revision 2012 maths

27 SOALAN ULANGKAJI SPM 2012

MathematicsAnalysis

[1449/1][1449/2]

TOPIC PAPER 1 PAPER 207 08 09 10 11 07 08 09 10 11

FORM

1-3

1 Polygon I, II 6, 7 7 6, 7 7 72 Transformations I, II 9, 10 9, 10 9, 10 9,10 9,103 Trigonometry I 114 Algebraic Expressions I, II, III 19 19 195 Algebraic Formulae 21 21 20 20 216 Algebraic Fractions 20 20 21 21 207 Linear Equations 22 22 22 22 22 2 4 2 2 28 Indices 23, 24 23 23, 24 23,24 23,249 Linear Inequalities 25 24 25 25 25 3

10 Graph of Functions I11 Solid Geometry I, II, III 6 5 11 4 712 Circles I, II 7 8 6 7 9

13 Statistics I, II 27 25,26, 27 26, 27 26,27 26,27

FORM

4

1 Standard Form 1, 2, 3 1, 2, 3, 4 1, 2, 3 1,2,3,

41,2,3,

42 Quadratic Expressions and Equations 19 19 1 3 3 3 3

3 Sets 29,30, 31

29,30, 31

29,30, 31

29,30, 31

29,30, 31 1 1 1

4 Mathematics Reasoning 8 6 7 5 55 The Straight Lines 32,33 32,33 32,33 32,33 32,33 5 10 5 6 66 Statistics III 26 14 14 16 14 147 Probability I 34,35 34,35 34,35 98 Circles III 8 8 8 8 8

9 Trigonometry II 11,12, 13 12, 13 11,12,

1311,12,

1311,12,

13

10 Angles of Elevation and Depression 15,16 15 15,16 15,16 14,15, 16

11 Lines and Planes in 3-Dimensions 14 14 14 14 4 2 4 11 4

FORM

5

1 Number Bases 4,5 5,6 4,5 5,6 5,62 Graph of Function II 28 28 28 28 28 12 13 1, 12 12 123 Transformations III 13 12 15 13 134 Matrices 39,40 39,40 39,40 39,40 29,40 11 11 9 8 8

5 Variations 36,37, 40

36,37, 40

36,37, 40

36,37, 40

36,37, 38

6 Gradient and the Area under a Graph 10 9 10 9 117 Probability II 34, 35 34, 35 7 8 10 108 Bearing 16 17 16 17 179 Earth as a Sphere 17,18 18 17,18 18 18 15 15 13 15 16

10 Plans and Elevations 16 16 16 16 15Total 40 40 40 40 40 16 16 16 16 16

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Mathematic Paper 1 [1449/1]

Paper 1 : Answer all questions 1 Round off 0.03967 correct to two significant figures. A 0.03 B 0.039 C 0.04 D 0.040

2 Express 98700 in standard form A 9.87 × 10-4 B 9.87 × 104 C 9.87 × 10-2

D 98.7 × 104

3 530000 + 6.1 × 10 4 = A 5.91 × 10 4 B 1.14 × 10 5 C 5.91 × 10 5

D 11.4 × 10 9

4 101012 – 10012 = A 10002 B 11002 C 10102 D 11102 5 What is the value of digit 3 in base five in the number 78 35410 ? A 1200 B 2020 C 2220 D 2200

6 In Diagram 6, PQRSTU is a regular hexagon. RQM is a straight line.

Diagram 6 The value of x + y =

A 90 C 100 B 94 D 110

7 In Diagram 7, PQRSUV is a regular hexagon PUT and RST are straight lines.

Find the value of x + y

A 170o B 200o C 180o

D 190o

8 Diagram 8 shows a circle with centre O and TPQ as a tangent to the circle at P. Given that SRQ is a straight line.

Find the value of x. A 20o B 50o C 30o

D 40o

xo

V

U

T

S R

Q

P

yo

Diagram 7

T xo

yo

50o

Y

R

Q

M

S


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9 Diagram 9 shows five quadrilaterals.Which is the image of P under a rotation of 90 anticlockwise about centre (4,0)?

10 Diagram 10 shows two triangles drawn on a square grid.

Triangle II is the image of triangle I under a translation .

Diagram 10

State the translation

A B C D

⎟⎟⎠

⎞⎜⎜⎝

⎛

kh

⎟⎟⎠

⎞⎜⎜⎝

⎛

kh

⎟⎟⎠

⎞⎜⎜⎝

⎛−

34

⎟⎟⎠

⎞⎜⎜⎝

⎛−

43

⎟⎟⎠

⎞⎜⎜⎝

⎛

− 34

⎟⎟⎠

⎞⎜⎜⎝

⎛

− 43

11 In Diagram 11, PTQ is a straight line. Given sin .

Find the value of sin STQ .

