Table of Content Topic Fraction/Decimal/Signi…can raction/Decimal/Signi…cantt Figures/Standard Form Substitution Changing the Subject of a Formula............. ormula...................... ................... .................... .................... ............... ..... Simplifying Expressions............... Expressions........................ ................... .................... .................... .................... ................... ......... Ratio:-Proportion-Direct-Inverse-Variation............................................. Factorisation-Simple-Grouping-Quadratic............................................... Simple Equations-Grouping-Brack Equations-Grouping-Bracket-F et-Fraction-W raction-Worded orded Problems............ Problems............ Simultaneous Equations....................................................................... Quadratic Equations............................................................................ Linear and Non-Linear Simultaneous Simultaneous Equations............... Equations......................... .................... ............ Consumer Arithmetic......................................................................... Mensuration/Measurements................................................................ Functions(1)....................................................................................... Geometry(Construction,Polygons, Similar & Congruent Shapes)............ Coordinate Geometry......................................................................... Sets.................................................................................................. Statistics........................................................................................... Vectors............................................................................................ Circle Theorem................................................................................. Trigonometry(Trig rigonometry(Trig Ratios/Sin-Cos Rule/Bearings)............. Rule/Bearings)....................... ................... ......... Matrices........................................................................................... Speed-Distance-Time........................................................................ Graphs-Curves................................................................................. Inequalities....................................................................................... Linear Programming......................................................................... Completing the Squares................. Squares........................... ................... ................... .................... .................... ............ .. Transformation & Matrix Transformation............ ransformation...................... .................... ................... ........... Sequence and Patterns............ Patterns...................... .................... .................... .................... ................... ................. ........ Trigonometrical Graphs.................................................................... Answer Sheets................................................................................
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Fraction/Decimal/Signi…can raction/Decimal/Signi…cantt Figures/Standard Form Jan 2006 1. (a) Using Calculators,or, otherwise, calculate (i) The exact value of
21 4 3 5
4 5 1 2
(ii) correct to, 3 signi…cant …gures the value of 18 of 18::75
2
(2: (2:11)
[7 marks]
May/Jun 2006 2. (a) Using a calculat calculator, or, or otherwise otherwise,, determin determinee the value value of 2 (12: (12:3) (0: (0:246 3) and 3) and write the answer. (i) exactly (ii) correct to two signi…cant …gures [2 marks] Jan 2007 3.(a) Using a calculator, or otherwise, evaluate (i) 5 (i) 5::24(4 1:67) :68 (ii) 1:51 1:45
2
[5 marks] May/Jun 2007 4. (a) Using a calculat calculator, or, or otherwise otherwise,, determin determinee the exact value value of 2 (3: (3:7) (6: (6:24 1:3) [3 marks]
Jan 2008 5. (a) Calculate the EXACT value of 1 (i) 2 1 7 1 2
(ii) 2
3 4 1 5
0:24 0:15
[6 marks]
May/Jun 2008 6. (a) (i) Using Using a calculat calculator or or otherwise, otherwise, determine determine the EXACT value value of (3: (3:9 0:27) + 0:6724 (ii) Express as a single fraction
p
21 2
4 5
3 4
[5 marks] Jan 2009 7. (a) Calculate the EXACT value of 33 4 2
1 3
5 6
giving your answer as a fraction [3 marks] May/Jun 2009 8.(a) Using a calculator, or otherwise,calculate the EXACT value of 2 +1 (i) 6 giving your answer as a common fraction 2 3
1 5
2 5
q (ii)
0:0256 25
giving your your answer in standard form [6 marks]
Jan 2010 9. (a) Using a calculat calculator or or, otherwise otherwise,, calculate calculate the exact value of 2:76 2 + 8: 8 :7 0:8 [3marks] May/Jun 2010 10. (a) Determine Determine the EXACT EXACT value of: 1 (i) 4 1 2 2 5
2 5 3 4
2
:89 (ii) 2: 2 :52 217 giving yor answers correcto to 2 signi…cant …gures
[6 marks] Jan 2011 11. (a) Calculate the exact value of (i) (5 (i) (5::82 + 1: 1:02) 2:5 2 3 (ii) 4 7 4 9 2 3
[5 marks]
May/Jun 2011 12. (a) Using a calculator calculator,, or otherwise, otherwise, determin determinee the EXACT value value of 2 +1 (i) 4 expressing your answer as a fraction 1 4
1 8
1 2
(ii) 3: 3 :96
0:25 p 0:0256
[6 marks]
Jan 2012 13. (a) Using a calculator calculator,, or otherwise, otherwise, calculat calculatee the EXACT value value of 3 2 1 (i) (1 (i) (1 4 ) 3 2 , expressing your answer as a fraction (ii) 0:0529 + 0: 0:216 216,, expressing expressing your answer in standard standard form [6 marks]
p
May/Jun 2012 14. (a) Calculate the EXACT value of 31 5
2 3
24 5
giving your answer as a fraction in its lowest term [3 marks] Jan 2013 15. (a) Using a calculator calculator,, or otherwise, otherwise, calculat calculatee the exact value of 2 (2: (2:67 4:1) 1:3 [3 marks]
May/Jun 2013 16. (a) Using a calculator calculator,, or otherwise, otherwise, calculat calculatee the EXACT value value of 1 (i) 2 4 5
1 3
2 5
(ii)
p 1:5625 + (0: (0:32)
2
[4 marks]
Substitution Jan 2006 1. (a) Given that m =
2 and n = 4, calculate the value of (2 of (2m m + n)(2m )(2m n) [2 marks]
Jan 2007 2. (a) If a = a = 2; b = (i) ab (i) ab bc (ii) b (ii) b((a c)2
May/Jun 2007 3. (a) Given that a Evaluate (i) 4 (i) 4 8 (ii) 2 (ii) 2 (4 8)
3 and c = 4, evaluate [3 marks]
b = ab = ab
b a
[4 marks]
May/Jun 2008 4. (a) Find the value of EACH of the following when a = a = 2; b = 3
1; c = 3
(i) a (i) a((b + c) 2ac (ii) 4ab+b+ c 2
Jan 2009 5. (a) If a a
[3 marks] 2
b = a = a b;evaluate b;evaluate 5 5 2
Jan 2010 6. (a) Given that a = a = 6; b = a +b (i) cb 2
May/Jun 2010 7. (a) Given that a = a = (i) a (i) a + b + c (ii) b (ii) b 2 c2
[1 mark]
4; c = 8; calculate the value of [3 marks]
1; b = 2; c = 3, …nd the value of:
[2 marks]
May 2011 8.(a) The binary operation * is de…ned by a b = (a + b)2 2ab Calculate the value of 3 of 3 4
[2 marks]
Changing the Subject of a Formula Jan 2006 1. (a) Given the formula s = 21 (u + v)t; express t; express u u in terms of v; v ; s and t: [3 marks] Jan 2009 2. (a) Make t Make t the subject of the formula p = t+g r 2
q
[3marks]
Jan 2010 3. (a) The relationship between kinetic energy E ; mass m; mass m; velocity v velocity v , for a moving particle is E = = 21 mv2 (i) Express v Express v in terms of E of E and m and m (ii) Determine the value of v of v when E when E = = 45 and 45 and m = 13 [5 marks] Jan 2011 4. (a) Express p as the subject of the formula q = = p tr [3 marks] 2
Jan 2013 5. (a) Make r Make r the subject of EACH of the following formulae: (i) r (i) r h = rh = rh (i) V (i) V = r 2 h [4 marks]
May/Jun 2013 6. (a) (i) Make C the C the subject of the formula F = 59 C + + 32 (ii) Given that F = F = 113; 113; calculate the value of C of C [3 marks]
4
Simplifying Expressions
May/Jun 2006 3 x2 1.(a) Simplify x 3 5 a (b) Simplify a a+3+4 a4 2
2
Jan 2007 2.(a) Expand ( Expand (x x + 3)2 (x
[6 marks]
4), 4), writing writing your answer answer in descending descending powers powers of x: x: [3 marks]
May/Jun 2007 3.(a) Simplify, expressing your answer in its simplest form 5 p 3q
2
4 p
q
[2 marks]
May/Jun 2008 4. (a) Change the following statements into algebraic expressions: (i) Four times the sum of x of x and 5 (ii) 16 larger than the product of a of a and b (b) Simplify (i) x (i) x 2 x3 x4 (ii) a (ii) a b ab3
p
3 2
5 2
[6 marks]
Jan 2009 5. (a) Simplify, expressing your answer as a single fraction (i) 2nm 53m n [3 marks]
Jan 2010 6. (a) Simplify the expression (i) 3(x 3(x y ) + 4(x 4(x + 2y 2y ) 4x 3x (ii) 6x 2
4
3
[5 marks]
May/Jun 2010 7.(a) Write the following phrases as algebraic expressions: (i) seven times the sum of x of x and y (ii) the product of TWO consecutive numbers when the smaller number is y: y : [2 marks] Jan 2011 8. (a) Simplify the expression (i) 25x x3 expressing your answer as a single fraction (ii) 7 p5 q 3 2 p2 q
[4 marks]
May/Jun 2011 9.(a) Write as a single fraction in its lowest terms x 2
3
x+1 4
[3 marks]
5
RatioRatio-Proportion Jan 2007 1. (a) A sum of money money is share shared d between between Aaron Aaron and and Betty Betty in the ration ration 2 : 5. Aaron Aaron recei receive ved d $60. How How much much money was shared altogether? shared altogether? (b) In St. Vincent, 3 liters of gasoline cost EC$10.40. (i) Calculate the cost of 5 liters of gasoline in St. Vincent, stating your answer correct to the nearest cent. (ii) How many liters of gasoline cant be bought for EC$50.00 in St. Vincent? Give your answer correct to the nearest whole number [7 marks] May/Jun 2007 2.(a) A total of 1200 students attend Top View High School. The ratio of teachers to students is 1 : 30 (i) How many teachers are there at the school? Two-…fths of the students own personal computers. (ii) How many students do NOT own personal computers? Thirty percent of the students who own personal computers also own play stations (iii) (iii) What What fraction of fraction of the students in the students in the school own play stations? Express your answer in its lowest terms. lowest terms. [8 marks] Jan 2010 3. (a) The ingredien ingredients ts for making making pancakes pancakes are shown below.
