1. The equation x
^{2} + 4x + k = 0 has 2 equal roots.
(a) Calculate the value of k.
(b) Calculate the root.
[k=4, x=2]
2. Solve, for x, the equation 2
^{x+3} = 4
^{x1}[x=5]
^{}3. Evaluate (6/5)
^{2}[25/36]
4. A chess board has 64 squares. On the 1st square, 1 grain of rice is placed. On the 2nd square, 2 grains of rice are placed. The number of grains placed is doubled for each subsequent square. State the last digit of the number of grains of rice placed on the 10th square.
[1]
5a. Solve the inequality 16 ≤ 14 – 3p ≤ 2. Illustrate your answer with a number line.
[p ≤ 10, p ≥ 4]
5b. x is an integer such that 8 ≤ x < 4. y is an integer such that 3 ≤ y ≤ 2
Calculate
(i) the greatest value of x + y.
(ii) the least value of xy.
(iii) the greatest value of x
^{2} + y
^{2}[6, 24, 73]
6a. A baby elephant has a volume of 2m³. Its tail is 0.5m long. As it grows up, it maintains the same shape exactly. Calculate its volume when the tail is 1m long.
6b. The baby elephant has a leg of length 60cm. Calculate the length of the leg when it has grown up.
6c. The baby elephant’s ear has an area of 40cm². Calculate the area of the ear when it has grown up.
[vol = 16 m
^{3}, length = 123 cm, area = 160 m
^{2}]
7. A man bought a machine at $100 to produce buns. Each bun’s ingredients cost $3. Each bun is sold for $5. Let the number of buns produced and sold be x.
a. If the total cost of production of x buns is $ C, write down an expression linking C and x. (Omit the $ sign)
b. Sketch and label a graph of C against x.
c. Calculate the value of x for a profit of $100 to be made.
[C = 100 +3x, x = 100]
8. A car starts to move from rest and accelerates at a constant rate along a straight road. The distance travelled, s metres, varies directly as the square of time, t seconds, taken.
(a) Between t = 1s and t = 3s, the distance travelled was 16m. Find an expression for s in terms of t.
(b) Calculate the distance travelled after 2 seconds
(c) Calculate the average speed between t = 2s and t = 3s.
[k = 2, distance = 8m, v = 10m/s]
9. In the diagram, angle ABC = 70º and angle GAE = 35º. O is the centre of circle.
(a) Calculate angle OAC
(b) Calculate angle BAO
(c) Calculate angle AEO
(d) Calculate angle AHC
[20º, 35º, 105º, 105º]
10(a)
(i) In a triangle PQR, PQ is 7cm , QR is x cm and PR is (8 – x) cm.
Given that PRQ = 90º, form an equation involving x and show that it reduces to 2x
^{2}  16x + 15 = 0.
(ii) Use the equation to find the possible values of QR and PR.
[QR = 1.08cm, 6.92cm; PR = 6.92cm, 1.08cm]
(b) Factorise completely
(i) 4ax
^{2}  3b + 12bx
^{2}  a
(ii) 4xy
^{2}  (9/16)xz
^{2}[ (a + 3b)(2x  1)(2x + 1); x(2y  3/4z)(2y + 3/4z) ]
11. ∆ABC has vertices whose coordinates are A(6, 1), B(4, 2) and C(2, 3).
Calculate
(a) the length of AB
(b) the gradient of AB
(c) the equation of the line through C, parallel to AB
(d) the equation of the line which passes through the midpoint of BC and the point (2, 2)
[3.61, 3/2, y = 3/2x  6, y = 5/6x + 1/3 ]
12. A car starts from rest at Point A and increases its speed at a steady rate until it reaches 25m/s. It keeps at this speed for 30 seconds until it decreases its speed at a steady rate and stops at Point B. The whole journey takes 2 minutes. Sketch the speedtime graph, marking in all the given data and calculate the distance between Point A and Point B.
[1875m]
13. Simplify
Ans: 14.
Ans:
15.
[k = 3, greatest = 5, least = 1]
16.
(a) Given that x and y are positive integers, solve the equation 9x
^{2}  y
^{2} = 41.
(b) Express 2x
^{2} + 6x – 5 in the form of a(x + b)
^{2} . Find the values of a and b.
