Revision Spm Paper 2 Format July 2009

REVISION Additional Mathematics SPM

Paper 2 Format

1.

July 2009

Kepler’s third law states that the period of a satellite around a planetary body (T) is related to the distance of the satellite from the body (R) by the formula: T = aR b An experiment was conducted on a planet in a galaxy far far away and the following table gives the values of T corresponding to values of R. T(/h)

8

125

353

649

1000

R(km)

40

250

500

750

1000

(a)

Use the data above in order to draw, on graph paper, the straight line graph of lg T against lg R.

(b)

Use your graph to estimate (i) (ii) (iii)

the values of a and b, the value of T when R = 100 km, the value of R when T = 750 h.

[Ans: (b)(i) a = 0.03162, b = 1.5 (ii) 31.62 (iii) 944.1]

2.

The table below shows the experiment results of two related variables, x and y. x

0

100

400

900

1600

2500

y

20

30

40

50

60

70

It is expected that x and y are related by the equation 4k 2 x = ( y − c) 2 , where k and c are constants.

(a)

Draw the graph of y against

(b)

From your graph, find

x.

(i) the value of k and c, (ii) the value of x when y = 55, (iii) the value of y when x = 500. [Ans: (b)(i) k =

1 2

, c = 20 (ii) x = 1225 (iii) y = 42.5]

Sri Bintang Tuition Centre, Kuching

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1

REVISION Additional Mathematics SPM

3.

(a)

Given 7 x − 2,6 − 3 x and x – 2 are the first three terms of a geometric progression. Find (i) (ii)

(b)

the value of x, the sum to infinity of the progression.

Express the recurring decimal 0.272727…..as a single fraction in its lowest terms.

[Ans: (a)(i) x = 8 (ii) 40

1 2

3

(b)

11

4.

]

y y=

2 x

3

y=

1

x

2

A x

O

1 2 which intersects a straight line y = x at x 2 point A. Calculate the volume generated when the shaded region is revolved through 360o about the y-axis.

Diagram above shows a curve y =

[Ans: 4π unit 3 ]

5.

(a)

Find the equation of the tangent to the curve y =

x+3 at the point where x −1

y = 5.

(b)

Find the equation of the normal to the curve y = x −

3 at the point where x

x = 3.

(c)

The volume of a cone decreases at the rate of 20cm 3s −1 . Find the rate of change in the height of the cone if the radius of the base is fixed at 4 cm.

(d)

Given y =

16 dy 16 , find . Hence, find the approximate value of . 2 dx x (3.99 )2

[Ans: (a) y + 4 x − 13 = 0 (b) 4 y + 3 x − 17 = 0 (c) −

Sri Bintang Tuition Centre, Kuching

@biid09

15 4π

cms −1 (d) 1.005]

2

REVISION Additional Mathematics SPM

6.

Q 5 cm

P 70 o

8 cm

R

80 o

S

The diagram shows a quadrilateral PQRS. Given the area of triangle PQS is 15 cm 2 and ∠PQS is obtuse angle. Calculate (a) (b) (c)

the value of ∠PQS , the length, in cm, of PS, the area, in cm2, of triangle QRS. [Ans: (a) 131o 25' (b) 11.91 cm (c) 16.77 cm2]

7.

Building material A B C

Cost Price (RM) 2001 2002 200.00 220.00 230.00 264.50 250.00 262.50

Weightage 5 3 2

The above table shows the cost prices of three types of building materials in the years 2001 and 2002, together with their respective weightages. (a)

Calculate the index number of each of the materials A, B and C for year 2002 taking the year 2001 as the base year.

(b)

Calculate the composite index for the year 2002 taking the year 2001 as the base year.

(c)

The costs of materials A, B and C increased by 5%, 10% and 20% respectively from the year 2002 to year 2003. Using the same weightages as in (b), calculate the composite index for the year 2003 taking the year 2002 as the base year.

[Ans: (a) 110, 115, 105 (b) 110.5 (c) 109.5]

Sri Bintang Tuition Centre, Kuching

@biid09

3

REVISION Additional Mathematics SPM

8.

Table below shows the frequency distribution of number of books read by a group of students. The mean and variance of the distribution are 9.5 and 25.25 respectively. Number of books 1–5 6 – 10 11 – 15 16 – 20 Number of students

10

x

(a)

Find the values of x and of y.

(b)

Without ogive, calculate the median.

y

6

[Ans: (a) x = 14, y = 10 (b) 9.071]

9.

(a)

The probability of an adult at the age of forty suffered from short-sightedness is 0.25. Five adults whose age are forty undergone an eye test. Find the probability that (i) exactly 2 adults, (ii) at least one adult, suffered from short-sightedness.

(b)

The marks obtained by 4000 students in a Mathematics test are found to be normally distributed with a mean of 54 and a standard deviation of 12 marks. (i) (ii)

If the minimum mark for grade A is 75, find the number of students who obtained grade A. If 20% of the students failed the test, determine the minimum passing mark.

[Ans: (a)(i) 0.2637, (ii) 0.7627 (b) (i) 160, (ii) 44]

10.

(a)

Prove that cot x − cot 2 x = cos ec2 x .

(b)

It is given that cot θ = t , where t > 0 for 0 o ≤ x ≤ 360 o .

(i) (ii) [Ans: (b)(i)

Express cosec θ sec θ in terms of t. Hence, or otherwise solve the equation cosec θ sec θ = 2 . 1+ t2 t

, (ii) θ = 45 o ,225 o ]

Sri Bintang Tuition Centre, Kuching

@biid09

4