(i) Ryan wishes to mak makee 12 pancakes pancakes using the instruction instruction given above. above. Calculate Calculate the number of cups of pancake pancake mix he must use (ii) Neisha Neisha used 5 cups of milk to make pancakes pancakes using the same instruction instruction.. How many many pancakes pancakes did she mak make? e? [5 marks] Ratio-Direct/Inverse Variations May/Jun 2007 1. Two variables x and y are related such that ’y ’ y varies inversely as the square of x of x’. ’. (a) Write an equation in x; y and k to describe the inverse variation, where k is the constant of variation x 3 1.8 f (b) y 2 r 8 Using the information in the table above, calculate the value of (i) k (i) k,, the constant of variation (ii) r (ii) r (iii) f [8 marks] 6
Jan 2008 2. (a) The volume, V ; of a gas varies inversely as the pressure ; P; when the temperature is held constant (i) Write an equation relating V and P and P (ii) If V If V = 12: 12 :8 when P when P = = 500; 500;determine the constant of variation (iii) Calculate the value of V of V when P when P = = 480: 480: [6 marks] May/Jun 2010 3.(a) When y When y varies directly with the square of x of x,, the variation equation is written y = kx = kx 2 ; where k where k is a constant (i) Given Given that y that y = 50 when 50 when x x = = 10; 10; …nd the value of k: k : (ii) Calculate the value of y of y when x when x = 30: 30:
May/Jun 2011 4.(a) The table below shows corresponding values of the variables x and y, y , where y where y varies directly as x x 2 5 b y 12 a 48 Calculate thevalues of a of a and b [3 marks] May/Jun 2013 5. (a) The incomplete incomplete table below shows shows one pair of values values for A and R where A is directly proportional to the sqaure of R R A 36 196 R 3 5 (i) Express A Express A in terms of R R and a constant k constant k (ii) Calculate the value of the constant k (iii) Copy and complete the table [5 marks] Factorising-Simple-Grouping-Quadratic actorising-Simple-Grouping-Quadratic Jan 2006 1. (a) Factorise completely (i) 4 (i) 4x x2 25 (ii) 6 (ii) 6 p p 9 ps + 4q 4q 6qs 2 (iii) 3x 3 x + 4x 4
[6 marks]
May/Jun 2006 2. (a) Factorise (i) x (i) x 2 5x (ii) x (ii) x 2 81
[2 marks]
Jan 2007 3. (a) Factorise completely (i) 2 (i) 2 p p2 7 p + 3 (ii) 5 (ii) 5 p p + 5q 5q + p + p2 q 2
[3 marks]
Jan 2008 4. (a) Factorise completely (i) x (i) x 2 xy (ii) a (ii) a 2 1 (iii) 2 p 2 p 2q p2 + pq
[4 marks]
May/Jun 2008 7
5. (a) Factorise completely (i) 6 (i) 6a a2 b3 + 12a 12a4 b 2 (ii) 2 (ii) 2m m + 9m 9m 5
[4 marks]
Jan 2009 6. (a) Factorise completely (i) 3 (i) 3x x 6y + x2 2xy
[2 marks]
May/Jun 2009 7. (a) Factorise completely (i) 2 (i) 2ax ax + 3ay 3ay 2bx 3by 2 (ii) 5x 20 (iii) 3x 3 x2 + 4x 4x 15
[6 marks]
May/Jun 2010 8. (a) Factorise completely (i) 4 (i) 4yy 2 z 2 (ii) 2 (ii) 2ax ax 2ay bx + by (iii) 3x 3 x2 + 10x 10x 8
[5 marks]
Jan 2011 9. (a) Factorise completely (i) a (i) a 2 b + 2ab 2ab [1 mark] May/Jun 2011 10. (a) Factorise completely (i) xy (i) xy 3 + x2 y (ii) 2 (ii) 2mh mh 2nh 3mk + mk + 3nk 3 nk
[4 marks]
Jan 2012 11. (a) Factorise completely (i) x (i) x 2 16 (ii) 2 (ii) 2x x2 3x + 8x 8x 12
[3 marks]
May/Jun 2012 12.(a) Factorise completely (i) 2 (i) 2x x3 y + 6x 6 x2 y 2 (ii) 9 (ii) 9x x2 4 (iii) 4x 4 x2 + 8xy 8xy xy 2y2
[5 marks]
Jan 2013 13. (a) Factorise completely (i) 25 (i) 25m m2 1 (ii) 2n 2 n2 3n 20
May/Jun 2013 14. (a) Factorise completely (i) 2 (i) 2x x3 8x (ii) 3 (ii) 3x x2 5x 2 = (3x (3x + 1) 1) (x
[4 marks]
2)
[4 marks]
8
Simple Equations-Grouping Equations-Grouping-Brack -Bracket-F et-Fraction-W raction-Worded orded Problems Jan 2007 1.(a) Solve for x where x where x Z x x (i) 2 + 3 = 5 (b) The cost of ONE mu¢n is $m $ m The cost of THREE cupcakes is $2m $2 m (i) Write an algebraic expression in m for the cost of: (a) FIVE mu¢ns (b) SIX cupcakes (ii) Write an equation in terms of m of m to represent the following information. The TOTAL cost of 5 mu¢ns and 6 cupcakes is $31.50 [6 marks]
2
Jan 2008 2.(a) The table below shows the types of cakes availabe at a bakery, the cost of each cake and the number of cakes sold for a given day. TYPE TYPE OF CA CAKE KE Sponge Chocolate Fruit
COST($ COST ($)) k+5 k 2k
NO. NO. OF CAKE CAKES S SOLD SOLD 2 10 14
(i) Write an expression in terms of k of k;; for the amount of money collected for the sale of sponge cake for the day (ii) Write an expression in terms of k of k;; for the TOTAL amount of money collected The total amount of money collected for the day was $140.00 (iii) Determine the value of k [5 marks] May/Jun 2008 3. (a) Solve the equation (i) 15 (i) 15 4x = 2(3x 2(3x + 1)
[2 marks]
Jan 2009 4. (a) A drinking straw of length 21cm is cut into three pieces. the length of the …rst piece is xcm. x cm. The second piece is 3cm shorter than the …rst piece The third piece is twice as long as the …rst piece (i) State, in terms of x; of x; the length of EACH of the pieces (ii) Write an expression in expression in terms of x x to represent the sum of the lengths of the three pieces of drinking straw (iii) Hence (iii) Hence calculate calculate he value of x of x [5 marks] Jan 2011 5. (a) The students students in a class sells sells donuts donuts to raise money for their school school project. The donuts donuts are in small and large large boxes. The number number of donuts in EACH type type of box is given given in the table below: Type Type of Box Box Number Number of Don Donuts uts per per Box Box Small Box x Large Box 2x + 3 The students sold 8 small boxes and 5 large boxes in all. (i) Write an expression in terms of x of x to represent the TOTAL number of donuts sold (ii) The total total number number of donuts sold is 195. Calculate Calculate the number number of donuts donuts in a (a) small Box (b) large Box [6 marks] Jan 2012 6. (a) Tickets for a football match are sold at $30 for EACH adult and $15 for EACH child. 9
A company bought 28 tickets. (i) If x x of these tickets were for adults, write in terms of x: of x: (a) the number of tickets for children (b) the amount spent on tickets for adults (c) the amount spent on tickets for children (ii) Show that the TOTAL amount spent on the 28 tickets is $(15x $(15x + 420) (iii) Given that the cost of the 28 tickets was $660, calculate the number of adult tickets bought by the company. [6 marks] May/Jun 2012 7. (a) Solve for x (i) 2x33 52 x = 3
[3 marks]
Jan 2013 8. (a) Solve for p (i) 2( p + 5) 7 = 4 p
[2 marks]
May/Jun 2013 9. (a) 500 tick tickets ets were were sold for for a concert. concert. Of these these x tickets were sold at $6 each, and the remainderat $10 each. (i) Write an expression, in terms of x of x,, for (a) the number of tickets sold at $10 each (b) the TOTAL amount of money collected for the sale of the 500 tickets (ii) The sum of $4108 was collected for the sale of the 500 tickets Calculate the number of tickets sold at $6 each. [5 marks]
Simultaneous Equations Jan 2006 1. (a) Solve the simultaneous equations (i) 5 (i) 5x x + 6y 6y = 37 2x 3y = 4
[4 marks]
May/Jun 2006 2. (a) Two cassettes and three CD’s cost $175 while four cassettes and one CD cost $125. (i) Given that one cassette cost $x $ x and one CD cost $y $y, write two equations in x and y to represent the information (ii) Calculate the cost of one cassette [4 marks] May/Jun 2007 3. (a) A stadium has two sections. A and B Tickets for Section A cost $ a each. Tickets for Section B cost $b $ b each. Johanna paid $105 for 5 Section A tickets and 3Section B tickets. Raiyah paid $63 for 4 Section A tickets and 1 Section B ticket. (i) Write two equations in a and b to represent the information above (ii) Calculate the values of a of a and b: [5 marks] May/Jun 2009 4. (a) One packet of biscuit cost $ x and one cup of ice-cream cost $y $y One packet packet of biscuit and two cups of ice cream cost $8.00, while while three packets packets of biscuit and one cup of ice cream cost $9.00. 10
(i) Write a pair of simultaneous equation in x and y to represent the given information above. (ii) Solve the equations obtained above to …nd the cost of one packet of biscuit and the cost of one cup of ice-cream [6 marks] May/Jun 2010 5.(a) Solve the pair of simultaneous equations: (i) 2 (i) 2x x + y = 7 x 2y = 1
[3 marks]
Jan 2012 6. (a) Solve the pair of simultaneou simultaneouss equations equations (i) 3 (i) 3x x + 2y 2y = 13 x 2y = 1
[3 marks]
May/Jun 2012 7. (a) Solve the simultaneous equations (i) 3 (i) 3x x 2y = 10 2x + 5y 5y = 13
[4 marks] Jan 2013 8. (a) A candy store packages lollipops and to¤ees in bags for sale 5 lollipops and 12 to¤ees have a mass of 61 grams 10 lollipops and 13 to¤ees have a mass of 89 grams (i) If the mass of one lollipop is x grams and the mass of one to¤ee is y grams, write two equations in x in x and y to represent the above information. (ii) Calculate the mass of (a) One lollipop (b) One to¤ee [6 marks] Quadratic Equations May/Jun 2006 1.(a) A strip of wire of length 32cm is cut into two pieces. One piece is bent to form a square of side xcm. x cm. The other piece is bent to form a rectangle of length l cm and width 3cm
(i) Write an expression, in terms of l of l and x for the length of the strip of wire (ii) Show that l = 13 2x The sum of the areas of the square and the rectangle is represented by S (iii) Show that S that S = = x2 6x + 39 (iv) Calculate the values of x of x for which S which S = = 30: 30:25 [10 marks]
May/Jun 2007 2. (a) The length of the rectangle below is (2x (2 x
1)cm 1)cm and its width is (x ( x + 3)cm. 3)cm. 11
(i) Write an expression in the form ax 2 + bx + c for the area of the rectangle (ii) Given that the area of the rectangle is 294cm 2 ; determine the value of x of x (iii) Hence, state the dimensions of the rectangle, in centimeters. [8 marks] Jan 2008 3.(a) The lengths, in cm, of the sides of the right angled triangle shown below are a; (a
(i) Using Pythagoras theorem, write an equation in terms of a of a to represent the relationship among the three sides. (ii) Solve the equation for a (iii) Hence, state the lengths of the THREE sides of the triangle. [9 marks] Jan 2013 4. (a) Solve the equation (i) 3x2 5x + 1 = 0, 0, expressing your answer correct to two decimal places. [4 marks]
Linear and Non-Linear Simultaneous Equations Jan 2006 1. (a) Solve the pair of simultaneou simultaneouss equations equations (i) 2 (i) 2x x + y = 7 x2 xy = xy = 6
[6 marks]
May/Jun 2006 2. (a) Solve the pair of simultaneou simultaneouss equation equation (i) y (i) y = x = x + 2 y = x = x 2 [5 marks] May/Jun 2008 3. (a) Solve the following following pairof equations equations for x and y : (i) y (i) y + + 4x 4 x = 27 xy + xy + x = 40 [6 marks]
12
7); 7);and ( and (a a + 1): 1):
May/Jun 2009 4. (a) Solve the pair of simultaneou simultaneouss equations equations (i) y (i) y = 4 2x y = 2x2 3x + 1
[4 marks]
May/Jun 2011 5. (a) Solve the pair of simultaneou simultaneouss equations equations (i) y (i) y = x = x 2 x + 3 y = 6 3x
[5 marks]
May/Jun 2012 6.(a) Solve the pair of simultaneous equations (i) y (i) y = 8 x 2x2 + xy = xy = 16 (ii) State, giving the reason for your answer, whether the line y = 8 to the curve 2x 2 x2 + xy = xy = 16 [7 marks]
x is a tangent
Consumer Arithmetic Jan 2006 1.(a) A loan of $12000 was borrowed from a bank at 14% per annum. Calculate (i) the interest on the loan at the end of the …rst year (ii) the total amount owing at the end of the …rst year A repayment of $7800 was made at the start of the second year Calculate (iii) the amount still outstanding at the start of the second year (iv) the interest on the outstanding amount at the end of the second year [5 marks] May/Jun 2006 2. (a) The table below gives information on the values and the rates of depreciation in value of two motor vehicles. Motor Motor Vehicle ehicle Taxi Private Car
Initia Initiall Valu Value e $40000 $25000
Yearly early Rate Rate of Depr Depreci eciati ation on 12% q %
Value alue after after One One Year Year $ p $21250
Calculate (i) the value of p of p and q (ii) the value of the Taxi after 2 years (b) GUY $1.00 = US $0.01 and EC $1.00 = US $0.37 Calculate the value of (i) GUY $60000 in US $ (ii) US $925 in EC $ [10 marks] Jan 2008 3. (a) The cash price price of a bicycle is $319.95. $319.95. It can be bought on hire purcha purchase se by making making a deposit deposit of $69.00 and 10 monthly monthly installmen installments ts of $28.5 $28.50 0 EACH. EACH.
13
(i) What is the TOTAL hire purchase price of the bicycle? (ii) Calculate the di¤erence between the total hire purchase price and the cash price. (iii) Express your answer in (ii) above as a percentage of the cash price [5 marks] May/Jun 2008 4. (a) In this question, use CAN$1.00 = JA$72.50 (i) On a vacation in Canada, steve used his credit card to buy a camera for CAN$250.00 What is the value of the camera in Jamaican dollars? (ii) Steve Steve credit car limit is $30000.00. $30000.00. After After buying the camera, camera, how many Canadian dollars does he have left on his credit card for spending? [5 marks] b. Annie went to the post o¢ce and bought a collection of SIX of each of the following stamps.
(i) What is the TOTAL cost of the stamps? (c) She had to post parcels parcels and the total cost cost of the postage was was $25.70. $25.70. What stamps stamps can she select from the collection, to make up this amount if she must use. (i) as many as $4.00 stamps as possible (ii) all her $1.00 stamps (d) (i) What is the largest number of stamps that she can use from the collection to post the parcels? (ii) List the selction of stamps she can use [10 marks]
Jan 2009 5. (a) BDS$ mean Barbados dollars EC$ mean Eastern Caribbean dollars (i) Karen exchanged exchanged BDS$2000.00 BDS$2000.00 and received received EC$2700.00. EC$2700.00. Calculate the value of one BDS$ in EC$ (ii) If Karen exchanged EC$432.00 for BDS$, calculate the amount of BDS$ which Karen would receive [Assume the buying and selling rates are the same] [4 marks] (b) A credit union union pays 8% per annum compund compund interest interest on all …xed deposits. deposits. A customer customer deposited $24000 in an account. Calculate the TOTAL amount of money in the account at the end of two years. [4 marks] May/Jun 2009
14
6. (a) The basic wage earned by a truck driver for a 40-hour week is $560.00 (i) Calculate Calculate his hourly rate For overtime the driver is paid one and a half the basic hourly rate (ii) Calculate his overtime wage for 10 hours of overtime (iii) Calculate the TOTAL overtime wages earned by the truck driver by for a 55-hour week [6 marks] Jan 2010 7.(a) In a certain company, a salesman is paid a …xed salary of $3140 per month plus an annual annual commission commission of 2% on the TOTAL TOTAL value value of cars sold for the year. If the salesman sold cars valued at $720000 in 2009, calculate. (i) his …xed salary for the year (ii) the total amount he received in commission for the year (iii) his TOTAL income for the year [3 marks] May/Jun 2010 8. (a) Mrs Jack bought 150 for $1920 from a factory (i) Calculate the cost of ONE T-shirt
The T-shirts T-shirts are sold at $19.99 $19.99 each. each. Calculate (ii) the amount of money Mrs Jack received after selling ALL the T-shirts (iii) the TOTAL pro…t made (iv) the pro…t made as a percentage of the cost price, giving your answer correct to the nearest whole number. [4 marks] Jan 2011 9. (a) A company pays its employees a basic wage of $9.50 per hour for a 40-hour week. (i) Calculate the basic weekly wage for ONE employee. Overtime is paid at a rate of time and a half (ii) Calculate the overtime wage for an employee who works 6 hours overtime in a certain week. In a certain week the company paid its 30 employees a total of $12084.00 in basic and overtim overtimee wages. Calculate Calculate for that week: week: (iii) the TOTAL paid in overtime wages (iv) the TOTAL number of overtime hours worked by employees [6 marks] May/Jun 2011 10.(a) 10.(a) The table below below shows shows Pamela’s Pamela’s shopping bill. bill. Some of the informati information on was not included. included.
