SULIT 1 3472/2

3472/2 MatematikTambahanKertas 22 ½ jamMei 2015

SEKOLAH MENENGAH KEBANGSAAN JASIN,JASIN, MELAKA.

PEPERIKSAAN PERTENGAHAN TAHUNSIJIL PELAJARAN MALAYSIA 2015

TINGKATAN 5

MATEMATIK TAMBAHAN

Kertas 2

Dua jam tiga puluh minit

JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU

1. This question paper consists of three sections : Section A, Section B and Section C.

2. Answer all question in Section A , four questions from Section B and two questions from Section C.

3. Give only one answer / solution to each question..

4. Show your working. It may help you to get marks.

5. The diagram 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 on pages 2 to 3.

8.A booklet of four-figure mathematical tables is provided.

9. You may use a non-programmable scientific calculator.

Kertas soalan ini mengandungi 14 halaman bercetak

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The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used.

ALGEBRA

1 x =

2 am an = a m + n

3 am an = a m - n

4 (am) n = a nm

5 loga mn = log am + loga n

6 loga = log am - loga n

7 log a mn = n log a m

8 logab =

9 Tn = a + (n-1)d

10 Sn =

11 Tn = ar n-1

12 Sn = , (r

1)

13 , <1

CALCULUS

1 y = uv ,

2 , ,

3

4 Area under a curve

= dx or

= dy

5 Volume generated

= dx or

= dy

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2

5 A point dividing a segment of a line

( x,y) =

6. Area of triangle =

1 Distance =

2 Midpoint

(x , y) = ,

3

4

GEOM ETRY

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STATISTICS

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3

TRIGONOMETRY

1 Arc length, s = r

2 Area of sector , A = 3 sin 2A + cos 2A = 1

4 sec2A = 1 + tan2A

5 cosec2 A = 1 + cot2 A

6 sin2A = 2 sinAcosA

7 cos 2A = cos2A – sin2 A = 2 cos2A-1 = 1- 2 sin2A

8 tan2A =

9 sin (AB) = sinAcosB cosAsinB

10 cos (AB) = cos AcosB sinAsinB

11 tan (AB) =

12

13 a2 = b2 +c2 - 2bc cosA 14 Area of triangle =

1 =

2 = 3 = =

4 = =

5 M =

6

7 8 9

10 P(AB)=P(A)+P(B)-P(AB)

11 p (X=r) = , p + q = 1

12 Mean , = np

13 SD, 14 z =

SECTION A

[40 marks]

Answer all questions in this section .

1. Given that (5, h) is a solution of the simultaneous equations

h(x – y) = x + y – 1 = hx2 – 11y2.Find

a. the value of hb. the other solution of the equations.

[5 marks]

2. Diagram 1 shows a side elevation of lamppost of a parabolic shape of garden-lamp which can be represented by the quadratic function f(x) = –x2 + 6x – 5. The height of the lamp is k meter from the ground.

a. By using completing the square, express f(x) in the form of f(x) = a(x + p)2 + q b. Hence, state the value of k.c. Given that AB = 60 cm, find the height, in meter, of point A from the

ground.

[5 marks]

3. Given that 7 is the mean of 8, 4, 8, 4, 4, 12, x and y

a. Show that x + y = 16b. Find the mode for the above set of numbers, if

i. x = y

4

60 cmA

Ground

y

0 x

k

B

Diagram 1

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ii. x ≠ y[4 marks]

4. Diagram 2 shows a trapezium ABCD. It is given that

a. Express and in term of x and y [3 marks]

b. Hence, show that A, F and C are collinear. [2 marks]

5. An Oil Drilling Company produced 18000 oil barrels per month starting January 2012. The company increases 500 oil barrels productions for each of following month, compare to the previous month.

