Mathematical Formulae

Compound Interest

Total amount = 100

r 1

+n

P

Mensuration Curved surface area of a cone = rl

Surface area of a sphere = 4 2r

Volume of a cone = hr 231

Volume of a sphere = 334 r

Area of a triangle = 21 absin C

Arc length = r , where is in radians

Sector area = 221 r , where is in radians

Trigonometry

Cc

Bb

Aa

sin

sin

sin==

a 2 = b 2 + c 2 2bc cos A

Statistics

Mean =

ffx

Standard deviation = 22

ffx

ffx

3

Answer all the questions.

1 (a) Evaluate the following and give your answers correct to 2 significant figures:

(i) )952.4346.7()3421626784( + [1]

(ii) 804.1456.2197.3

3

[1]

(b) Simplify the expression xy

yxy3

)(94 22

. [3]

(c) It is given that c

axmxp )( += . Express x in terms of a, c, m and p. [2]

2 For housing loan, a bank charged an interest rate of 1.5% per annum for the first $100 000 and 2.5% per annum for any amount above $100 000. The interest is charged on any outstanding loan amount at the end of each year. On 1 Jan 2008, Larry took a loan of $ 400 000 from the bank. He paid the bank $20 000 on the first day of each subsequent year. Calculate

(a) the interest charged by the bank at the end of the first year, [1] (b) the interest charged by the bank at the end of the second year, [2] (c) the amount that Larry owed to the bank immediately after the third repayment. [3]

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4

3 A sign outside a restaurant in a hotel reads:

Adults Children Monday to Friday Set Lunch $20.00 $12.00 Set Dinner $32.00 $18.00 Saturday & SundaySet Lunch $30.00 $14.00 Set Dinner $42.00 $20.00

All prices include service charge and GST The prices for the various meals, on a weekday, may be represented by the

matrix P =

18321220

.

(a) Represent the prices for having set lunch and set dinner, at the restaurant on a weekend, by a matrix Q. [1]

(b) A tour group consisting of 35 adults and 15 children stays in the hotel for a week and consume set lunch and set dinner, daily, at the restaurant.

(i) Evaluate the matrix S = P

1535

. [1]

(ii) State what the elements of S represent. [1]

(iii) Evaluate the matrix T = Q

1535

. [1]

(iv) Evaluate the matrix A = 5S+2T. [1] (v) Evaluate the matrix B = (1 1)A. [1]

(vi) State what B represents. [1]

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5

4 The cumulative frequency curve below illustrates the marks obtained, out of 50, by 30 students in a Mathematics test in Term 1.

(a) Using the graph, find

(i) the median mark, [1]

(ii) the interquartile range of the marks. [2]

(b) The same students took another Mathematics test in Term 2 and their marks, out of 50, are represented in the stem and leaf diagram below.

0 | 8 9

1 | 1 1 2 3 5 6 7 8

2 | 2 4 4 5 5 5 8 8 9

3 | 0 0 1 2 4 6 9

4 | 3 5 6 6

(i) Find the modal mark. [1]

(ii) Find the median mark. [1]

(iii) Calculate the mean mark. [1]

(iv) Calculate the standard deviation. [2]

(c) State, with a reason, which test did the students perform better in. [1]

Marks

Cumulative Frequency

5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

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6

5 (a) A player throws three darts at a target, one at a time. The probability that he is successful in

hitting the target with his first throw is 81

. For each of his second and third throws, the

probability of success is

twice the probability of success on the preceding throw if that throw was successful,

the same as the probability of success on the preceding throw if that throw was unsuccessful.

The probability tree is constructed below to show this information. S represents success, i.e.

the dart hits the target, and U represents otherwise. 1st throw 2nd throw 3rd throw (i) Write down the values of p, q, r and s. [2]

(ii) Find the probability that all three throws are successful, [1]

(iii) Find the probability that at least one throw is successful. [2]

(b) Two fair die, one red and the other green, are thrown. (i) Find the probability that the score on the red die is divisible by 3 . [1]

(ii) Find the probability that the sum of the scores of the two die is 9. [2]

p

q

43

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81

87

41

21

r

81

87

41

87

s

81

S

S

U

S

U

S

U

S

U

S

U

S

U

U

7

6 Lucy made x litres of lemonade from a budget of $30.00. She then sold the lemonade in 0.2-litre cups, each cup selling at 50 cents more than what she paid for. (a) Write down an expression, in terms of x, for the cost, in cents, of each cup of lemonade. [1]

(b) Write down an expression, in terms of x, for the selling price, in cents, of each cup of lemonade. [1]

(c) At the end of the day, she discarded 1.3 litres of unsold lemonade.

Write down an expression, in terms of x, for the number of cups of lemonade she sold. [1]

(d) She made a profit of $10.30.

Write down an equation to represent this information, and show that it simplifies to

078027150 2 = xx . [2] (e) Solve the equation 078027150 2 = xx . [2] (f) Find the selling price of each cup of lemonade. [2]

7 (a) A steel structure has a mass of 0.572 giga grams.