15
Item Itemss Rice Potatoes Milk
Quan Quanti tity ty Unit Unit Pric Price( e($) $) 1 62 2.40 4 bags X Y cartons 2.35 Sub Total Z%VAT TOTAL Calculate the values of W of W;; X;Y and Z: and Z:
Total otal Cost Cost($ ($)) W 52.80 14.10 82.50 9.90 92.40 [5 marks]
Jan 2012 11. (a) A typist is paid a a basic wage of $22.50 per hour for a 40-hour week. (i) Calculate Calculate the typist’ typist’ss basic hourly rate rate Overtime is paid at one and a half times the basic hourly rate. (ii) Calculate the overtime wage for ONE hour of overtime work To earn some extra money, the typist decided to work overtime Calculate (iii) the wage she would earn for overtime if she worked for a TOTAL of 52 hours during a given week (iv) the number of overtime hours she must work during a given week to earn a TOTAL wage of $1440. [6 marks] May/Jun 2012 12. (a) The table below shows the cost price, selling price and pro…t or loss as a percentage of the cost price Copy and complete the table below, inserting the missing values at (i) and (ii) Cost Co st Pric Price e $55 (ii)_______
Sell Sellin ing g Pric Price e $44 $100.00
Perce ercent ntag age e Pro… Pro…tt or Loss Loss (i)________ 25% Pro…t
(b) The table below shows some rates of exchange US $1.00 = EC $2.70 TT $1.00 = EC $0.40 Calculate the value of (i) EC $1.00 in TT $ (ii) US $80 in EC $ (iii) TT $648 in US $ [6 marks] Jan 2013 13. (a) Mr Harry who lives lives in St. Kitts Kitts is planning to travel travel to Barbados. Barbados. A travel travel club o¤ers the rates shown below Petty’s Travel Club Holiday in Barbados Retu Return rn Air Air Far Fare e US $356 $356.0 .00 0 Hote Hotell Acco Accomm mmod odat atio ion n US $97. $97.00 00 per per nigh nightt (i) Calculate the TOTAL cost of airfare and hotel accommodation for 3 nights using the rates o¤ered by Petty’s Travel Club (ii) Another travel club advertises the following package deal. Angie’s Travel Club Holiday in Barbados 3 Nights Hotel Accommodation plus Return Air Fare EC $1610.00 Calculate, in US dollars, the cost of the trip for 3 nights as advertised by Angie’s travel club. US $1.00 = EC $2.70 16
(iii) State, giving a reason for your answer, which travel club (Petty’s or Angie’s) has the better o¤er (iv) The EC $1610.00 charged by Angie’s Travel Club includes a sales tax of 15%. Calculate the cost of the trip for three nights BEFORE the sales tax was added. [8 marks] May/Jun 2013 14. (a) Smiley Orange Juice is sold in cartons of two di¤erent sizes at the prices shown in the table below Cart Carton on Size Size Cost Cost 350 ml $4.20 450ml $5.13 Which size carton of orange juice is the BETTER buy? Justify your answer (b) Faye borrowed $9600 at 8% per annum compound interest (i) Calculate the interest on the loan for the …rst year. At the end of the …rst year, she repaid $4368 (ii) How much did she still owe at the beginning of the second year? (iii) Calculate the interest on the remaining balance for the second year. [7 marks] Mensuration/Measurements Jan 2006 1.(a)
The diagram above shows the position of three cities A, B and C on the northcoast of Africa. Africa. The scale scale of the map is 1:20 1:20 000 000 (i) Measure and state, in centimeters the length of the line segment, BC (ii) Hence, calculate in kilometers, the actual shortest distance from City B to City C. (iii) Using protractor, determine the bearing of B from A [8 marks] (b) The curved surface area of a cylinder= cylinder = 2rh 2 rh,, where r where r is the raduis and h is the height, and the 2 surface area of sphere is 4r 4 r
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The diagram above, not drawn to scale, shows a solid a solid glass glass paperweight which consist of a hemisphere mounted on a cylinder The radius of the hemisphere is 3cm, the radius of the cylinder is 3cm and its height is 8cm. Calculate using = 3:14 (i) the curved surface area of the cylinder (ii) the surface area of the hemisphere (iii) the TOTAL surface area of the sold paperweight [6 marks] Jan 2007 2.(a) The diagram below, not drawn to scale, shows a prism of length 30cm. The cross-section W X Y Z is is a square with area 144cm 2 :
Calculate (i) the volume in cm 3 of the prism (ii) the total surface area in cm 2 , of the prism (b) The diagram below, not drawn to scale, shows the sector of a circle with center O. angle M angle M ON = 45 0 and O and ON N = 15cm 15 cm
Calculate giving your answer correct to 2 decimal places (i) the length of the minor arc M N (ii) the perimeter of the …gure M ON (iii) the area of the …gure M ON [12 marks] May/Jun 2007 3.(a) The diagram below shows a map of a golf course drawn on a grid of 1cm squares The scale of the map is 1:4000.
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Using the map of the golf course, …nd (i) the distance to the nearest m, from South Gate to East Gate (ii) the distance, to the nearest m, from North Gate to South Gate (iii) the area on the ground represented by 1cm 2 on the map (iv) the actual area of the golf course, giving the answer in square meters. (b) The diagram below, not drawn to scale, shows a prism of volume 960cm 3 : The cross-sect cross-section ion ABCD is a square. square. The length length of the prism is 15cm. 15cm.
Calculate (i) the length of the edge AB, in cm (ii) the total surface area of the prism, in cm 2 [11 marks] Jan 2008 4. (a) The diagram below, not drawn to scale, scale, shows a circle with center O and a square OPQR. The radius of the circle is 3.5cm.
Calculate the area of: (i) the circle (ii) the square square OPQR (iii) the shaded region 19
[7 marks] May/Jun 2008 5. (a) The diagram below, not drawn to scale, represents the plan of a ‡oor. The broken line RS , RS , diveides the ‡oor into two rectangles A and B :
(i) Calculate Calculate the length of RS of RS (ii) Hence state the value of x of x (iii) (iii) Calculate Calculate the perimeter perimeter of the entire ‡oor (iv) Calculate the area of the entire ‡oor (v) Section A Section A of the ‡oor is to be covered with ‡ooring board measuring 1m by 20cm. How many ‡ooring boards are needed for covering Section A? A ? [12 marks] (b) In the diagram below not drawn to scale, O is the center of the circle of radius 8.5cm and AB and AB is a chord of length 14.5cm
(i) Calculate the value of of to the nearest degree (ii) Calculate the area of triangle AOB AO B (iii) (iii) Hence, Hence, calculate calculate the area of the shaded region. [Use = 3:14] (iv) Calculate the length of the major arc AB [10 marks] Jan 2009 6. A compnay compnay makes makes cereal boxes boxes in the shape of a right prism. prism.
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EACH large box has dimensions 25cm, by 8cm, by 36cm (a Calculate the volume in cubic centimeters, of ONE large cereal box. (b) Calculate the total surface area of ONE large cereal box. (c) The cereal from ONE large box can exactly …ll six small boxes, each of eqaul volumes. (i) Calculate the volume of ONE small cereal box (ii) If the height of a small box is 20cm, list TWO di¤erent pairs of values which the company can use for the length and width of a small box. [10 marks] May/Jun 2009 7. (a) The map shown below is drawn to a scale of 1:50 000
(i) Measure, and state in centimeters, the distance on the map from L to M along M along a straight line (ii) Calculate the actual distance in kilometers, from L to M : (iii) (iii) The actual actual distance distance between between the two two points in 4.5km. Calculate Calculate the number number of centimeters that should be used to represent this distance on the map. [7 marks] Jan 2010 8. (a) The diagram below shows a map of a playing …eld drawn on a grid of 1cm squares. The scale scale of the map is 1: 1250 1250
21
(i) Measure and state in centimeters the distance from S to F to F on on the map. (ii) Calculate the distance, in meters, for S to F to F on on the ACTUAL playing …eld. (iii) (iii) Daniel Daniel ran the distance from S to F to F in in 9.72 seconds. Calculate his average speed in (a) m/s (b) km/h giving your answers correct to 3 signi…cant …gures. [6 marks] May/Jun 2010 9. (a) The diagram below, not drawn to scale, scale, shows a wooden toy in the shape of a prism, with the cross-section ABCDE: F is F is the midpoint of E of E C , and angle B angle B AE = = angle C angle C BA = BA = 900 :
Calculate (i) the length of E of E F (ii) the length DF D F (iii) the area of the face ABCDE [6 marks] Jan 2011 10. (a) Fresh Farms Dairy sells milk in cartons in the shape of a cuboid with internal dimensions 6cm, by 4cm, by 10cm.
22
(i) Calculate in cm 3 ; the volume of milk in EACH carton. (ii) A recipe for making ice-cream ice-cream requires requires 3 liters of milk. How many many cartons of milk should be bought to make the ice-cream? (iii) One carton of milk is poured into a cylindrical cup of internal diameter 5cm. What is the height of the milk in the cup? Give your answer to 3 signi…cant …gures [Use = 3:14] [9 marks] May/Jun 2011 11. (a) In this question, use = 22 7 A piece of wire is bent to form a square of area 121cm 2 Calculate (i) The length of each side of the square (ii) The perimeter of the square (b) The same piece of wire is bent to form a circle Calculate (i) The radius of the circle (ii) The area area of the circle circle [7 marks] Jan 2012 12. (a) The diagram below, not drawn to scale, shows a cuboid with length 13cm, width 4cm and height h height h cm.
(i) State, in terms of h; of h; the area of the shaded face of the cuboid. (ii) Write an expression, in terms of h of h,, for the volume of the cuboid. (iii) If the volume of the cuboid is 286cm 3 calculate the height, h, h , of the cuboid. [4 marks] May/Jun 2012 13. (a) The diagram below, not drawn to scale, scale, shows the cross-section of a prism in the shape of a sector sector of a circle, circle, center center O, and radius 3.5cm. 3.5cm. The angle at the center center is 270 0 [Use [Use = 22 ] 7
23
Calculate (i) the length of the arc ABC AB C (ii) the perimeter of the sector OABC (iii) the area of the sector OABC (b) The prism is 20cm long and is made of solid tin. Calculate (i) the volume of the prism (ii) the mass of the prism, to the nearest kg, given that 1cm3 of the tin has a mass of 7.3kg [10 marks] Jan 2013 14. The diagram below, not drawn to scale, shows a hollow cylinder with height 8cm and diameter 12 12cm cm.. Use Use = 3:14
(a) Calculate for the cylinder: (i) The radius (ii) The circumference of the cross section (b) The rectangle shown below, not drawn to scale, represents the net of the curved surface of the cylinder shown above.
24
(i) State the values of a of a and b: (ii) Hence, calculate the area of the curved surface of the cylinder (c) If 0.5 liters of water is poured into the cyclinder, calculate, correct to one decimal place, the height of water in the cylinder. [11 marks]
Functions(1) Jan 2007 1.(a) f 1.(a) f and g and g are functions de…ned as follows f : x 7x + 4 1 g : x : x 2x
! !
Calculate (i) g (i) g (3) (ii) f ( f ( 2) (iii)f (iii)f 1 (11)
[5 marks] May/Jun 2007 2. (a) Given that g( g (x) = 2x5+1 and f and f ((x) = x + 4: 4: (i) Calculate the value of g of g(( 2) (ii) Write an expression for g f ( f (x) in its simplest form (iii) Find the inverse function g 1 (x)
[7 marks] May/Jun 2008 3. (a) If f ( f (x) = 2x (i) f (i) f (2) (2) (ii) f (ii) f 1 (0) (iii) f 1 f (2) f (2)
3; …nd the value of [5 marks]
Jan 2009 4.(a) Two functions are de…ned as follows f : x x 3 g : x : x x2 1 (i) Calculate f Calculate f (6) (6) (ii) Find f Find f 1 (x) (iii) Show that f that f g(2) = f = f g ( 2) = 0
! !