A B

C

D

12 Given sin θ = - 0∙3201 and 90° ≤ θ ≤ 270°. Find the value of θ.

A 108° 41´ B 251° 19´ C 161° 19´ D 198° 40´

13 Diagram 13 shows a cuboid with base PQRS.

Name the angle between line LP and the plane KLRQ.

A

B

C

D

135

=∠QPR

∠

54

−

53

−

53

54

M L

P

J S R

Q

K

PLQ∠

LPR∠

PLR∠

LPQ∠

Diagram 13

Q

T

R

P S

5 cm

3 cm

8 cm

DIAGRAM 11 Diagram 11



14 Diagram 14 shows a cuboid with a horizontal base PQRS.

State the angle between the planes WQV and WSRV.

A B C D

15 Diagram 15 shows the position of points P, Q and R on a map. Given that bearing of R from P is 080°.

Find the bearing of Q from R

A 060° C 240° B 300° D 270° 16 In Diagram 16, N and S are the North and South Poles

respectively. The latitude of P is 40° N and PM = MS.

Find latitude of M.

A 25° S C 35° S B 30° S D 40° S

W V

U T

S R

Q P

Diagram 14

5 cm

12 cm

13 cm

Diagram 14

Diagram 15

QVR∠

WQP∠

WQS∠

VQR∠

140°

Q

P

R

DIAGRAM 15

P

N

S

40°N

M

Diagram 16

17 In Diagram 17, N is the North Pole, S is the South Pole, NOS is the axis of the earth.

The position of point P is

A (60o N, 80o W) B (60o N, 100o W) C (80o N, 60o E) D (80o N, 100o E)

18 Diagram 18 shows two vertical poles, JK and LM, on a horizontal plane

The angle of elevation of vertex L from vertex J is 48°. Calculate the height, in m, of the pole JK.

A 13.51 C 16.65 B 16.49 D 13.34

19

A C B D

20 Express as a single fraction in its simplest form.

A C

B D

21 Given that , then p in terms of w is

A C

B D

600

1000

S

O

P

Diagram 17

N

Greenwich Meridian

Equator

Diagram 17

Diagram 18

J

K

L

M

30 m

15 m

( ) =+− pqqp 2 22 qpqp ++22 qpqp +−

22 3 qpqp ++ 22 qpqp −−

mnnm

mm +−

+5

5

551 m−

nn5

5−

mnmmnn

5510 −+

mnmmn

55−

2251 pw −

=+

223 wp −

=

27 wp −

=

23 wp −

=

245 wp −

=



22 Solve the equation + 3 = k – 2.

A C 3 B 3 D 7

23 Simplify

A n3 C p5 n3

B n5 D p5 n5

24 Simplify

A 4p C 3p B 4p2 D 3p2

25 Find the solution of the simultaneous inequalities and

A C B D

26 The frequency distribution shows the scores obtained by a group of pupils in a game.

If 2 is the modal score, the minimum value of k is

A 11 C 9 B 10 D 6

27

The pie chart shows the hobbies of a group of teenagers. The number of teenagers whose hobby is drawing is 50% of the number whose hobby is swimming. Calculate the ratio of the number of teenagers whose hobby is swimming to the number whose hobby is fishing

A 2 : 1 C 3 : 1 B 2 : 3 D 4 : 3

28 It is given that the universal set, = P Q R, set P = { H, A, R, M, O, N, I }, set Q = { M, U, R, N, I } and

set R = { M, U, K, A}. The elements of the set P Q R =

A { A, H, I, K, M, N, O, R, U } B { A, I, K, M, N, R, U } C { A, I, K, M, N, O, R, U } D { A, H, I, K, M, N, U }

72k

71

43

21

( )12

6

31

−÷×⎟⎟⎠

⎞⎜⎜⎝

⎛pnnpn

( ) ( )31 42 2p p p− −× ÷

231

≤x1161 −<− x

2≥x 62 ≤< x

6<x 62 <≤ x

Score 1 2 3 4 5 6Frequency 6 k 9 9 8 6

Swimming

Reading Fishing

144o 72o

Drawing

29 Diagram 29 shows a sketch of a function

What is the equation of the function?

A y = x2 – 3 C y = – x2 – 3 B y = – x3 – 3 D y = x3 – 3

30 The diagram shows the number of elements in sets P, Q and R.

Find

A 4 C 6 B 11 D 8

31 The diagram is a Venn Diagram with the universal set Set M ={Pupils who read

Malay newspapers} and Set E = {Pupils who read English newspapers}.

Given that n(M) = 80, n(R) = 104 , and the number of pupils who do not read any newspaper is

4, find the total number of pupils in Year Five.