(i) Express 0.572 giga grams in grams, giving your answer in standard form. [1]

(ii) Given that the density of steel is 7.86 g/cm3, calculate the volume of the steel structure, giving your answer in standard form. [1]

(b) The speed of light is 81000.3 m/s. The distance between planets and Earth is measured by light years, which is the distance travelled by light in a year.

(i) Taking 1 year = 365 days, express 1 light year in terms of kilometers, in standard form. [2]

(ii) An observatory notices that Planet X which is 2 light years from Earth, revolves round the Earth in a circular motion. It takes the planet 40 hours to complete a cycle. Express the speed of Planet X in km/h. [2]

(c) A student constructed a model of the observatory by joining a

hemisphere of radius 10 cm to a cylinder of radius 10 cm and height 20 cm. O is the centre of the solid hemisphere and a portion of the sphere has been cut out as shown in the diagram.

(i) Find the volume of the model. [2] (ii) Find the total surface area of the model. [3]

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O 120 10 cm

20 cm

8

8 In the diagram, X is the intersection of lines AC and BD. AB = 6p, DC = 4p and AC = 7q .

(a) Prove that triangles AXB and CXD are similar. [2] (b) Express BD in terms of p and q. [2]

(c) Find

(i) CXD triangleof AreaAXB triangleof Area

[1]

(ii) CXD triangleof AreaAXD triangleof Area

[1]

(iii) ABCD quad of Area

AXD triangleof Area [2]

9 Three points A, B and C, lie on a horizontal field.

Angle BAC = 75 and the bearing of C from A is 236. AB = 3.60 km and AC = 3.00 km (a) Calculate

(i) the distance BC, [2] (ii) the bearing of B from A, [1] (iii) the angle ABC, [2] (iv) area of triangle ABC. [2] (b) A kite, K is at a point directly above B. The angle of

elevation of K from A is 6.0. Calculate (i) the distance BK, [2]

(ii) the greatest possible angle of elevation of K from a point on AC. [3]

A B

CD

6p

4p

7q X

North

A

B

C

3.00 km

3.60 km

75

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9

10 A metal trough is shown in Figure 1. below:

Figure 2 shows the top view of the trough, with AB = CD = 120 cm and BC = AD = 60 cm at the top; EF = GH = 80 cm and FG = EH = 40 cm at the base and height 20 cm.

(a) Let M, N, P, Q, S and T are the mid-points of AD, BC,

EH, FG, PQ and MN respectively as shown in Figure 3 on the right. Show that VS is 40 cm. [3]

(b) A farmer uses the trough to fill water for his animals. Show that the amount of water required

to fill the empty trough to its brim is 30400031 cm3. [3]

(c) At the end of the day, the animals consumed 80 000 cm3 of water. Calculate the water level in

the trough. [5]

120 cm

80 cm

40 cm 60 cm

Figure 2

D

A

C

B

F

G

E

H

Figure 1

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Figure 3

P Q S

V

M N 20 cm

T

B

C

A

D

E F

G H

V

10

11 Answer the whole of this question on a sheet of graph paper.

The variables x and y are connected by the equation x

xy 816

2

+= . The table below gives some values of x and the corresponding values of y correct to 1 decimal place.

x 1 2 3 4 5 6 7

y 8.1 4.3 3.2 3.0 3.2 p 4.2

(a) Find the value of p correct to 1 decimal place. [1] (b) Using a scale of 2 cm to represent 1 unit on each axis, draw a horizontal x-axis for 70 x and a vertical y-axis for 100 y . On your axes, plot the points given in the table and join them with a smooth curve. [3] (c) By drawing a tangent, find the gradient of the curve where x = 2. [2]

(d) By drawing a suitable line, use your graph to find the values of x in the range 71 x for

which .0128643 =+ xx [3] (e) State the range of values of x for 71 x for which the gradient of the curve is positive. [1]

(f) The graph of 2x + y = k touches the curve x

xy 816

2

+= at a point. By drawing a suitable tangent to your curve, find the value of k. [2]

End of Paper

11

4

2ampcax +=

20421430

1390880

17701260

104906920

Ans Key 1ai) 90000 ii) 1.6 b) 3x-5y c) 2a) $9000 b) $8725 c) $366 168.13 3a) bi) ii) Each element shows the amount that will cost the tour group for lunch and dinner on a weekday. iii) iv) v) 17410 vi) The amount that the tour group has to pay the restaurant for lunch and dinner, for a week. 4ai) 24.5 ii) 6 bi) 25 ii) 25 iii) 25.7 iv) 11.1 v) Median for the second test is 25 marks. Hence the students perform better in the second test. 5ai) p = q = r = s = aii) 1/64 iii) 169/512 bi) 1/3 ii) 1/9 6a) 600/x b) 600/x + 50 c) 5(x-1.3) e) 7.5 f) $1.30 7ai) 81072.5 g ii) 71028.7 cm3 bi)

121046.9 km ii) 121097.2 km/h ci) 7680 ii) 2250 8b) 7p-10p ci) 9/4 ii) 3/2 iii) 6/25 9ai) 4.05 ii)311 iii) 45.7 iv) 5.22 bi) 0.378 km ii) 6.2 10c) 5.79 11a) 3.6 c) -1.75 d) 2.2, 6.75 f) 8.3