May/Jun 2009 5.(a) Given that f ( f (x) = 2x (i) f (i) f (( 2) (ii) gf g f (1) (1) (iii) f (iii) f 1 (3)
[5 marks]
2
5 and g( g (x) = x 31, 31, calculate the value of [5 marks]
May/Jun 2010 6. (a) The functions f and g and g are de…ned as f as f ((x) = 2x Calculate the value of (i) f (i) f (4) (4) (ii) g (ii) g f (4) (4) (iii) Find f 1 (x)
5 and g( g (x) = x [5 marks] 25
2
+3
Jan 2011 7. (a) The arrow arrow diagram shown shown below represen represents ts the relation relation f : x x 3; 4; 5; 6; 7; 8; 9; 10
2f
g
2
! x k; where k; where
Calculate the value of (i) k (i) k (ii) f (ii) f (3) (3) (iii) x when f when f ((x) = 95 [6 marks] May/Jun 2011 8. (a) The functions f and g and g are de…ned by 2 f ( f (x) = 6x + 8; g(x) = x 3 (i) Calculate the value of g( 12 ) (ii) Write an expression for gf g f ((x) in its simplest form (iii) Find the inverse function f 1 (x) [6 marks] Jan 2012 9. (a) Make x Make x the subject of the formula y = 2xx+3 4 (b) Hence, determine the inverse of f of f ((x) = (c) Find the value of x of x for which f ( f (x) = 0
2x+3 x 4
where x = 4 ; where x
6
[6 marks] Jan 2013 10. (a) The functions f and g and g are de…ned as follows: f (x) = 2x + 5 3 g(x) = x 2 Evaluate (i) f (i) f 1 (19) (ii) g (ii) gf f (3) (3) [4 marks] May/Jun 2013 11. (a) Given that f ( f (x) = (i) f (i) f g (2) (ii) f (ii) f 1 (3)
2x+1 3
and g and g (x) = 4x + 5; 5; determine the value of
[6 marks] Geometry(Construction,Polygons, Similar & Congruent Shapes) May/Jun 2006(Polygons/Constructions) 2006(Polygons/Constructions) 1. (a) In the quadrilateral KLMN; not KLMN; not drawn to scale, LM = LN = LK; 0 = 140 ; and \ LK N = 40 0 \ K LM =
26
Giving the reason for each step of your answer, calculate the size of (i) \ LN K (ii) \ N LM (iii) \ K N M [6 marks] (b) Using a ruler, a pencil and a pair of compasses construct the triangle ABC in which AB = AB = 8cm = 600 ; and AC = 5cm 5 cm \ B AC = (Credit will be given for a neat, clear diagram) (i) Measure and state the length of B of B C: (ii) Find the perimeter of ABC: (iii) Draw on your diagram the line C D which is perpendicular to AB and meets AB at D at D:: (iv) Determine the length of C of C D: (v) Calculate the area of ABC giving ABC giving your answer to 1 decimal point. [12 marks]
4 4
4 4
Jan 2007(Constructions) 2. (a) Using a pencil, ruler and a pair of compasses only, construct ABC AB C with B with BC C = 6cm 6 cm and and AB = AC = AC = = 8cm: All construction lines must be clearly shown (b) Draw a line segment AD such that AD meets BC at D and is perpendicular to BC (c) Measure and state (i) the length of the line segment AD (ii) the size of angle ABC AB C [7 marks]
4
May/Jun 2007(Polygons) 2007(Polygons) 3. (a) The …gure below, not drawn to scale, scale , is a regular octagon with center X , X , and X and X Y = 6cm 6 cm
27
Calculate (i) the size of angle X Y Z (ii) the area of the triangle Y X Z , expressing your answer correct to one decimal place (iii) the area of the octagon [6 marks] Jan 2008(Polygons) 4. (a) The diagram below, not drawn to scale, scale, shows a quadrilateral ABCD with AB = AD, 0 0 = 90 and \ DBC D BC = = 42 , AB is parallel to DC \ B AC =
Calculate giving reasons for your answers, the size of EACH of the following angles (i) \ABC (ii) \ABD (iii) \BAD [6 marks] May/Jun 2008(Construction) 5. (a) Using (a) Using a pair of compasses, ruler and a pencil, pencil , construct a parallelogram ABCD in which AB which AB = AD AD = = 7cm and cm and the angle B AD AD = = 600 (b) Measure and wirte down the length of the diagonal AC [6 marks] Jan 2009(Constructions) 6. (a) The diagram below not drawn to scale, shows a rectangle PQRS with PQ = 7.0cm and QR = 5.5cm.
(i) Using a ruler, pencil and a pair of compasses only, cnonstruct the rectangle. [6 marks] May/Jun 2009(Constructions) 7. (a) The diagram below, not drawn to scale, shows a rhombus, PQRS , with the diagonal P R = 6cm, cm, and angle RP angle RP Q = 600
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(i) Using (i) Using a ruler, a pencil, and a pair of compasses , construct the rhombus PQRS accurately accurately (ii) Join QS , QS , Measure and state, in centimeters, the length of QS: of QS: [6 marks] Jan 2010(Constructions) 8. (a) Using (a) Using pencil, ruler, and a pair of compasses only: (i) Construct, accurately, the triangle ABC shown below, where AC = 6cm 6 cm 0 \ AC B = 60 AB = 600 \ C AB =
(ii) Complete the diagram to show the kite, ABCD; in ABCD; in which AD which AD = 5cm (iii) Measure and state the size of \ DAC D AC [6 marks] May/Jun 2010(Constructions/Polygons) 2010(Constructions/Polygons) 9. (a) Using a ruler, pencil and a pair of compasses, construct triangle E F G with EG = EG = 6cm EG = = 600 \ F EG and \ E F G = 900 (b) Measure and state (i) the length of E of E F (ii) the size of \EF G [7 marks] (c) The diagram diagram below, below, not not drawn to scale, scale , shows two straight lines, P Q and RS , RS , intersecting 29
a pair of parallel lines, T U and V and V W:
Determine, giving a reason for EACH of your answers, the value of (i) x (i) x (ii) y (ii) y [4 marks] Jan 2011(Constructions) 10. (a) Using pencil, ruler and a pair of compasses only, construct a triangle LM N with angle LM angle LM N = 60 0 , M N = 9cm 9 cm and and LM = 7cm 7 cm ALL construction lines MUST be clearly shown (b) Measure and state the size of \M N L (c) On the diagram, show the point, K , K , such that KLMN that KLMN is is a parallelogram [7 marks] May/Jun 2011(Constructions/Similar 2011(Constructions/Similar Triangles) 11. (a) Using a ruler, pencil, a pair of compasses and protractor, draw accurately a quadrilateral EFGH quadrilateral EFGH using using the measurements: 0 0 EF = 8cm 8 cm F G = 4cm EH = EH = 7cm \EF G = 125 \H EF = 70 (b) Measure and state in centimeters, the length of GH of GH [6 marks] 0
0
(c) The diagram below, not drawn to scale, shows OMN; and OMN; and its image, OM N under an enlargement with center, O, O , and scale factor, k. k . Angle O Angle ON N M = 90 0
4
Using the dimensions shown on the diagram, calculate (i) the value of k of k;; the scale factor of the enlargement (ii) the length of O of OM M (iii) the length of O of OM M 0
30
4
[4 marks] Jan 2012(Constructions/Polygons) 2012(Constructions/Polygons) 12. (a) Using (a) Using a pair of compasses, a ruler and a pencil (i) construct a triangle triangle C C DE in DE in which D which DE E = = 10cm; 10cm; DC = = 8cm and cm and 0 = 45 \ C DE = (ii) construct a line, C F; perpendicular F; perpendicular to DE D E such such that F that F lies on DE: D E: (iii) Using a protractor, measure and state the size of \DCE [7 marks] (b) The diagram below, not drawn to scale, shows TWO triangles, J KL and MLP; with MLP; with 0 J K parallel parallel to ML:LM = MP; KLP is is a straight line, angle J LM = 22 and angle LM angle LM P = 36 0
Calculate, giving reasons for your answers, the measure of EACH of the following: (i) \ M LP (ii) \ LJ K (iii) \ J KL (iv) \ K LJ [4 marks] (c) The diagram diagram below, below, not drawn to scale, scale , shows a regular hexagon with center, O, O , and AO and AO = 5cm
(i) Determine the size of angle AOB (ii) Calculate, to the nearest whole number, the area of the haxagon [5 marks] May/Jun 2012(Constructions) 13. (a) Using a ruler, pencil and a pair of compasses, construct a triangle P QR with QR with 0 0 P Q = 8cm; \P QR = QR = 60 \QP R = 45 (b) Measure and state the length of RQ of RQ [5 marks] Jan 2013(Similar Triangle) 14. (a) In the diagram below, not drawn to scale, ABC AB C is is an isosceles triangle with AB = AC = AC and 0 angle AB angle ABC C = = 54 . DE is DE is parallel to B C:
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Calculate, giving a reason for your answer, the measure of: (i) \ B AC (ii) \ AED AE D (iii) Explain why triangles ABC AB C and AD and ADE E are are similar but not congruent. [6 marks] May/Jun 2013(Similar Triangles/Construction) riangles/Construction) 15. (a) In the diagram below, not drawn to scale, AEC AE C and ADB and ADB are straight lines. 0 ABC = \ADE = = 90 \ABC = AC = = 10m;AB 10m;AB = = 8m and DB D B = 3:2m
(i) Calculate the length of BC (ii) Explain why triangles ABC AB C and AD and ADE E are are similar (iii) Determine the length of D of DE: E: [6 marks] (b) The diagram below shows an isosceles triangle CDE: G is the midpoint of C of C D
(i) Measure and state in centimeters, the length of D of DE E (ii) Measure and state, in degrees, the size of \EC D (iii) Determine the perimeter of the triangle C DE (iv) Calculate the area of the traingle CDE: [5 marks]
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Coordinate Geometry Jan 2006 1. (a) The equation of the line l is y = 4x + 5 (i) State the gradient of any line that is parallel to l (ii) Determine the equation of the parallel line to l that passes through the point (2; (2 ; 6) [4 marks]
Jan 2007 2. (a) P (a) P the point (2 point (2;; 4) and 4) and Q is the point (6 point (6;; 10) Calculate (i) the gradient of P of P Q (ii) the midpoint midpoint of P of P Q [4 marks] May/Jun 2008 3. (a) The diagram shows the graph of a straight line passing through the point A and B
The equation of the line above is y = mx = mx + c (i) State the value of c of c (ii) Determine the value of m of m (iii) Determine the coordinates of the mid-point of the line segment AB (b) The point ( point ( 2; k) lies lies on the line. Determin Determinee the value value of k of k (c) Determine the coordinates of the point of intersection of the line y = x = x and the line shown above [12 marks]
2
Jan 2009 4. The diagram below represents represents the graph graph of a staright staright line which passes through the points P and Q and Q
33
(a) State the coordinates of P of P and and Q Q (b) Determine (i) the gradient of the line segment P Q (ii) the equation of the line segement P Q (c) the point ( 8; t) lies on the line segment P Q:Determine Q:Determine the value of t of t The line segment, AB , is perpendicular to P Q; and Q; and passes through (6; (6 ; 2) (d) Determine the equation of AB of AB , in the form y = mx = mx + c [11 marks]
Jan 2010 5. (a) A straight line passes through the point T (4 T (4;; 1) and 1) and has a gradient of 53 : Determine the equation of this line. [3 marks] Jan 2011 6. The equation of a straight line is given by: 3y = 2x 6 Determine (i) the gradient of the line (ii) the equation of the line which is perpendicular to 3y 3 y = 2x 6, and passes through the point (4 point (4;; 7) [5 marks]
May/Jun 2011 7. (a) The diagram below shows the line segment which passes through the points A and B
34
Determine (i) the coordinates of A of A and B (ii) the gradient gradient of the line segment segment AB (iii) the equation of the line that passes through A and B [6 marks] May/Jun 2012 8. (a) The line l passes through the points S points S (6; (6; 6) and 6) and T (0; (0; 2): 2): Determine
(i) the gradient of the line l (ii) the equation of the line l (iii) the midpoint of the line segment S T (iv) the length of the line segment, T S [7 marks] Jan 2013 9. (a) A line segment GH has GH has equation 3 equation 3x x + 2y 2y = 15 (i) Determine the gradient of GH of GH (ii) Another line segment, J K , is perpendicular to GH and GH and passes through the point (4; (4; 1): 1):Determine the equation of the line J K [4 marks] May/Jun 2013 10. (a) A (a) A ( 1; 4) and 4) and B (3; (3; 2) are 2) are the end points of a line segment AB: AB : Determine (i) the gradient of AB of AB (ii) the coordinates of the midpoint of AB of AB (iii) (iii) the equation of the perpendicular perpendicular bisector bisector of AB of AB [7 marks]
Sets Jan 2006 1. (a) On a certain day, 300 customers visited a bakery that sells bread and cakes 70 customers bought cakes only 80 customers bought neither bread nor cakes 2x customers bought bread only x customers bought both bread and cakes
35
(i) U (i) U represents represents the set of customers visiting the bakery on that day, B represents the set of customers who bought bread, and C represents C represents the set of customers who bought cake. Copy and complete the Venn diagram to illustrate the information. (ii) Write an expression in x to represent the TOTAL number of customers who visited the bakery on that day. (iii) Calculate the number of customers who bought bread ONLY. [8 marks] May/Jun 2006 2. (a) In a survey of 39 students, it was found the 18 can ride a bicycle, 15 can drive a car, x can ride a bicycle and drive a car, 3x can do neither.
B is the set of students in the survey who can ride a bicycle, and C the C the set of students who can drive a car. (i) Copy and complete the Venn diagram to represent the information (ii) Write an expression in x for the number of students in the survey. (iii) Calculate the value of x of x [5 marks] Jan 2007 3. (a) Describe, using set notation only, the shaded regions in each Venn diagram below. …rst one is done for you
36
The
(b) The following information is given. U = 1; 2; 3; 4; 5; 6; 7; 8; 9; 10 P = prime numbers Q = odd numbers Draw a venn diagram to represent the information above
f f f
g
g
g
(c) The Venn diagram below shows the number of elements in each region
Determine how many elements are in EACH of the following sets: (i) A (i) A B (ii) A B (iii) ( (iii) (A A B) (iv) U (iv) U [10 marks]
[ \ \
0
May/Jun 2007 4.(a) The Venn Diagram below represents information on the type of games played by members mem bers of a youth club. club. All members members of the club play play at least one game.
