrongly, then SS-1

from full marks.

22 3 * 4 2 4 0x x x 2

4 42* 3* 4 0

2 2

y yy

2( 3) ( 3) 4(1)( 1)

2(1)x

2( 2) ( 2) 4(1)( 12)

2(1)y

3.303, 0.303x

2.606, 4.606y

or

or

5

P1

K1

K1

N1

N1

1

SULIT 2 3472/2

3472/2 (C) 2016 Hak Cipta SM Zon B Kuching Sarawak SULIT


Mk

Full

Mk

64a

14

r

23

1π (2) 1 256πR

1 2R

641

14

S

2563

or 1

853

or 85.33 6

K1

N1

K1

N1

K1

K1

3 (a)

(b)

1 1( ) 1

2gf x g x

1 14 1

2 2x

12 1

2x

12 2

2x

12 2

2x

1( ) ( )gh x f x

1 14 ( ) 1

2 2h x x

1 12 2

x

14 ( )

2h x x

1( )

8h x x

or 5

22

x 2.5 2xor

6

K1

N1

K1

N1

2 (b) (i)

(ii)

SULIT 3 3472/2


SULIT


Mk

Full

Mk

d0

d

y

x

23 6 dy x x x 3 23x x c

3 212 (1) 3(1) c

10c

3 23 10y x x

Equation of the curve

At turning point,

23 6 0x x

3 2 0x x

0, 2x x

When

3 2(0) 3(0) 10y

14

0,x 2,x

3 2(2) 3(2) 10y

10

When

2

2

d6 6

d

yx

x Second derivative,

When 0,x 2,x When

2

2

d6(0) 6

d

y

x

6 0

0, 10

2, 14

6 0

2

2

d6(2) 6

d

y

x

is the maximum point.

is the minimum point. 8

K1

N1

K1

N1

K1

K1

K1

N1

4 (a)

(b)

SULIT 4 3472/2



Mk

Full

Mk

Shape of sine graph

Number of solutions = 3

sin sintan tan

cos cosA B

A BA B

sin( )

cos cos

A B

A B

sin cos cos sincos cos

A B A BA B

Amplitude = 2

1.5 cycle for 3π

02

x

22

πy x or equivalent

Correct gradient or correct y -intercept

y

xπ

π2

O 3π2

2sin2y x2

π4

3π4

5π4

2

22

πy x

1

8

K1

P1

N1

K1

P1

K1

P1

P1

5 (a)

(b) (i)

(ii)

SULIT 5 3472/2


SULIT


Mk

Full

Mk

40 2cos

2 60 3

1 2cos 48.19

2 3

1.682 rad

2 48.19 96.38

3.14296.38

180

21 1

60 2 3.142 *1.682 60 60 sin*96.382 2

Area of major sector POR

Area of triangle POR

21

60 2 3.142 *1.6822

160 60 sin*96.38

2

Area of immersed cross-sectional region

210072.45 cm 7 N1

N1

K1

K1

K1

K1

K1 6 (a)

(b)

SULIT 6 3472/2



Mk

Full

Mk

2(5) 1(2) 2(4) 1( 2),

1 2 1 2T

4,2 (shown)

Area of QOR 0 2 5 01

2 0 2 4 0

10 8 0 0 10 0

2

29 unit

1

4 ( 2)2

5 2m

212

m

12 4

2y x

14

2y x

2 2( 4) ( 2) 3x y

2 28 16 4 4 9x x y y

2 2 8 4 11 0x y x y

2

22 1m

10

N1

N1

K1

K1

K1

K1

N1

N1

K1

K1

7 (a)

(b)

(c)

(d)

SULIT 7 3472/2


SULIT


Mk

Full

Mk

2,4B

4,0A

*2

20

( 4) dA x x x

Area under the curve from 0x to 2x

*2 2

04 dx x x

*232

0

23x

x

3

2(*2)2(*2) 0

3

163

Area of the region required by the question

16 1* *2 *4

3 2

43

4 2

2π dy x

Volume generated

4 22

2π 4 dx x x

4 4 3 2

2π 8 16 dx x x x

5 3 5 34 4(4) 16(4) (2) 16(2)

π 2(4) 2(2)5 3 5 3

256π

15 or

117 π

15

45 34

2

16π 2

5 3x x

x

10

K1

K1

N1

N1

N1

K1

K1

K1

K1

N1

8 (a)

(b)

(c)

SULIT 8 3472/2



Mk

Full

Mk

Range 41 50 1 10

2 2

The height of the bars are proportional to respective frequencies

45.5 5.5

40

Midpoints 5.5, 15.5, 25.5, 35.5, 45.5

3(5.5) 8(15.5) 10(25.5) 12(35.5) 7(45.5)

3 8 10 12 7x

Mean,

114028.5

40

Variance,

2 2 2 2 22 23(5.5) 8(15.5) 10(25.5) 12(35.5) 7(45.5)

28.53 8 10 12 7

23813028.5

40

141

New variance *141

Boundaries or midpoints or class intervals are correctly labeled

(bars are of the same width)

Correct way of finding mode

Mode 33.5 10

K1

N1

N1

K1

P1

P1

K1

N1

P1

P1

9 (a)

(b)

(ii)

(i)

(c)

SULIT 9 3472/2


SULIT

0

f

x0.5 10.5 20.5 30.5 40.5 50.5 60.5

2

4

6

8

10

12

14

mode 33.5

SULIT 10 3472/2



Mk

Full

Mk

10n 25

p 35

q or or

( 8) ( 8) ( 9) ( 10)P X P X P X P X

8 2 9 1 10 0

10 10 108 9 10

2 3 2 3 2 35 5 5 5 5 5

C C C

0.01229

0.010616832 0.001572864 0.000104857

50 62( 50)