[x = 7, y = 20, a = 2, b = 3/2 ]
17. In the diagram, A is due north of D and B is due east of D. Given that the bearing of B from A is 150º and the bearing of C from B is 225º, find
(a) angle ABD,
(b) the bearing of B from C,
(c) the area of ∆BCD if the distance of BC is √8 km.
[60º, 045º, 2km
^{2}]
18. If cosß = 12/13 and 0º ≤ ß ≤ 180º, find
(a) sinß  tanß,
(b) cos (180º  ß).
[125/156, 12/13]
19. In the diagram, BC = 10cm, angle CBD = 120º and angle ADB = 40º. Area of ∆BCD is 15√3 cm
^{2}. Using as much of the information below as is necessary, calculate
(a) BD,
(b) AD.
(sin400 = 0.643, cos 400 = 0.766, tan 400 = 0.839)
[6cm, 4.596cm]
20. The figure shows a triangular prism of height 4 cm. AB = 4 cm, BC = x cm, and angle BAC = 90º.
(a) Find AC in terms of x.
(b) Find the total surface area of this figure in terms of x.
(c) Given that the total surface area is 80 cm
^{2}, form an equation in terms of x and show that it reduces to 3x
^{2} + 32x – 320 = 0.
(d) Find the length of BC by solving the equation 3x
^{2} + 32x – 320 = 0.
[√x
^{2}  16, 4[2(√x
^{2}  16) + 4 + x, x = 6.29, 17.0, BC = 6.29]
21. A map is drawn to a scale of 1 : 40 000. The area of a field in the map measures 50 cm². Calculate the area in cm² which represents the same field on a second map whose scale is 1 : 20 000.
[200cm
^{2}]
22. The following was reported about a particular day in January this year over different parts of the United States.
In Maine, temperature dipped to 53ºC.
Philadelphia got 10 cm of snow, at a temperature higher than Boston by 10ºC.
Boston experienced only a dusting of snow, with a temperature of 22ºC.
New York City reached a high of just 9ºC with 12.7 cm of snow.
On the same day in Singapore, the temperature was at 32ºC
(a) Arrange the 5 places named above in descending order of their temperatures.
(b) Find the difference in temperature between Maine and Philadelphia.
[Singapore, New York City, Philadelphia, Maine; 41ºC]
23. In the diagram, the bearing of B from A is 072º and the bearing of C from B is 146º.
If AB = BC, find the bearing of
(a) A from B,
(b) A from C.
[252º, 289º]
24. Simplify the following algebraic expression
[ 1/(x  2)
^{2} ]
25. Simplify the following expressions:
(a)
giving your answer in positive index form
(b)
[ c
^{2}/a
^{7}b; 8]
26. The brightness of an object, B, varies inversely as the square of the distance, d m, of the object from a light source.
(a) What happens to B when d is doubled?
(b) Given that d = 2.5 m when B = 0.8, form an equation connecting B and d.
(c) Find the value of d when B = 0.2.
[B is 1/4 of its original value, B = 5/d
^{2}, 5 ]
27. ABCD is a parallelogram whose diagonals meet at E. The coordinates of A, B and
C are (9, 2) , (2, 3) and (3 , 8) respectively. Find
(a) the coordinates of E,
(b) the gradient of AC,
(c) the coordinates of D,
(d) the equation of the line passing through E and parallel to AD.
[ (3,5), 1/2, (4,7), y + x =9 ]
28. The diagram below shows a cone of height h cm and radius 5 cm. The top portion of the cone which has a base radius of 3cm is removed to form a remaining truncated cone. Given that the height of the truncated cone is 4cm, calculate
(a) the original height, h of the cone.
(b) Another similar truncated cone which is 12 times the weight of the one shown below is made. Calculate the height of this truncated cone.
[10cm, 9.16cm ]
29. The following is a table of values for the function y = x
^{3} + 3x
^{2} + 1.
x  1  0.5
 0
 0.5
 1
 1.5
 2
 2.5
 3

y  5  1.88
 1
 ∂
 3
 4.37
 5
 4.12
 1

(a) Calculate the value of ∂
(b) Using a scale of 4cm to represent 1 unit on both axes, draw the graph of y = x
^{3} + 3x
^{2} + 1 for 1 ≤ x ≤ 3.