S represents represents the set of members who play squash. T represen represents ts the set of mem members bers who play play tennis tennis H represe H represents nts the set of members members who play hockey hockey Leo, Mia and Neil are three members of the youth club. State what game(s) is/are played by (i) Leo (ii) Mia (iii) Neil (iv) Describe in words the members of the set H S: [5 marks] 0
\
Jan 2008 5. (a) S (a) S and T and T are are subsets of a universal set U such U such that: U = k;l;m;n;p;q;r S = = k;l;m;p
f f
g
g
37
T = k;p;q (i) Draw a ven diagram to represent this information List using set notation, the members of the set (ii) S T (iii) S (iii) S [6 marks]
f
g
[ 0
May/Jun 2008 6. (a) A uiversal set, U , U , is de…ned as: U = 15; 15; 16; 16; 17; 17; 18; 18; 19; 19; 20; 20; 21; 21; 22; 22; 23; 23; 24; 24; 25 Sets M Sets M and N and N are are subsets of U such U such that: M = Prime numbers and N = Even numbers (i) Draw a venn diagram to represents the sets M ; N and U and U (ii) List the elements of the set (M ( M N ) N ) [6 marks]
f
g
f
g
f
g
[ [
0
Jan 2009 7. (a) A school has 90 students is in Form 5 54 students study Physical Education 42 studetnts study Music 6 students study neither Physical Education nor Music x students study both Physical Education and Music (i) Copy and complete the Venn diagram shown below
(ii) Show on your Venn diagram the information relating to the students of Form 5 (iii) Calculate the number of students who study BOTH Physical Education and Music [6 marks] May/Jun 2009 8. (a) In a survey of 50 students , 23 owned cellular phones 18 owned digital camera x owned cellular phones and digital camera 2x owned neither Let C Let C represents represents the number of students in the survey who owned cellular phones, and D the set the set of students who owned digital cameras (i) Copy and complete the Venn diagram below to represent the information obtained from the survey
38
(ii) Write an expression in x for the TOTAL number of students in th survey (iii) Calculate the value of x: of x: [5 marks] Jan 2010 9. (a) T (a) T and E and E are are subsets of a universal set, U ,such U ,such that: U = 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 10; 11; 11; 12 T = multiples of 3 E = = even number (i) Draw a Venn diagram to represent this information List the elements of the set (ii) T E (iii) (T (T E ) [6 marks]
f f f
\ \ [ [
g g
g
0
May/Jun 2010 10. (a) A survey survey was conducted conducted among among 40 tourist. tourist. The results results were: were: 28 visited Antigua(A) 30 visited Barbados(B) 3x visited both Antigua and Barbados x visited neither Antigua nor Barbados (i) Copy and complete the Venn diagram below to represent the given information above
(ii) Write an expression, in x, x , to represent the TOTAL number of tourists in the survey, (iii) Calculate the value of x: of x: [6 marks] Jan 2011 11. (a) The universal set, U; is given as: U = Whole numbers from 1 to 12 H is H is a subset of U of U ,, such that H that H = = Odd numbers between 4 and 12 : (i) List the members of the set H J is J is a subset of U ,such U ,such that J = Prime numbers
f
g f f
g
g
39
(ii) List the members of the set J (iii) Draw a Venn diagram to represent the sets U; H and J; and J; showing ALL the elements in the subsets. [5 marks] May/Jun 2011 12. (a) The Venn diagram below shows the number of students who study Music and Art in a class class of 35 students students U = students in the class M = students who study Music A = students who study Art
f f f
g
g
g
(i) How many students study neither Art nor Music (ii) Calculate the value of x of x (iii) Hence, state the number of students who study Music only. [5 marks] Jan 2012 13. (a) A universal set U; is de…ned as: U = 51; 51; 52; 52; 53; 53; 54; 54; 55; 55; 56; 56; 57; 57; 58; 58; 59 A and B are subsets of U , U , such that: A = odd numbers B = prime numbers (i) List the members of the set A (ii) List the members of the set B (iii) Draw a Venn diagram to represent the sets A; B and U and U [5 marks]
f f f
g
g
g
May/Jun 2012 14. (a) In a survey of 36 students, it was found that 30 play tennis x play volleyball ONLY 9x play BOTH tennis and volleyball 4 play niether tennis nor volleyball (i) Copy and complete the Venn diagram below to show the number of students in the subset marked y marked y and z and z
40
(ii) Write an expression in x to represent the TOTAL number of students in the survey (iii) Write an equation in x to represent represent the total number number of students students in the survey and hence solve for x [5 marks] Jan 2013 15. (a) There are 50 students students in a class. Students Students in the class were were given awards awards for Mathemat Mathematics ics or Science 36 students received awards in either Mathematics or Science 6 students received awards in BOTH Mathematics and Science 2x students received awards for Mathematics only x students received awards for Sciecne only In the Venn Diagram below U = all the students in the class M = students who received awards in Mathematics S = = students who received awards for Science
f f f
g
g
g
(i) Copy and complete the Venn Diagram to represent the information about the awards given, showing the number of students in EACH subset. (ii) Calculate the value of x of x [6 marks] May/Jun 2013 16. (a) A survey of the 30 students in Form 5 showed that some students used cameras (C) or mobile phones (M) to take photographs. 20 students used mobile phones 4x students used ONLY cameras x students used BOTH mobile phones and cameras 2 students did not use either cameras or phones (i) Copy and Venn diagrams below and complete it to show, in terms of x of x,, the number of students in each region.
(ii) Write an expression, in terms of x of x,, which represents the TOTAL number of students in the survey (iii) Determine the number of students in Form 5 who used ONLY cameras [6 marks] 41
Statistics Jan 2006 1. (a) The data below are the lengths, to the nearest centimeter, of the right foot of the 25 students in a class
(i) Copy and complete complete the following following grouped frequency frequency table for the data above above
(ii) State the lower boundary of the class interval 14 16 (iii) State the width of the class interval 20 22 (iv) A student’s student’s right foot measured 16.8cm. 16.8cm. State the class interv interval al in which which this length length would lie. (v) A student was chosen at random from the group, and the length of his right foot was measured. Calculate the probability that the length was GREATER THAN or EQUAL to 20cm (vi) State the modal length of a student’s right foot (vii) Calculate an estimate of the mean length of a student’s right foot using the midpoint of the class intervals in (i) above. [11 marks]
May/Jun 2006 2. (a) In an agricultural experiment, the gains in mass of 100 cows during a certain period were recorded in kilograms as shown in the table below Gain Gain in Mass Mass ( kg) kg) Frequen requency cy Mid-In Mid-Inter terv val Valu Values( es(kg) kg) 5 9 2 7 10 14 29 12 15 19 37 17 20 24 16 25 29 14 30 34 2
(i) Copy and complete the mid-interval values column (ii) Calculate an estimate of the mean gain in mass of the 100 cows Hint: EACH of the 29 cows in the "10-14" interval is assumed to have a mass of 12kg (iii) On your answer sheet, complete the drawing of the frequency polygon for the gain in mass of the cows. (iv) Calculate the probability that a cow chosen at random from the experimental group gained 20 kg or more. 42
[11 marks] Jan 2007 3. The table below shows a frequency distribution of the scores of 100 students in an examination.
(i) Copy and complete the table above to show the cummulative frequency for the distribution. (ii) Using a scale of 2cm to represent a score of 5 cm on the horizontal axis and a scale of 2cm to represent 10 students on the vertical axis, draw a cummulative frequency curve of the scores. scores. Start your your horizon horizontal tal scale scale at 20. (iii) Using the cummulative frequency curve, determine the median score for the distribution. (iv) What is th probabiltiy that a student chosen at random has a score greater than 40. [12 marks] May/Jun 2007 4. (a) A class of 32 students participated in running a 400m race in preparation for their sports day. The time, in seconds, taken by each student is recoreded below 83 51 56 58 62 65 61 64 72 71 54 62 81 80 78 77 71 55 70 54 82 59 71 62 83 63 65 72 78 73 68 75 (i) Copy and complete complete the frequency frequency table to represen representt this data Time Time in second secondss Frequen requency cy 50 54 3 55 59 4 60 64 6 65 69 70 74 75 79 80 84
(ii) Using the raw scores, determine the range for the data. (iii) Using a scale of 2cm to represent 5 seconds on the horizontal axis and a scale of 1cm to represent 1 student on the vertical axis, draw a frequency polygon to represent the data. NOTE: An NOTE: An empty interval must be shown at each end of the distribution and the polygon closed (iv) To qualify for the …nals, a student must complete the race in less than 60 seconds. What is the probability that a student from this class will qualify for the …nals? [12 marks] Jan 2008 43
5. In a survey, all the boys in a Book Club were asked how many books they each read during the Easter Easter holiday holiday. The results results are shown shown in the the bar graph below. below.
(a) Draw a frequency table to represent the data shown in the bar graph. (b) How many boys are there in the Book Club? (c) What is the modal number of books read? (d) How many books did the boys read during the Easter holiday. (e) Calculate the mean number of books read. (f) What is the probability that a boy chosen at random read THREE OR MORE books? [12 marks] May/Jun 2008 6. (a) At a career guidance seminar, a survey was done to …nd out the type of careers that Form 5 students were likely to choose The results are shown in the table below.
There were 1080 students surveyed. (i) Calculate the value of t; of t; the number of students who were interested in becoming doctors. (ii) The data above above are to be represen represented ted on a pie chart. chart. Calculate Calculate the size of the angle in each sector of the pie chart. (iii) Using a cirlce of radius 4cm, construct the pie chart to represent the data. [10 marks] Jan 2009 7. The table below shows the distribution of the marks on a test for a group of 70 students.
44
(a) Copy and complete the table to show the cummulative frequency for the distribution (b) (i) Using Using a scale of 1cm to represent represent 5 marks on the horizontal horizontal axis and 1cm to represent represent 5 students on the vertical axis, draw the cummulative frequency curve for the scores (ii) What assumptions have your made in drawing your curve through the point (0,0)? (c) The pass mark for the test was 47. Use your graph to determine determine the number of students students who passed passed the test. (d) What is the probability that a student chosen at random had a mark less than or equal to 30? [12 marks] May/Jun 2009 8. The table below shows the time, to the nearest minute, that 80 students waited to be served at a school’s canteen.
(a) Copy and complete the table, showing the cummulative frequency (b) Use the values from your table to complete the cummulative frequency curve. (c) Use (c) Use your graph in (b) above to estimate (i) the median of the data (ii) the numberof students who waited for no more than 29 minutes (iii) the probability that a student chosen at random from the group waited for no mo more re than than 17 17 minutes. [12 marks] Jan 2010 9. A class of 26 students students each each recorded the distance distance travelle travelled d to school. school. The distance, distance, to the nearest km, is recorded below 21
11
3
22
6
32
22 45
18
28
26 39
16 17
17 22
34 24
12 30
25 18
8 13
19 23
14
(a) Copy and complete the frquency table to represent this data Distance Distance in kilometer kilometerss Frequency requency 1 5 1 6 10 2 11 15 4 16 20 6 21 25 26 30 31 35 36 40
(b) Using (b) Using a scale of 2cm to represent 5km on the horizontal axis and a scale of 1 cm to represent 1 student on student on the vertical axis, draw a histogram to represent the data. (c) Calculate Calculate the probability probability that a student student chosen at random random from this class recorded recorded the distance distance travelle travelled d to school as 26km or more. (d) The P.T.A P.T.A plans to set up a transportation transportation service service for the school. Which Which average, average, mean , mode or median median is the most appropriate appropriate for estimati estimating ng th cost of the service? Give Give a reaon for your answer. [11 marks] May/Jun 2010 10. A class of 24 threw the cricket cricket ball at sports. The distance distance thrown thrown by each student student was was measured measured to the nearest nearest meter. The results results are shown below. below. 22 50 35 52 47 30 48 34 45 23 43 40 55 29 46 56 43 59 36 63 54 32 49 60 (a) Copy and complete the table for the data shown above Distan Distance( ce(m) m) 20 29 30 39
Frequen requency cy 3 5
(b) State the lower boundary for the class interval 20 29 (c) Using a scale of 1cm of 1cm on the axis to represent 5 meters , and a scale of 1cm of 1cm on the y-axis to represent 1 student, draw a histogram to illustrate the data. (d) Determine (i) the number of students who threw the ball a distacne recorded as 50 meters or more. (ii) the probability that a student chosen at random threw the ball a distance recoreded as 50 meters or more. [11 marks]
Jan 2011 11. (a) The line graph below shows the monthly sales, in thousands of dollars, at a school cafeteria for the period January to May 2010.
46
(i) Copy and complete the table below to show the sales for EACH month
(ii) Between which TWO coonsecutive months was there the GREATEST decrease in sales? (iii) Calculate the mean monthly sales for the period January to May 2010 (iv) The TOTAL TOTAL sales for the period January January to June was $150 000. Calculate Calculate the sales sales in dollars for the month of June. (v) Comment on how the sales in June compared with the sales in the previous …ve months. [11 marks] May/Jun 2011 12. The table below below shows the distributi distribution on of the masses of 100 packages packages Mass(k Mass(kg) g) 1 10 11 20 21 30 31 40 41 50
No. No. of Pack Package agess 12 28 30 22 8
Cummu Cummulat lativ ive e Frequen requency cy 12 40
(a) Copy and complete the table to show the cummulative frequency for the distribution (b) Using (b) Using a scale of 2cm to represent 10 kg on the x- axis and 1cm to represent 10 packages on the y axis, draw axis, draw the cummulative frequency curve for the data (c) Estimate Estimate from the graph (i) the median mass of the packages (ii) the probability that a package, chosen at random, has a mass which is LESS than 35kg [12 marks]
Jan 2012 13. The histogram histogram below shows shows the distribut distribution ion of height height of seedlings seedlings in a sample. sample.