A 172 C 200 B 176 D 196

Diagram 29

y

x

O

– 3

ξ ∪ ∪

∪ ∪

P

ξ Q

R

3 2

1

5 4

3

)'( RPn ∪

{Pupils in Year Five},ξ =

12)( =∩ RMn



32 The gradient of the straight line

A B C -7 D -2

33 In the diagram, PQ is a straight line

Find the gradient of PQ

A -2 C 2 B D

34 A jar contains 240 orange, lychee and coffee flavoured sweets. 90 of them were orange-flavoured sweets. A sweet is picked at random from the jar. The probability of picking a lychee-flavoured sweet is . Calculate the number of coffee-flavoured sweets.

A 70 C 10 B 80 D 50

35 A bag contains blue, red, green and yellow buttons. There are 2 blue buttons and 8 green buttons in the bag. A button is picked at random from the bag. The probability of picking a blue button is and the probability of picking a red button

is . Find the probability of picking a yellow button.

A C B D

1472 =− yx

72

27

−

12 1

2−

13

241

13

61

85

2411

2413

36 The relation between p, n and r is p . It is given that

p = 4 when n = 8 and r = 6. Calculate the value of p when n = 64 and r = 3

A 16 C 48 B 24 D 32

37 It is given that p varies inversely with w and p=6 when w = 2. Express p in terms of w.

A p = C p = B p = 3w D p = 12w

38 P varies directly with the square of R and inversely with Q. It is given that P = 2 when Q= 3 and R = 4. Express P in terms of R and Q.

A P = C P =

B P = D P =

39 Given , find matrix M.

A C

B D

40 Given , find the value of x

A 9 C -3 B 4 D -6

w3 w

12

QR

83 2

2332RQ

RQ

34

QR3

1 72

3 7M

−⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

45⎛ ⎞⎜ ⎟⎝ ⎠

42−⎛ ⎞⎜ ⎟−⎝ ⎠

42⎛ ⎞⎜ ⎟⎝ ⎠

45−⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( )181

3 =⎟⎟⎠

⎞⎜⎜⎝

⎛

−

xx

END OF QUESTION PAPER

∝ rn3



Mathematic Paper 2 [1449/2]

Section A[52 marks]

Answer all questions in this section.

1 On the graph provided, shade the region which satisfies these three inequaltities.

[ 3 marks]

2 Solve the following quadratic equation [ 4 marks]

Answer :

3 Calculate the value of k and n that satisfy the following simultaneous equations. [ 4 marks]

Answer:

2,2xy ≤ + 2 and 2.y x x≥ − <

2 9 13 3( 2)x x x+ − = −

3 4 162 8

k nk n− = −

+ =

y

2 x

y = 2 -x

2



4 Diagram 4 shows a solid cone with radius 9 cm and height 14 cm. A cylinder with radius 3 cm and height 7 cm is taken out of the solid. Calculate the volume, in cm3, of the remaining solid.

[ 4 marks]

Use .

5 In diagram 5, QRS and UT are arcs of two circles, centre P and S respectively.

Diagram 5 It is given that PUS is a straight line, PQ = 21 cm and US = 14 cm. Using , calculate a) the area, in cm2, of the shaded region b) the perimeter in cm, of the shaded region. [6 marks]

Diagram 4

⎟⎠

⎞⎜⎝

⎛ =722

π

722 =π



6 Diagram 6 shows a right prism with rectangle ABCD as its horizontal base. Right angled triangle FAB is the uniform cross-section of the prism. The rectangular surface BCEF is inclined.

Calculate the angle between the plane ABE and the base ABCD. [3 marks]

Diagram 6


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7 a) Determine whether the following sentence is a statement. “ 7 is not a factor of 40 “ b) Write two implications from the following statement. “ P R if and only if P R = P ” c) Make a general conclusion by induction for the following number pattern. 5 = 4(1) + 13

16 = 4(2) + 23

37 = 4(3) + 33 80 = 4(4) + 43

……………… [5 marks]



8 In Diagram 8, PQRS is parallelogram and O is the origin . It is given that the equation of the straight line PQ is 2y = x + 18.

Diagram 8

Find a) the equation of the straight line RS, b) the x-intercept of the straight line RS.

Answer :

y

x R( 7, 2)

S

Q

P

O0

[ 5 marks ]



9 Given that the inverse matrix of

a) Find the value of m and n b) Hence, using matrices, solve the following equation: [7 marks]

Answer :

,2131 is

3212

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎟⎟⎠

⎞⎜⎜⎝

⎛

−− nm

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

−− 17

3212

ed



10 Diagram 10 shows the speed-time graph of a particle for a period of 10 seconds. Given that the total distance travelled in the first 6 seconds is twice of the total distance travelled in the last 4 seconds.

a) Calculate the value of v. b) Calculate the rate of change of speed, in ms-2, in the first 3 seconds.