(c) Use your graph to find the values of x for which x
^{3} + 3x
^{2} + 1 = 2.5
(d) State the symmetry of the graph of y = x
^{3} + 3x
^{2} + 1
(e) By drawing a suitable straight line on the same axes, use your graph to find the solutions of
x
^{3} + 3x
^{2}  x + 1 = 0
30. Given that x is inversely proportional to y
^{2} and that x = 10 when y = 2,
(a) express x in terms of y,
(b) find the value of x when b = 3.
[x = 40/b
^{2}]
31. The temperature in Angel Town at 1 p.m. was 6ºC. According to the weather forecast, the temperature at midnight was expected to be 3ºC.
(a) By how many degrees Celsius was the temperature expected to fall?
(b) The temperature at midnight was 2ºC lower than expected. What was the temperature at midnight?
[9, 5ºC]
32. Andre buys an article at $50. If he wishes to sell the article at a gain of 20% after allowing a 25% discount on the mark price, at what price should the article be marked?
[$80]
33. Factorise completely (x – y)(2x – y) – (x – 2y)(y – x)
[ 3(x  y)
^{2} ]
34. Simplify
Express your answer as a single fraction.
[ (x
^{2}  5)/(x  2) ]
35. Given that 19x – 7(y + 2x) = 0, find the value of 2x/5y.
[21/25]
36. Make x the subject of the formula (y/x + z)(1 – y) = 1.
[ x = (y  y
^{2})/(1  z + yz) ]
37. The numbers 220 and 2500, written as the products of their prime factors, are 220 = 2
^{2} x 5 x 11 and 2500 = 2
^{2} x 5
^{4}. Find
(a) the largest integer which is a factor of both 220 and 2500,
(b) the smallest positive integer m for which 220m is a multiple of 2500,
(c) the smallest positive integer n for which
is a whole number.
[20, 125, 50]
38.
The equation of each of the above curves is of the form y = x
^{n}, where n is an
integer. State a possible value for n in each case.
[a: any positive even number; b: any positive odd number; c: any negative odd number]
39.
a. Simplify (2)
^{2n+1} + 2 (2)
^{2n}.
b. Given that 9
^{3} x 27
^{1} = 3
^{a}, find the value of a
[0, 3]
40. Given that z varies directly as y
^{2} and that y varies inversely as √x,
(a) write down an expression for z in terms of y and a constant m,
(b) write down an expression for y in terms of x and a constant n,
(c) show that x and z are in inverse proportion.
[z = my
^{2}, y = n/√x, z = m(n/√x)
^{2} = mn
^{2}/x where mn
^{2} is a constant]
42. The volume of a cone, A, is 4 cm
^{3}.
(a) A similar cone, B, has base radius two times that of cone A. Find the volume of cone B.
(b) Another cone, C, has 1/4 the height and three times the radius of cone A. Find the volume of cone C.
[32, 9]
43. Given that y varies inversely as (x
^{2 }– x  1) and that y = 4 when x = 1,
(a) express y in terms of x,
(b) find the values of x for which y = 4.
[y =  4/(x
^{2}  x  1), x = 1 or 2 ]
44. The graph of y = (x + p)(7 – 2x) cuts the xaxis at A and B. It cuts the yaxis at C and passes through the point D (3, 4).
Find
(a) the value of p,
(b) the length AB,
(c) the coordinates of C,
(d) the midpoint, M, of AD,
(e) the equation of the straight line CM.
[p = 1, AB = 4.5, C(0,7), M(1,2), y = 5x + 7 ]
45. In the diagram, AD is the diameter of the circle ABCDEF, centre O. PDQ is the tangent to the circle at the point D. FA is parallel to EO and OB is parallel to PQ.
Given that angle AOE = 110º and angle DQO = 40º, find
(a) angle OAC,
(b) angle BAC,
(c) angle FAE,
(d) angle AFE,
(e) angle ACE.
[25º, 20º, 35º, 125º]
46. ABC is an isosceles triangle with AC = AB = 10 cm and angle ACB = 35º.
Given that the bearing of C from A is 1750, and using as much of the information
given below as is necessary, find
[sin 700 = 0.94 cos 700 = 0.34 tan 700 = 2.75 ]
(a) the bearing of B from C,
(b) the bearing of A from B,
(c) the area of ∆ABC,
(d) the distance BC, leaving your answer in the form of a square root.