47
(a) Copy and complete the frequency table for the data in sample. Heig Height ht in cm MidMid-poi poin nt Frequ requen ency cy 1 10 5. 5.5 18 11 20 15.5 25 21 30 31 40 41 50 (b) Determine (i) the modal class interval (ii) the number of seedling in the sample (iii) the mean height of the seedlings (iv) the probability that a seedling chosen at random has a height that is GREATER than 30 cm. [12 marks]
May/Jun 2012 14. The table below shows the ages, to the nearest year, of the persons who visited the clinic during a particular week Age(yr Age(yrs) s) Number Number of person person Cummu Cummulat lativ ive e Freq Frequen uency cy 40 49 4 4 50 59 11 15 60 69 20 70 79 12 80 89 3 50 (a) Copy and complete the table to show the cummulative frequency (b) Using a scale of 2cm to represent 10 years on the x axis and 1cm to represent 5 persons on the y the y axis, draw the cummulative frequency curve for the data (c) Use your graph drawn at (b) above to estimate (i) the median age for the data (ii) the probability that a person who visited the clinic was 75 years or younger Draw the lines on your graph to show how these estimates were obtained [11 marks]
Jan 2013 15. The scores obtained by 100 children in a competition are summerized in the table below
48
Scor Score e Clas Classs midmid-poi point nt((x) Frequency( Frequency(f ) f x 0 9 4.5 8 36 10 19 14.5 13 188.5 20 29 25 30 39 22 40 49 20 50 59 12 Total 100 (a) (i) State the modal class interval (ii) State the class interval in which a score of 19.4 would lie (b) Copy and complete the table to show (i) the class mid-point (ii) the values of "f " f x xx" (iii) Calculate the mean score for the sample (c) Explain why the value of the mean obtained in (b) (ii) is only an estimate of the true value. (d) In order to qualify for the next round of the competition a student must score AT LEAST 40 points What is the probability that a student selected at random quali…es for the next round? [10 marks]
May/Jun 2013 16. The table below shows the amount, to the nearest dollar, spent by a group of 40 students at the school canteen during a period of one week. Amount Amount Spent($) Spent($) Number Number of Students Students Cumulati Cumulative ve Frequency requency 1 10 3 3 11 20 7 10 21 30 9 19 31 40 11 41 50 8 51 60 2 (a) Copy and complete the table to show the cummulative frequency (b) Using a scale of 1cm to represent $5 on the horizontal axis and 1cm to represent 5 students on the vertical axis, draw the cummulative frequency graph for the data (Marks will be awarded for axes appropriately labelled, points correctly plotted, and a smooth curve carefully drawn) (c) Use your graph to estimate (i) the median amount of money spent (ii) the probability that a student chosen at random spent less than $23 during the week. Show on your graph, using broken lines, how these estimates were determined [11 marks]
Vectors Jan 2006 1. The points A points A(1 (1;; 2); 2); B (5; (5; 2); 2); C (6; (6; 4) and 4) and D(2 D (2;; 4) are 4) are the vertices of a quadrilateral ABCD (a) Express in the form
x
y (i) the position vectors OA; OB; OC and OD where OD where O O is the origin (0; (0 ; 0) (ii) the vectors AB and DC
! ! ! ! ! !
!
j!j
!
(b) Calculate the AB and hence deteremine the unit vector AB (c) Using the answers in a(ii) (i) state TWO geometrical relationship between the line segments AB and D and DC C (ii) explain why ABCD is ABCD is a parallelogram (d) Using a vector method, determine the position vector of G of G,, the midpoint of AC: of AC: 49
Hence, state the coordinates of the point of intersection of the diagonals AC and B and B D of parallelogram ABCD parallelogram ABCD [15 marks] May/Jun2006 2. The diagram below shows the position vectors of two points, A and C , C , relative to an origin O
(a) Copy and complete the diagram to show (i) the point B such that OABC that OABC is is a parallelogram (ii) the vector u where u where u= = OA + OC x (b) Write as a column vector, in the form y the vector (i) OA (ii) OC (iii) AC (c) Given that G is the midpoint of O of OB B , use a vector method to (i) determine the coordinates of G of G (ii) prove, using a vector method, that A; G and C lie C lie on a straight line [15 marks]
! !
! ! ! ! !
Jan 2007 3. In the diag diagram ram belo below, w, M M is is the midpoint of ON
!
50
!
(a) (i) Sketch the diagram above in your answer booklet and insert the point X on OM such OM such that 1 OX = = 3 OM (ii) Produce P X on Q on Q such that P X = = 4X Q (b) Write down the following in terms of r of r and s and s (i) OM (ii) P X (iii) QM (c) Show that P N = 2 P M + + OP [15 marks]
!
! !
! ! !
!
!
!
! !
May/Jun 2007
!
!
4. OK and O and OM M are are position vectors such that OK =k = k and OM =m (a) Sketch Sketch the diagram diagram above. Show Show the approximat approximatee positions of points R and S such S such that R is the mid-point of O of OK K S is is a point on OM O M such such that OS = = 31 OM (b) Write down in terms of k of k and m and m the vectors (i) M K (ii) RM (iii) KS (iv) RS (c) L (c) L is the mid-point of RM of RM : Using a vector method, method , prove that RS that RS is is parallel to K L [15 marks]
! !
!
! ! ! ! !
Jan 2008 5. (a) In triangle ABC AB C , not drawn to scale. P and Q and Q are the mid-points of AB of AB and B and B C respectively.
(i) Make a sketch of the diagram and show the points P and Q and Q (ii) Given that AB = AB = 2x and BC = BC = 3y; write y; write in terms of x x and y; y ;an expression for (a) AC (b) P Q
! ! !
!
!
51
!
!
(iii) Hence show that P Q = 21 AC (b) The position vectors of the points R; S and T and T relative relative to the origin are 3 1 5 OR = OR = OS = = OT = 4 6 2 a (i) Express in the form the vectors b
!
! ! ! !
!
!
(a) RT (b) SR (ii) (a) The point F is F is such that RF that RF = F T :Use a vector method to determine the position vector of F of F (b) Hence, state the coordinates of F of F [15 marks] May/Jun 2008 6. The position vectors A and B relative to the origin are a and b respectively.
! ! ! !
The point P point P is is on OA OA such that OP = 2 P A The point M is M is on B on B A such that BM = M A (a) Copy the diagram and complete it to show the points P and M and M (b) O (b) OB B is produced to N such N such that O that OB B = BN B N (i) Show the position of N on N on your diagram (ii) Express in terms of a of a and b the vectors AB; P A and P M (c) Use a vector method to show that P; P ; M and N and N are are collinear (d) Calculate the length of AN of AN if 6 1 a = and b and b = 2 2 [15 marks]
! !
Jan 2009 7. The diagram below shows the position vectors OP and OQ
!
!
52
!
(a) Write Write as column column vector, vector, in the form
! (i) OP ! (ii) OQ
x y
(b) The point R point R has coordinates (8 coordinates (8;; 9)
x ! (i) Express QR as QR as a vector in the form y ! ! (ii) Using a vector method, show that, OP OP is parallel to QR ! ! (iii) Determine the magnitude of the vector P R (c) The point S point S has has the coordinates (a; ( a; b) (i) Write QS as as a column vector, in terms of a of a and b (ii) Given that QS = = OP , OP , calculate the value of a of a and the value of b of b (iii) Using a vector method, show that OPSQ is OPSQ is a parallelogram [15 marks]
! !
! ! !
May/Jun 2009
2 ! 4 ! 8. (a) The points A and B have position vector OA = OA = and OB = OB = where O where O is the 5 2 ! = AB ! origin (0 origin (0;; 0): 0): The point G point G lies on the line AB such that AG AG = 1 3
Express in the form
x
y ! ! ! (i) the vectors AB and AG ! (ii) the position vector OG (b) In the diagram below not drawn to scale, scale, B is the mid-point of OX of OX;; C is is the mid-point of AB; AB ; and D is such that OD O D = 2DA: DA: The The vectors a vectors a and b are such that OA = OA = 3a and OB = OB = b
!
!
53
Write in terms of a and b (i) AB (ii) AC (iii) DC (iv)DX (c) State TWO geometrical relationship between DX D X and D and DC C (d) State ONE geometrical relationship between the points D; D ; C and X and X [15 marks]
! ! ! ! !
Jan 2010 9. (a) The …gure not drawn to scale shows the points O(0 O (0;; 0); 0); A(5; (5; 0) and 0) and B ( 1; 4) which are the vertices of a triangle OAB O AB
(i) Express in the form
! ! ! !
a b
the vectors
(a) OB (b) OA + OB (ii) If M ( M (x; y ) is the mid-point of AB of AB;; determine the values of x of x and y (b) In the …gure below not drawn to scale OE;EF and M and M F F are straigh straightt lines. lines. The point point H is H is such that E F = 3EH: 3 EH: The point G point G is such that M that M F = 5MG: 5 MG: M is is the mid-point of OE
!
!
The vector OM = v and EH = u
(i) Write in terms of u of u and.or v and.or v , an expression for: (a) H F (b) M F (c) OH (ii) Show that OG = OG = 53 (2v (2v + u) (iii) Hence prove that O; O ; G and H lie H lie on a straight line [15 marks]
! ! !
!
May/Jun 2010 10. The diagram below not drawn to scale shows scale shows triangle J triangle J KL
54
M and N and N are are points on J on J K and J and J L respectively such that J M = 31 J K and J and J N = 31 J L (i) Copy the diagram in your answer booklets and show the points M and N and N (ii) Given Given that that J M = u and u and J N = v write in terms of u of u and v; v ; an expression for (a) J K (b) M N (c) KL (iii) Using your …ndings in (ii), deduce TWO geometrical relationship between KL and M N [8 marks]
! ! ! !
!
Jan 2011 11.(a) OP and OR are OR are position vectors with respect to the origin O 4 P is P is the point (2; (2 ; 7) and 7) and P R = 3 a Write in the form the vectors b (i) OP (ii) OR
!
!
! !
! !
(b) A point S point S has has coordinates (14 coordinates (14;; 2) (i) Find RS (ii) Show that P; P ; R and S are S are collinear
!
[8 marks] May/Jun 2011 12. (a) W (a) W XY X Y V is is a parallelogram in which V Y = a and V W = b S is is a point on W Y Y such that W that W S : : SY S Y = 1 : 2
!
!
Write in terms of a and b; an expresssion for: (i) W Y (ii) W S
! !
55
!
(iii) SX (b) R (b) R is the mid-point of V of V W: Prove W: Prove that R; that R; S and X and X are are collinear [8 marks] Jan 2012 13.(a) The diagram below shows two position vectors OA and OA and OB
! !
Write as a column vector, in the form
! ! (i) OA ! (ii) OB ! (iii) BA
!
x y
(b) Given that G is the mid-point of the line AB; AB ;write as a column vector in the form
! !
x y
:
(i) BG (ii) OG
[6 marks] May/Jun 2012
6 ! 3 ! 12 ! ! 14. (a) The points A; B and C and C have have position vectos OA = OA = OB = OB = and OC = = respectively 2 4 2 Express in the form
! !
x y
the vector
(i) BA (ii) BC (iii) State ONE geometrical relationship between B A and B C (iv) Draw a sketch to show the relative positions of A; of A; B and C and C [7 marks] Jan 2013 15. (a) The diagram below, not drawn to scale, shows a parallelogram OKLM where O where O is the origin. origin. The points points S is S is on K on K M such M such that M that M S = = 2SK: OK = = v and OM = u
!
56
!
Express EACH of the following in terms of u of u and v (i) M K (ii) SL (iii) OS [5 marks]
! ! ! !
May/Jun 2013 16. (a) In the diagram below, not drawn to scale, P and Q and Q are the midpoints of O of OA A and AB respectively OA = OA = 2a and OB = OB = 2b
!
!
Express in terms of a a and b the vectors (i) AB (ii) P Q (iii) State TWO geometrical relationships that exist between OB O B and P and P Q Give reasons for your answers [6 marks]
! !
57
Circle Theorem Jan 2006 1.