Answer :

11 Two students are chosen at random to arrange books in the library. They are chosen from a group of 2 boys and 3 girls. List the sample space, hence, find the probability that a) two girls are chosen, b) at least one boy is chosen.

[ 5 marks ]

[ 6 marks ]

DIAGRAM 5

Speed (ms 1− )

Time (s)

v

4

0 6 10 Diagram 10



Section B[48 marks]

Answer any four questions in this section.

12 a) Complete the table in the answer space for the equation . [2 marks] b) For this part of the question, use the graph paper provided on page 41. You may use a flexible curve rule. 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 for −4.5 ≤ x ≤ 4.5. [4 marks]

c) From your graph, find (i) the value of x when y = 7, (ii) the value of y when x = 1.5. [2 marks] d) Draw a suitable straight line on your graph to find the positive values of x which satisfy the equation for – 4.5 ≤ x ≤ 4.5. State the values of x. [4 marks] Answer : a)

b) Refer graph

c) (i) x = ______________________________________ (ii) y = ______________________________________

d) _____________________________________________

15yx

= −

15yx

= −

23 5 15x x= +

x -4.5 -3 -1.5 -1 -0.5 0.5 1 2.5 3 4.5y 3.3 5.0 15.0 30.0 -30.0 -15.0 -5.0 -3.3



Graph for Question 12



13 a) Transformation U is a translation Transformation V is reflection on the line y = 1 State the coordinates of the image of the point (3, 4) under the following transformations : (i) U, (ii) UV. b) Diagram 13 shows three quadrilaterals, ABCD, HGFE dan KJIE on a Cartesian plane.

HGFE is an image of ABCD under transformation W and KJIE is an image of HGFE under transformation Z. (i) Describe completely, (a) transformation W, (b) transformation Z.

(ii) Given that trapezium HGFE represents an region of area 26m2, calculate the area represented by hexagon KJIFGH.

⎟⎠

⎞⎜⎝

⎛−

23

[3 marks]

[9 marks]

Diagram 13



14 The data below shows the Chemistry test marks scored by a group of 40 students in Form 4 Wawasan.

68 40 65 66 60 46 53 56 49 5853 57 73 53 69 51 56 64 58 6247 65 67 66 67 36 56 53 38 3762 61 37 74 48 41 75 63 69 55

a) Using the given data and a class interval of 10, complete Table 2 in the answer space.

b) From the table in (a), i) state the modal class, ii) calculate the mean mark. c) For this part of the question, use the graph paper. By using a scale of 2 cm to 10 marks on the horizontal axis and 2 cm to 2

students on the vertical axis, draw a frequency polygon for the data. Answer:

a)

TABLE 2 (b) (i)

(ii)

[4 marks]

[4 marks]

[4 marks]

Marks scored Midpoint Frequency 20 – 29 30 – 39



Graph for Question 14



15a) Diagram 15(i) shows a solid right prism with rectangular base ABCD on a horizontal table. The surface ABLKGF is the uniform

cross section of the prism. Rectangle EFGH is an inclined plane and rectangle JKLM is a horizontal plane. AF, GK and BL are vertical edges and GK = 2 cm.

Draw full scale, the elevation of the solid on a vertical plane parallel to AB as viewed from X.

[ 3 marks ]

b) Another solid right prism is joined to the prism in Diagram 15(i) at the vertical plane CVJQ. The combined solid is as shown in Diagram 15(ii). Quadrilateral PQCW is the uniform cross section of the prism. rectangle WCRU is an inclined plane and the square TPWU is a horizontal plane. The base ABRSQD lies on a horizontal table, KL = JM = 3 cm and GK = 2 cm.

Draw full scale, (i) the plan of the combined solid (ii) the elevation of the combined solid on a vertical plane parallel to BCR as viewed from Y

Diagram 15 (i)

Diagram 15 (ii)

[5 marks][4 marks]


SOALAN ULANGKAJI SPM 201246 SOALAN ULANGKAJI SPM 2012

16 P (25ON, 60OE ), Q and R are three points on the surface of the earth. PR is the diameter of the earth.

a) State the longitude of R. [2 marks]

b) PQ is the diameter of the parallel of latitude 25ON .

(i) State the position of Q .

(ii) Calculate the shortest distance, in nautical mile, from P to Q measured along the surface of the earth. [4 marks]

c) An aeroplane took off from P and flew due west to Q along the common parallel of latitude and then flew due south to R. Calculate (i) the distance, in nautical miles, from P to Q measured along the common parallel of latitude. (ii) the time taken, in hours, for the whole flight if the average speed of the whole flight is 650 knots.

[6 marks]


SOALAN ULANGKAJI SPM 2012

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revision 2012 maths

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