[320º, 105º, 0.94, √268 ]
47. A motorist is travelling from A to C in a remote part of the country. He travels for 50km at a constant speed of x km/h, until he reaches point B.
(i) Write down an expression, in terms of x, for the time, in hours, that he took to travel from A to B.
He then travels the remaining 6 km from B to C at a constant speed of (x – 16)km/h.
(ii) Write down an expression, in terms of x, for the time, in hours, taken for him to walk from B to C.
(iii) Given that the total time for the whole journey from A to C is 50 minutes, form an equation in x and show that it reduces to 5x
^{2}  416x + 4800 = 0.
(iv) Solve the equation, giving both answers correct to two decimal places.
[50/x, 6/(x  16) ]
48. At Singapura Petroleum Company the petrol pump prices for two different grades of petrol are as follows:
Economy Grade $1.52 per litre.
Premium Grade $1.86 per litre.
(a) Mr Tan’s car has a fuel tank capacity of 50 litres. If it is 1/4 tank full now, how much would it cost Mr Tan to fill up to full tank if he pumps
(i) Economy Grade petrol,
(ii) Premium Grade petrol
(b) Mr Tan’s car can travel about 9 km per litre of Economy Grade petrol.
(i) How far can the car travel on a full tank?
(ii) How much would it cost Mr Tan for the car to travel 1 km on Economy Grade petrol?
(c) Singapura Petroleum Company claims that its premium grade petrol can boost mileage by 15%. Assuming that the claim is true, calculate the cost Mr Tan would have to pay for travelling 1 km on Premium Grade petrol. Give your answer correct to the nearest tenth of a cent.
(d) The prices quoted include a government imposed tax of 125% on all petroleum products. Calculate, to the nearest dollar, the amount of petrol tax that Mr Tan would have paid when he pumps 50 litres of Premium Grade petrol.
(e) If the petrol prices at the Singapura Petroleum Company before tax have been marked up by 40%, calculate the actual cost for 1 litre of the Premium Grade petrol before tax, Give your answer correct to the nearest tenth of a cent.
[$57, $69.75, 50l  450 km, $0.17, 18.0 cents, $52, 59.3 cents]
49. Answer the whole of this question on a sheet of graph paper.
The diagram shows a solid which is made up of a rectangular cuboid of base x cm by (6 – 2x) cm and height x cm with a triangular based prism standing on one of the faces of the cuboid. The cross section of the prism is a triangle of height 3 cm.
(a) Show that the volume of the solid, y cm
^{3} is given by y = 2x
^{3} + 3x
^{2} + 9x.
(b) The table below gives some values of x and the corresponding values of y that satisfy the equation
y = 2x
^{3} + 3x
^{2} + 9x.
x  0
 0.5  1  1.5  2  2.5  3 
y  0  5  10  r  14  10  0 
Calculate the values of r.
(c) Using a scale of 2 cm to represent 0.5 unit on the xaxis and a scale of 1 cm to represent 1 unit on the yaxis, draw the graph of y = 2x
^{3} + 3x
^{2} + 9x for the values of x shown in the table.
(d) Use your graph to estimate
(i) the volume of the solid if x = 2.8,
(ii) the possible heights of the solid if the volume of the solid is 12 cm
^{3}.
(e) The solid is melted down and made into a solid rectangular block of dimensions 4 cm by 0.5 cm by 2x cm. On the same axes, draw a suitable straight line to find the value of x.
[r = 13.5, 4.9, x = 1.25 or 2.32, draw y = 4x > x = 2.5 ]
50. In the diagram, ABC represents a horizontal triangular field and AP represents a vertical flagpole. B is 85 m from A on a bearing 025º and C is 170 m from A. Length of BC is 180 m and the height of the flagpole is 12 m.
(a) Calculate
(i) the bearing of A from B;
(ii) angle ABC;
(iii) the angle of depression of C from P.
(b) If the cost of the plot of land is $50 per m
^{2}, find the cost of the land ABC.
(c) A man walks along BC. Calculate
(i) the shortest distance the man is from A as he walks along BC;
(ii) the greatest angle of elevation of the top of the flagpole when viewed
by the man as he travels from B to C.
[205º, 69.5º, 4.04º, $358275, x = 79.6m, 8.6º]