In the diagram above not drawn to scale. O is the center of the circle \AOB = AOB = 1300 0 = 30 :and AE and AEC C and B and B ED are chords of the circle. \DAE = (a) Calculate the size of EACH of the following angles, giving reasons for EACH step of your answers (i) \ACB (ii) \CB D (iii) \AED (b) Show that BC E and ADE are ADE are similar (c) Given that C E = = 6cm; EA = EA = 9:1cm and cm and DE D E = = 5cm (i) Calculate the length of E of E B (ii) Calculate correct to 1 decimal place the area of AED [15 marks]
4
4
4
May/Jun 2006 2. (a) The diagram below, not drawn to scale, shows a circle, center O: O : The lines B D and DCE D CE are tangents to the circle, and angle B CD = 700
Calculate, giving reasons for each step of your answer, (i) \OCE OC E (ii) \BAC (iii) \BOC (iv) \BDC
58
[8 marks] Jan 2007 3.(a) Two circles with centers P and Q and Q and radii 5cm and 2cm respectively are drawn so that they can touch each other at T and T and a straight line X Y at S at S and R: and R:
State with reasons (i) why P why P T Q is a straight line (ii) the length P Q (iii) why P why P S is is parallel to QR (b) N (b) N is is a point on P S such such that QN that QN is is perpendicular to P S Calculate (i) the length of P of P N (ii) the length of RS of RS (c) In the diagram below not drawn to scale, scale, O is the center center of the circle circle.. The measure measure of 0 angle LO angle LOM M is is 110
(ii) \LMO LM O Calculate, giving reasons for your answers, the size of EACH of the follwing angles. (i) \M N L [15 marks] May/Jun 2007 4. In the diagram below, not drawn to scale, LM is LM is a tangent to the circle at the point, T : O is the center of the circle and angle M T S = = 230
59
Calculate Calculate the size of each each of the following following angles, giving reasons for your your answer answer (a) angle T angle T P Q (b) angle M angle M T Q (c) angle T angle T QS (d) angle S angle S RQ [9 marks] Jan 2008 5. (a) In the diagram below, not drawn to scale, O is the center of the circle W XY and
\W XY
Calculate, giving a reason for EACH step of your answer, (i) \W OY (ii) \OW Y [4 marks] May/Jun 2008 6. (a) In the diagram below, not drawn to scale, scale, P Q is a tangent to the circle, center O:PR is O:PR is 0 parallel to OS O S and and angle S angle S P R = 26
60
= 50 0
Calculate giving reasons for your answer, the size of (i) angle P angle P T S (ii) angle RP angle RP Q [4 marks] Jan 2009 7. (a) The diagram below, not drawn to scale, scale, shows a circle with center, O: O : EA and E B are tangents to the circle, and angle AEB AE B = 480
Calculate, giving Calculate, giving a reason for your answer, answer , the size of EACH of the following angles: (i) \OAE (ii) \AOB (iii) \ACB (iv) \ADB [9 marks] May/Jun 2009 8. The diagram below, not drawn to scale, shows a circle, center O. O . The line DCE D CE is a 0 0 tangent tangent to the circle. circle. Angle Angle AC E = = 48 and angle O angle OCB CB = 26
61
Calculate (i) \ABC (ii) \AOC (iii) \BC D (iv) \BAC (v) \OAC (vi) \OAB [6 marks] Jan 2010 9. (a) The diagram below not drawn to scale shows two circles. C is C is the center of the smaller circle, GH circle, GH is is a common chord and DEF D EF is is a triangle. Angle GC Angle GC H = = 880 and angle GH angle GH E = = 1260
Calculate, giving Calculate, giving reasons for the answer, answer , the measure of angle (i) GF (i) GF H (ii) GD GDE E (iii) DEF D EF (b) Use (b) Use = 3:14 in 14 in this part of the question 62
Given that GC that GC = = 4cm, cm, calculate the area of (i) triangle GC triangle GC H (ii) the minor sector bounded by the arc GH and GH and radii GC radii GC and H and H C (iii) the shaded segment [15 marks] May/Jun 2010 10. (a) In the diagram below, not drawn to scale, scale, P Q is a tangent to the circle P T S R so R so that 0 \RP Q = 46 0 \RQP = 32 and T and T RQ is RQ is a straight line
Calculate, giving a reason for EACH step of your answer (i) \P T R (ii) \T P R (iii) \T SR [7 marks] Jan 2011 11.(a) The diagram below, not drawn to scale, scale, shows a circle cneter, O SGH is SGH is a tangent to the circle \F OG = OG = 1180 and \DGS = = 650
Calculate, giving Calculate, giving reasons for EACH step of your answer, answer , the measure of (i) \OF G (ii) \DEF [5 marks]
63
May/Jun 2011 12. (a) In the diagram below, not drawn to scale, scale, W; X ,Y and Z and Z are are points on the circumference of a circle, center O: O: T Y V is is a tangent to the circle at Y ; \XW Z = = 640 and \ZY V = 23 0
Calculate, Calculate, giving reasons for your answer, answer , the measure of angle (i) X (i) X Y Z (ii) Y (ii) Y XZ (iii) OX O X Z [7 marks] May/Jun 2012 13. The diagram below, not drawn to scale, shows a circle center, O. O . The like U V W is a tangent to the circle, ZOXW is is a straight line and angle U OX = OX = 700
Calculate, showing working, where necessary, the measure of angle (a) O (a) OU U Z (b) U V Y (c) U (c) U W O (d) Name the triangle in the diagram which is congruent to triangle (i) Z (i) Z OU (ii) Y (ii) Y XU [9 marks] Jan 2013 14.(a) The diagram below, not drawn to scale, scale , shows a circle, center O. O . RQ is RQ is a diameter and P M M and P / N N are tangen tangentt to the circle circle.. Angle Angle M P N = 54 0 and angle RQM angle RQM = 20 0
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Calculate, Calculate, giving reasons for your answer, answer , the measure of (i) \M RQ (ii) \P M R (iii) \P M N [7 marks] May/Jun 2013 15. (a) The diagram below, not drawn to scale, shows a circle with center O: EBC is is a tangent 0 0 to the circle. \OBA OB A = 40 and \OBF OB F = 35
Calculate, giving reasons for your answer, the measure of (i) \EBF EB F (ii) \BOA (iii) \AF B (iv) \OAF [7 marks]
65
Trigonometry(T rigonometry(Trig rig Ratios/Sin-Cos Ratios/Sin-Cos Rule/Bearings) May/Jun 2006 1. (a) A man walks xkm, x km, due north, from point G to point H: H :He then walks (x ( x + 7)km 7)km due east from H from H to to point F point F::The distance along a straight line from G to F is F is 13km. The diagram below, not drawn to scale, scale , shows the relative positions of G; G; H and F and F:: The direction of north is also shown.
(i) Copy the diagram and show on the diagram, the distances xkm, x km, ( (x x + 7)km 7)km and 13km (ii) From the information on your diagram, write an equation in x which satis…es Pythagoras’ Theorem. Theorem. Show Show that the equation equation can be simpli…ed simpli…ed to give give x 2 + 7x 7x 60 = 0 (iii) Solve the equation and …nd the distance GH (iv) Determine the bearing of F of F from G: from G: [11 marks]
(b) The diagram below, not drawn to scale, shows a vertical tower, F T ; and a vertical antenna, T W; mounted mounted on the top top of the tower. tower. A point P point P is is on the same horizontal ground as F; F ; such that 0 0 P F = 28m; 28 m; and and the angles of elevation of T of T and W and W from P from P are are 40 and 54 repectively.
(i) Copy and label the diagram diagram clearly showing showing (a) the distance 28m (b) the angles of 40 0 and 540 (c) any right angles (ii) Calculate the length of the antenna T W [7 marks] (c) The diagram below, not drawn to scale, shows parallelogram EFGH in in which E F = 6cm 6 cm 0 EH = 4: 4 :2cm; and cm; and angle F angle F EH = 70
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Calculate (i) the length of the diagonal H F (ii) the area of the parallelogram EFGH [5 marks] Jan 2007 2. A boat leaves a dock at a point A and travels for a distance of 15km to a point B on a bearing of 1350 The boat then changes changes course and then then travels travels for a distance of 8km to a poin p ointt C on C on a bearing of 060 0 (a) Illustrate the above information in a clearly labelled diagram The diagram should show the (i) north direction (ii) bearings 1350 and 0600 (iii) (iii) distances distances 8km and 15km (b) Calculate (i) the distance AC (ii) \BAC (iii) the bearing of A of A from C [15 marks] May/Jun 2007 3.
(a) Three towns, P; P ; Q and R are such that the bearing of P P from Q from Q is 0700 . R is 10km due east of Q Q and P Q = 5km (i) Calculate, correct to one decimal place, the distance P R: (ii) Given that \QP R = 1420 , state the bearing of R of R from P from P:: [6 marks]
p
(b) Given that sin = 23 Express in fractional or surd form (i) cos cos (ii) tan tan (iii) Hence, determine the exact value of
sin cos
[7 marks] Jan 2008 4. (a) The diagram below, not drawn to scale, shows a vertical pole, AD, and a vertical 67
tower, tower, BC standing on horizontal horizontal ground XABY. XABY. The height height of the pole is 2.5 meters. meters. 0 From the point D, the angle of depression of B is 5 and the angle of elevation of C is 20 0 DE is a horizontal line.
Calculate, to one decimal place (i) the horizontal distance AB (ii) the height of the tower, BC. [6 marks] May/Jun 2008 5.(a) In the diagram below not drawn to scale, GH is GH is a vertical pole standing on a horizontal plane and H; J and K and K are are points on the horizontal plane
GH = 12m 12 m and the angle of elevation of the top of the pole G from J from J and K and K are are 320 0 and 27 respectively. (i) Copy the diagram and insert the angles of elevation (b) Calculate to one decimal place (i) the length of H of H J (ii) the length of J of J K [6 marks] (c) A ship leaves Port R; sails to Port S Port S then then to Port T Port T 0 The bearing of S from S from R R is 112 The bearing of T T from S from S is is 0330 The distance RT distance RT is is 75 km ans the distance RS is RS is 56 km (i) Draw a diagram showing the journey of the ship from R to S to T to T Show on your diagram (ii) the North direction (iii) the bearing 112 0 and 0330 (iv) the points R, R , S and T and T (v) the dista distance ncess 75 km and 56 km (d) Calculate (i) the size of angle RS T (ii) the size of angle RT S (iii) the bearing of R of R from T from T 68
(e) The ship leaves Port T and T and travels due west at a point X which X which is due north of R of R (i) show on your diagram the journey from T to X to X (ii) Calculate the distance T X [15 marks]
Jan 2009 6.(a) The diagram below not drawn to scale shows two straight wires, LJ and N and N K; supporting vertical, M J . J . LMN LM N is is a straight straight line line on the horizonta horizontall plane. Angle Angle 0 0 J M L = 90 , angle K angle K M N = 30 ; LJ = 13m; 13 m; LM = 5m 5 m and K N = 10m 10 m
Calculate in meters the length of (i) M (i) M K (ii) J (ii) J K [6 marks] (b) On the diagram below not drawn to scale, scale, T U = 8m 8 m; T W = 10m 10 m and V W = 11m: 11 m: 0 0 angle U angle U T W = 60 and angle W angle W U V = 40
Calculate (i) the length of U of U W (ii) the size of angle U V W (iii) the area of triangle T U W [6 marks] May/Jun 2009 7. (a) The diagram below not drawn to scale, shows the position of two hurricanes tracking stations, P stations, P and Q and Q,, relative to a point O:P O :P is is on a bearing of 025 0 from O from O;; 0 and O and OP P = = 400km: 400km: Q is Q is on a bearing of 080 from P from P and P and P Q = 700km 700km
(i) Copy the diagram diagram above. above. On your diagram diagram label the angles that show the bearings 69
of 0250 and 0800 Calculate (ii) \OP Q (iii) the length to the nearest kilometer of O of OQ Q (iv) the bearing of Q of Q from O from O [9 marks] Jan 2010 8. (a) The diagram below not drawn to scale, shows a triangle LMN;with LMN;with LN LN = 12cm; 12 cm; N M = xcm and xcm and \N LM = 0 The point K point K on LM on LM is is such that N K is is perpendicular to LM;NK = LM;NK = 6cm 6 cm and and K M = 8cm 8 cm
Calculate the value of (i) x (i) x (ii) [5 marks] (b) The diagram below, not drawn to scale, shows three stations P; P ; Q and R; such that the bearing of 0 0 Q from R from R is 116 and the bearing of P P from R from R is 242 . The veri verical cal line line at at R shows the North direction
70
(i) Show that angle P angle P RQ = RQ = 1260 (ii) Given that P R = 38 meters 38 meters and QR = 108 meters, 108 meters, calculate the distance P Q Give your answer to the nearest meter (c) K; (c) K; L and M are M are points along a straight line on a horizontal plane, as shown below
A vertical pole, S K is is positioned such that the angles of elevation of the top of the pole S from L from L and M are M are 210 and 140 respectively The height of the pole, S K; is K; is 10 meters (i) Copy and complete the diagram to show the poles S K and and the angles of elevation of S from S from L L and M Calculate correct to one deicmal place (i) the length of K of K L (ii) the length of LM of LM [15 marks] May/Jun 2010 9. (a) The diagram below, not drawn to scale, shows a vertical ‡agpole, F T with T with its foot F;on F;on the horizontal plane EFG: ET and GT and GT are are wires which support the ‡agpoles in its position. position. The angle angle of elevatio elevation n of T of T from G is 550 , E F = 8m 8 m F G = 6m and 0 \EF G = 120
Calculate giving your answers correct to 3 signi…cant …gures (i) the height F T ; of the ‡agpole (ii) the length E G (iii) the angle of elevation of T of T from E from E [8 marks] Jan 2011 10. (a) J; (a) J; K and L and L are three three sea ports. ports. A ship began began its journey journey at J sailed J sailed to K then K then to L to L and and returned to J: The bearing of K K from J from J is is 0540 and L and L is due east of K K J K = = 122 km 122 km and K and K L = 60 km 60 km (i) Draw Draw a carefully carefully diagram diagram tor represent represent the above informat information. ion. Show Show on 71
the diagram (ii) the north/south direction (iii) the bearing 054 0 (iv) the distance 122 km and 60 km Calculate (v) the measure of anlge J KL (vi) the distance J L (vii) the bearing of J of J from from L L [10 marks] May/Jun 2011 11. (a) The diagram below, not drawn to scale, shows PQR; which PQR; which represents the cross section of a roof. QS is is perpendicular to P SR 0 P Q = 12: 12:6 meters QR = QR = 8:4 meters \QP R = 15
4
Using the dimensions shown on the diagram, calculate, correct to 3 signi…cant …gures (i) the length of QS of QS (ii) the measure of \RQS (iii) the area of P QR [8 marks] (b) The diagram below, not drawn to scale, shows the route of an aeroplane ‡ying from Portcit Portcity(P) y(P) to Queenstown(Q) Queenstown(Q) and then to Riversda Riversdale(R) le(R).. The bearing of Q from P is 0 0 132 and the angle P QR is QR is 56
4 4
(i) Calculate the value of x of x,, as shown in the diagram (ii) The distance from Portcity( Portcity (P ) P ) to Queenstown( Queenstown(Q) is 220 kilometers and the distance from Queenstown to Riversdale (R ( R) is 360 kilomete kilometers. rs. Calculate Calculate the distanc distancee RP (iii) Determine the bearing of R from P [8 marks] Jan 2012 72
12. (a) The diagram below, not drawn to scale, shows a vertical pole, P L; standing L; standing on a horizontal plane, KLM: The angle of elevation elevation of P of P from K from K is is 280 , K L = 15m 15m 0 LM = 19m 19 m and \KLM = KLM = 115
(i) Copy Copy the diagram diagram.. Show Show the angle angle of elev elevatio ation, n, 28 0 and ONE right angle (b) Calculate, Calculate, giving giving your your answer answer to 2 signi…can signi…cantt …gures, …gures, the measure of (i) P (i) P L (ii) K (ii) K M (iii) the angle of elevation of P of P from M from M [8 marks] May/Jun 2012 13. (a) The diagram below, not drawn to scale, scale, shows the journey of a ship which started at port P port P;; sailed 15 km due south to port Q;then Q; then a further 20 km due west to port R
(i) copy the diagram and label it to show the points Q and R, R , and the distances 20 km and 15 km. (ii) Calculate P R the shortest distance of the ship from the port where the journey started. (iii) Calculate the measure of angle QP R, giving your answer to the nearest degree. [7 marks] (b) The diagram below not drawn to scale, shows a quadrilateral QRST in QRST in which QS which QS = = 7cm 0 0 ST = 10cm 10 cm;; QT = 8cm 8 cm , , \SRQ = SRQ = 60 and RS and RS Q = 48
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Calculate (i) the length of RS of RS (ii) the measure of \QT S [6 marks] Jan 2013 14.(a) The diagram below is a scale drawing showing the line RT and RT and the north direction on a playground It is drawn to a scale of 1centimeter:30 of 1centimeter:30 meters
Using the answer sheet provided (i) measure and state, in centimeters, the length of RT of RT as as drawn on the diagram (ii) measure and state, in degrees, the size of the angle that shows the bearing of T (iii) calculate the actual distance, in meters, on the playground that RT represents RT represents (b) A point M on M on the playground is located 300 meters from R on a bearing of 120 0 On the same answer sheet (i) calculate, in centimeters, the length of RM of RM that that should be used on the scale drawing (ii) using a ruler and a pair of compasses, draw the line RM on RM on the scale drawing (iii) mark and name the angle in the scale drawing that measures 120 0 [12 marks] (c) The diagram below, not drawn to scale, shows th position of three points A; B and C and C on on a horizontal plane.