8P X P Z

1.5P Z

0.9332

1 1.5P Z

1 1.5P Z

1 0.0668

Number of students 0.9332 400

373

( ) 0.1P X a

620.1

8a

P Z

621.281

8a

72.25a

1.281 8 62a

The minimum marks to get grade A, 10

N1

K1

K1

K1

P1

K1

N1

K1

K1

N1

10 (a)

(b)

(ii)

(i)

SULIT 11 3472/2


SULIT


Mk

Full

Mk

1x 2 3 4 5 6 7

10log y 1.0792 1.3802 1.6812 1.9823 2.2833 2.5843

1.075 1.375 1.675 1.975 2.275 2.575

Note:

1. In the table, the decimal number being rounded off must not be less precise

than the nearest 0.025 for Y.

2. The N mark is given for all correct X and Y.

3. If the value(s) of X and Y are not shown through a table, then the N mark is

given for all the correct points on the graph.

At least one *point correct, using correct axes

10log against 1y x in correct direction and orientation,

and with uniform scales

All 6 *points plotted accordingly

Line of best fit

Notes:

1. Graph can only be read with precision until half of the smallest grid

division.

2. Do not accept readings obtained by calculation or from the wrong graph.

3. SS-1 from full marks if not using the given scale.

(or equivalent) 10 10 10log log 1 logy a x b

*0.475

*0.47510a

2.985

10 10log log -intercepta y

Note: 2.985 3.162a

10log b m

0.3

0.310b

1.995

2.7 0.67.4 0.4

10

N1

N1

K1

K1

N1

K1

N1

P1

K1

N1

11 (a)

(b)

(ii)

(i)

(c)

SULIT 12 3472/2


0

10log y

1x

0.475c

1 2 3 4 5 6 7

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.4, 0.6

7.4, 2.7

SULIT 13 3472/2


SULIT


Mk

Full

Mk

ddv

at

1 4t

0 1 4t

14

t

2

max1 1

6 24 4

v

16

8

At maximum velocity, 0a .

or 498

6.125or

0 1 4(0)a

1

At initial acceleration, 0t .

When stopping instantaneously, 0v .

22 6 0t t

2 3 2 0t t

3, 2

2t t

0t

2t 2

0 dv t

22 3

0

26

2 3t t

t

2 3(2) 2(2)

6(2) 02 3

263

Distance from point O

28

3or 8.667or 10

K1

K1

K1

N1

K1

N1

K1

N1

K1

K1

12 (a)

(b)

(c)

SULIT 14 3472/2



Mk

Full

Mk

2 2 27 18 2(7)(18)cos75PR

17.54PR

*17.54 8.5sin110 sin QPR

1 8.5 sin110sin

17.54QPR

27.09

Note: 27.08 27.09QPR

or 27 5'

180 110 *27.09QRP

42.91

Area of PQR 1

(8.5)(17.54)sin 42.912

Area of PRS 1

(7)(18)sin752

Area of quadrilateral PQRS

1 1(8.5)(17.54)sin 42.91 (7)(18)sin75

2 2

50.75386719 60.85332706

111.61

Let the shortest distance from point Q to the line PR be h .

sin 42.918.5h

8.5sin42.91h

5.787

OR 1 1

(17.54) * (8.5)(17.54)sin 42.912 2

h

Note: 111.61 Area 111.63

Note: 5.787 5.788h 10

K1

K1

K1

N1

K1

N1

K1

N1

K1

N1

13 (a)

(b)

(c)

(i)

(ii)

SULIT 15 3472/2


SULIT


Mk

Full

Mk

I: 2y x

1II:

4y x

At least one straight line correct from the *inequalities

involving x and y

Draw all three lines correctly from the *inequalities

Shade the correct region

min 8y

35 25x y k

Objective function

Optimum point

max 35 43 25 11k

RM 1780

43,11

When 30, 7.5 20x y

Note: N0 if answer min 7.5y

Notes: 1. For (c)(i) and (c)(ii), accept the coordinates only if they are integers.

2. SS-1 if

in (a), symbol "<", ">" is used instead of "≤", "≥" or more than three

inequalities are given (except x ≥ 0 or y ≥ 0)

OR

in (b), not using the given scale or axes are reversed or not using graph

paper

2 3 120x y or III: 20 30 1200x y

10

N1

N1

N1

N1

K1

K1

N1

N1

N1

N1

14 (a)

(b)

(c) (i)

(ii)

SULIT 16 3472/2


0

y

x10 20 30 40 50 60 70

5

10

15

20

25

30

35

40 2y x

14

y x

2 3 120x y

R

35 25 875x y

max35 25x y k

43,11

SULIT 17 3472/2


SULIT


Mk

Full

Mk

7.50100

6.00x

125x

150 1004.00

y

6.00y

7.70140 100

z

5.50z

48(125) 24(150) 16(125) 12(140)

48 24 16 12I

13280100

132.8

14Food132.8 100

60.00

14Food RM 79.68

16,10 152.72I

16,10

132.8 115100

I

or equivalent

10

N1

N1

N1

K2

K1

N1

N1

K1

N1

15 (a)

(b)

(c) (i)

(ii)

modul peningkatan prestasi matematik tambahan …sulit 1 3472/2 3472/2 (c) 2016 hak cipta sm zon b...

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