Calculate

a. the production of oil barrels in the month of August 2012.[2 marks]

b. total oil barrels productions from January 2012 until January 2013.[2 marks]

c. the value of n such that the total production until the n th month exceed

500,000 oil barrels.[3 marks]

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A D

F

BC

E

•

•

Diagram 2

5

6. Solution to this question by scale drawing will not be accepted.

In Diagram 3, and the equation of straight line BC is 2y + x + 6 = 0.

a. Findi. the equation of the straight line ABii. the coordinates of B

[4 marks]

b. A point P moves such that its distance from point A is always 5 units. Find the equation of the locus of P

[2 marks]

7. Sketch the graph y = 3 sin 2x + 2 for the domain 0 ≤ x ≤ 2 . [3 marks]

a. Hence, use your graph to determine

i. the range of y corresponding to the given domain.ii. the number of complete cycle and the period of the graph.

[2 marks]

b. Using the same axes, draw a suitable straight line to find the number of solutions to the equation

for .

State the number of solutions.[3 marks]

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2y + x + 6 = 0

A(–4, 9)

B

0

C

x

y

Diagram 3

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SECTION B

[40 marks]

Answer four questions from this section.

8. a. If log 8 x = p, state log 2 x in term of p [2 marks]

b. Find the value of p if 5 log p 6 – log p 96 = 4 [3 marks]

c. Given that log k (xy) = 3 and log k (x2 y3) = 4.Find

i. the value of log k x and of log k y.

ii. the value of x and of y when k = 2[5 marks]

9. Use graph paper to answer this question.

A decaying radioactive substance has an initial mass of 200 milligrams (mg). At time t days later, the mass of the radioactive substance has been reduced to y mg, which is given by the formula y = y0ekt . Show that y0 = 200.Table 1 shows the values of y measured at difference subsequent times t.

a. Plot log 10 y against t, using a scale of 2 cm to 1 unit on the t-axis and 2 cm to

0.5 unit on the log 10 y-axis. Hence, draw the line of best fit.[4 marks]

b. Given that log10 e = 0.4343, use your graph in (a) to find the value of

i. kii. the number of days taken for the radioactive substance to

reduce by half.

[6 marks]

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t (days) 0 1 2 4 6 7

y (mg) 200 73.58

27.1 3.66 0.50 0.18

Table 1

7

10. Diagram 4, shows a semicircle ABC with centre O. Given that AOB is 1.2 radian, the length of OA is 7 cm and OP is perpendicular to CB.

Find,

a. BOC in degrees and minutes, [1 mark]

b. the length, in cm, of OP. [2 marks]

c, the perimeter of the shaded region. [4 marks]

d. the area of the shaded region. [3 marks]

11. Azmi bought x number of T-Shirts for RM 704. He sold 20 of them at a profit of RM 6 each.

a. Write down an expression, in term of x, for the selling price of each T-Shirt.

[2 marks]

b. Azmi sold the remaining T-Shirts at RM 30 each. Write down, in term of x, the total amount of money he received for all the T-Shirts.

[3 marks]

c. Given that Azmi received RM 920 altogether, form an equation in x and show that it reduces to 3x2 – 140x + 1408 = 0

[2 marks]

d. Solve the equation and hence find the number of T-Shirts Azmi bought.

[3 marks]

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7 cm

1.2 radian

OA

B

C

Diagram 4

P

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12. Diagram 5 shows the straight line 2y + x = 9 intersecting the curve y = x 2 + 3

at the point A. k is x-intercept of the straight line 2y + x = 9. The straight AB

parallel to the x-axis.

a. State the value of k [1 mark]

b. Find the coordinates of A and B [3 marks]

c. Find the area of the shaded region. [3 marks]

d. Find the volume generated, in terms of , when the region bounded by the y-axis, the curve y = x2 + 3 and straight line AB is revolved through 3600 about the y-axis.

[3 marks]

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Diagram 5

k

A

xO

y = x2 + 3

y

2y + x = 9

.

B

9

SECTION C

[20 marks]

Answer two questions from this section.

13. Table 2 shows the price indices in the year 2014 based to the year 2012, of four different materials P, Q, R and S, in the production of a type of a cordial. It also includes the division of the usages of the materials in the production of the cordial.

MaterialPrice Index,

I14/12

Weightage

P 125 4

Q 120 w

R 80 5

S 150 6

a. If the price of material P is RM 25 in the year 2014, calculate its price in 2012.

[2 marks]

b. If the composite index for the year 2014 based to the year 2012 is 120, find the value of w. [3 marks]

c. Find the price of the cordial in 2014 if its price in 2012 was RM 16.50[2 marks]

d. Given that the price of cordial is estimated to increase by 15 % from the year 2014 to 2015. Calculate the composite index of the cordial in the year 2015, based on the year 2012.

[3 marks]

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Table 2

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14. The Table 3 shows the prices and its indices of four ingredients for making a cake. The Diagram 6 shows a pie chart represent the relative quantity of materials used.

Ingredients

Price per kg (RM) Price Index for 2014 based on

2011Year 2011

Year 2014

P 0.80 1.00 x

Q 2.00 y 140

R 0.40 0.60 150

S z 0.40 80

a. Find the value of x, y and z [2 marks]

b. i. Calculate the composite index of the ingredients or making biscuit in

2014 based on the year 2011. [3 marks]

ii. Hence, calculate the corresponding cost of making these biscuits in the

year 2011 if the cost in the year 2014 was RM 2985.[2 marks]

c. The cost of making these biscuits is expected to increase by 50% from the year

2014 to the year 2015. Find the expected composite index for year 2015 based on the year 2011.

[3 marks]

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Table 3

Q1200

P

S650

R1000

Diagram 6

11

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15. Solutions by scale drawing is not accepted.

Diagram 7 shows a rectangle ABCD such that , AB = 9 cm, AC = 5.2 cm and CD = 8.3 cm. It is given that

a. Calculate

i. [2 marks]

ii. the length, in cm, of AD [3 marks]

iii the area of rectangle ABCD. [3 marks]

b. Point C’ lies on BC such that AC’ = AC

i. Sketch the triangle ABC’

ii. State [2 marks]

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B

340

9 cm

C

A

Diagram 7

D

8.3 cm

5.2 cm

13

16. a.

The Diagram 8 shows a cuboid PQRSTUVW which stands on its horizontal base. M is the midpoint of PQ.Find

i. the length WMii.iii. the area of ∆WRM

[5 marks]b.

Diagram 9 shows a trapezium KLMN. KN is parallel to LM. KN = 12.5 cm, LM = 5.6 cm, KM = 14 cm and .Find

i. ii. the length, in cm, of MN

[5 marks]

END OF QUESTION PAPER

Prepared by

Hj. Busrah Bin Md. SehSMKJ / Mei. 2015

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N

M

L

K

12.5 cm

14 cm

1000 5.6 cm

Diagram 9

M

S R

QP

V

8 cm

3 cm

T U

W

6 cm

Diagram 8

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SEKOLAH MENENGAH KEBANGSAAN JASIN, MELAKAPEPERIKSAAN PERTENGAHAN TAHUN

SPM TINGKATAN 5, 2015JADUAL SPESIFIKASI UJIAN MATEMATIK TAMBAHAN

(PAPER 2)

Bidang Topik

Tkn

.

KB

AT

Section A

Section B

Section C

No

Mark

s

No

Mark

s

No

Mark

s

Algebra

Quadratic Equations 4 / 11 10

Quadratic Functions 4 / 2 5

Simultaneous Equations 4 / 1 5

Indices and Logarithms 4 8 10

Progressions 5 / 5 4

Linear Law 5 9 10

GeometryCoordinate Geometry 4 6 6

Vectors 5 4 5

Statistics Statistics 4 3 4

Trigo-

nometry

Circular Measure 4 10 10

Trigonometric Functions 5 / 7 8

Calculus Integration 4 12 10

Technical Solution of Triangles 4 /

15 10

16 10

Social Science

Index Number 4 /

13 10

14 10

TOTAL 7 40 5 50 4 40

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15

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