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AB = AB = 174 meters, 174 meters, B B C = = 65 meters 65 meters and AC and AC = = 226 meters 226 meters Calculate (i) the measure of angle ABC AB C (ii) the area of triangle ABC AB C [4 marks] May/Jun 2013 15. (a) The diagram below, not drawn to scale, scale, shows three points R; S and F and F on on the horizontal ground. F T is T is a vertical vertical tower tower of height 25m. 25m. The angle of elevatio elevation n of the top of the tower, T , T , from R from R is 270 . R is due east of F F and S and S is is due south of F:SF of F:SF = 43: 43 :3 m
(i) Sketch separate diagrams of the triangle RF RF T ; T F S and and S S F R. Mark on EACH diagram diagram the given given measures measures of sides and angles angles (ii) Show, by calculation, that RF = 49: 49 :1m (iii) Calculate the length of S of S R correct to 1 decimal place. (iv) Calculate Calculate the angle of elevatio elevation n of the top of the tower, tower, T ; from S: from S: [8 marks]
Matrices Jan 2006 1. (a) The matrix L =
x 4
1 x (i) Calculate in terms of x of x,, the determinant of L of L (ii) Calculate the values of x of x given that L is singul singular ar [3m [3mark arks] s]
(b) The matrix M =
3 1 2 6
(i) Find M Find M 1 the inverse of M M (ii) Hence calculate the values of x of x and y for which M
x 12 y
=
8
[6 marks] May/Jun 2006 2. (a) The value of the determinant M =
2 3 1 x
is 9
(i) Calculate the value of x of x (ii) For this value of x of x,, …nd M …nd M 1 (iii) Show that M 1 M = I [ 7 marks] 75
Jan 2007 3. (a) Given that D =
1 9 p
is a singular matrix, determine the value(s) of p of p
p 4 (b) Given the linear equations
2x + 5y 5y = 6 3x + 4y = 8 (i) Write the equstion in the form AX = AX = B where A; where A; X and Bare and Bare matrices matrices (ii) (ii) a) Calcul Calculate ate the determ determina inant nt of the matrix matrix A b) Show that A that A 1 =
4 7 3 7
5 7 2 7
c) Use the matrix A 1 to solve for x and y [15 marks] May/Jun 2007 4. (a) A; (a) A; B and C and C are are three 2
2 matrices such that A = A =
a b c
d
; B =
5 3 3 2
; and C = C =
14 0 9
5
Find (i) 3A (ii) B1 (iii) 3A + B 1 (iv) the value of a;b of a;b;; c and d given that 3 that 3A A + B 1 = C [7 marks] Jan 2008
1 x 5. (a) A (a) A = = 2 1 , B = and C and C = 5 6 y 2 Given that AB that AB = C , C , calculate the of x of x and y. y . 2 values 1 (b) Given that the matrix R = R =
1 3 (i) Show that R is non-singular (ii) Find R Find R 1 the inverse of R R 1 (iii) Show that RR = I (iv) Using a matrix method, solve the pair of simultaneous equations 2x y = 0 x + 3y 3y = 7 [15 marks]
May/Jun 2008 6. (a) X and Y and Y are are two matrices where 2 0 4 1 X = = and Y and Y = 5 1 3 7
Evaluate X Evaluate X 2 + Y [4 marks] Jan 2009 7.
(a) (a) Calc Calcul ulat atee the the ma matr trix ix produ product ct 3 3AB AB = = 3AB; where A where A = =
1 2
1 3
and B and B = 2 1 2 5 [3 marks] (b) (i) Write the following simultaneous eqautions in the form AX = AX = B where A where A,, X and B and B are matrices 11x 11x + 6y 6y = 6 9x + 5y 5y = 7 (ii) Hence, Hence, write the solution for x and y as products of two matrices [4 marks]
Jan 2010 76
8. (a) L (a) L and N are N are two matrices where L =
3 2
1 4 Evaluate L Evaluate L
and N and N =
1 3 0
2
2
N
(b) The matrix, M , M , is given as M =
x 12 3
. Calculate the value of x of x for which M which M is is singular
x
[5 marks] May/Jun 2010 9. (a) A (a) A and B are two 2 A =
1
2 matrices such that 5 2 2 and B and B = 5 2 1
2 (i) Find AB Find AB (ii) Determine B 1 ; the inverse of B (iii) Given that 5 2 x 2 = 2 1 y 3 x Write as the product of TWO matrices y (iv) Hence, calculate the values of x of x and y
[7 marks] May/Jun 2011 10. (a) Determine the inverse of the matrix
3 5 2 4
[2 marks] Jan 2012 11. (a) L (a) L and M are M are two matrices where L =
3 2
1 4 Evaluate (i) L (i) L + 2M 2M (ii) LM
and M =
1 3 0
2
(b) The matrix, Q, Q , is such that Q =
4 2 1 1
(i) Find Q Find Q 1 amatrix method, …nd the values of x of x and y in the equation (ii)4 Using 2 x 8 1 1
y
=
3
[9 marks] May/Jun 2012
a 4 2 4 2 0 12. (a) (i) Calculate Calculate the the values values of a a and b such that = 1 b 1 3 0 2 2 4 (ii) Hence or otherwise, otherwise, write down the inverse inverse of 1 3 2 4 (iii) Use the inverse of 2 4 x 1123 to solve for x and y in the matrix equation 1
3
y
=
7
[8 marks]
Jan 2013 13.(a) 13.(a) A superstore sells sells 3 models models of cell phones. Model A cost $40 each, each, model B cost $55 77
each and model C cost $120 each The weekly sales for 2 weeks in June were: Week 1 Week 2 2 model A no model A 5 model B 6 model B 3 model C 10 model C (i) Wrtit Wrtitee down a matrix matrix of sizes 3 2 which represents the sales for the two weeks (ii) Write down a matrix of size 1 3 which represents the cost of the di¤erent models of cells cells phones phones (iii) Write down the multiplication of the two matrices which represents the superstore’s takings from the sale of cell phones for each of the two weeks [4 marks]
May/Jun 2013 14. (a) Given that M =
2 1 4 3
(i) Evaluate M Evaluate M 1 , the inverse of M of M : (ii) Show that M 1 M = I (iii) Use a matrix method to solve for r;s;t and r;s;t and u in the equation
2 1 r s 2 1 4 3
t
u
=
4
1
[9 marks]
Speed-Distance-Time Jan 2008 1. (a) John left Port A at 0730 hours and travels to Port B in the same time zone (i) He arrives arrives at Port B at 1420 hours. hours. How long long did the journey take? take? (ii) John travel travelled led 410 kilometers. kilometers. Calculate Calculate his average average speed in kmh 1 [3 marks] Jan 2009 2. (a) An answer sheet is provided for this question The distance-time graph below describes the journey of a train between two train stations, stations, A and B. Answer Answer the question question on on the answer sheet
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(i) For how many minutes was the train at rest? (ii) Determine the average speed of the train, in km/h, on its journey from A to B. The train continues its journey away from station A and B to another station C, which is 50 km from B. The averag averagee speed of th journey was was 60 km/h. (iii) Calculate the time, in minutes, taken for the train to travel from B to C. (iv) On (iv) On your answer sheet, sheet , draw the line segment which describes the journey of the train from B to C. [10 marks] May/Jun 2009 3. (a) The table below shows two reading s taken from an aircraft’s ‡ight record
For the period of times between the two readings, calculate. (i) the distance travelled in kilometers (ii) the average speed of the aircraft in km/h [4 marks] May/Jun 2010 4.(a) The diagram below shows th speed-time graph of the motion of an athelete during a race.
Using th graph determine (i) the MAXIMUM speed. (ii) the number of seconds for which the speed was constant. (iii) the TOTAL distance covered by the athlete during the race. During which time period of the race was (iv) the speed of the athlete in increasing. (v) the speed of the athlete decreasing. (vi) the acceleration of the athlete zero? [7 marks] May/Jun 2011 5.(a) The speed -time graph below, not drawn to scale, scale, shows the three stage journey of a racing car over a period of 60 seconds. 79
During the FIRST stage of the journey, the car increased its speed from 0m/s to 12m/s in x seconds accelerating at 0.6m/s2 (i) Calculate the value of x (ii) What is the gradient of the graph during the SECOND stage? Explain, in one sentence, what the car is doing during this stage. (iii) Calculate the distance travelled by the car on the THIRD stage of the journey. [6 marks] Jan 2012 6. (a) The follo following wing is an extract extract from a bus schedul schedulee . The bus begins begins its journey journey at Bellevie Belleview, w, travels to Chagville and ends its journey at St. Andrews. Town Arrive Depart Belleview 6:40 a.m Chagville 7:35 7:35 a.m a.m 7:45 7:45 a.m a.m St. Andrews Andrews 8:00 a.m (i) How long did the bus spend in Chagville? (ii) How long did the bus take to travel from Belleview to Chagville? (iii) The bus travelled at an average speed of 54 km/hour from Belleview to Chagville. Calculate, in kilometers, the distance from Belleview to Chagville. [4 marks]
MayJun 2013 7. (a) A car, travelling along a straight road at a constant speed of 54km/h, takes 20 seconds to travel the between two sign post. Calculate (i) the speed of the car in m/s (ii) the distance, in meters, between the two sign posts. [4 marks]
Graphs-Curves Jan 2006 1. (a) The path a ball thrown in the air is given by the equation h = 20t 20t 5t2 where h where h is the vertical distance above the ground (in meters) and t is the time (in seconds) after the ball was thrown. The table below shows some of the values of t of t and the values of h; h; correct to 1 decimal place.
80
(a) Copy and complete the table of values (b) Draw the graph of h of h = 20t 20t 5t2 for 0 t 4: You may proceed as follows: (i) Using a scale of 2cm to represent 0.5 seconds, draw the horizontal t-axis for 0 t 4 (ii) Using a scale 1cm to represent 1 meter, draw the vertical h axis for 0.0 h 21: 21:0 (iii) Plot the points from your table of values on the axes drawn. (iv) Join the points with a smooth curve.
(c) Using your graph calculate estimates of (i) the GREATEST height above the ground reached by the ball. (ii) the number of seconds for which the ball was MORE than 12 meters above the ground. (iii) the interval of time during which the ball was moving upwards. [11 marks] MayJun 2006 2. The diagram below shows the graph of the function f ( f (x) = x2 to the graph at (2, -3) is also drawn.
y
2x 3 for a x b. The tangent
5 4 3 2 1
-2
-1
1
2
3
-1 -2 -3 -4
Use the graph to determine the (a) values of a a and b which de…nes the domain of the graph (b) values of x x for which x 2 2x 3 = 0 (c) coordinates of the minimum point on the graph (d) whole number values of x for which x 2 2x 3 < 1 < 1 (e) gradient of f of f ((x) = x2 2x 3 at x = x = 2 [11 marks] Jan 2008 3. Given the y = x = x2 4x, copy and complete the table below. x -1 -1 0 1 2 3 4 5 (a) y 5 -3 -4 0 (b) Using (b) Using a scale of 2cm to represent 1 unit on both axes , draw the graph of the function y function y = x = x 2 4x for -1 x 5
81
4
x
(c) (i) On the same axes used in (b) above, draw the line y = 2 (ii) State the x - coordinates of coordinates of the two points at which the curve meets the line (iii) Hence, write the equation whoose roots are the x-coordinates stated in (c) (ii) [11 marks] May/Jun 2008 4. (a) The temperature, K; of a liquid t liquid t minutes after heating is given in the table below.
(i) Using a scale of 2cm to represent 10 minutes on the horizontal axis and a scale of 2cm to represnt 10 degrees on the vertical axis, construct a temperature-time graph to show how the liquid cools in the 60 minute interval Draw a smooth curve through all the plotted points. (b) Use your graph to estimate (i) the temperatue of the liquid after 15 minutes (ii) the rate of the cooling of the liquid at t = 30 minutes 30 minutes [7 marks] May/Jun 2009 5. (a) Given that y = x = x2 + 2x 2x 3 (i) Copy and complete the table below
(ii) Using scale of 2cm to represent 1uint on the x axis and axis and 1cm to represent 1 unit on the y axis, axis, draw the graph of y of y = x = x 2 + 2x 2x 3 for 4 x 2 [7 marks]
Jan 2010 6. (a) The graph shown below represents a function of the form f ( f (x) = ax2 + bx + c
82
Using the graph above determine (i) the value of f of f ((x) when x when x = 0 (ii) the values of x of x when f when f ((x) = 0 (iii) the coordinates of the maximum point (iv) the equation of the axis of symmetry (v) the values of x of x when f when f ((x) = 5 (vi) the interval within which x lies when f ( f (x) > 5 > 5:: Write your answers in the form a < x < b [11 marks] May/Jun 2010 7. (a) The diagram below shows the graph of y of y = x = x2 + 2x 2x
83
3 for the domain 4 x 2
Use the graph above to above to determine (i) the scale used on the axis (ii) the value of y of y when x when x = = 1:5 (iii) the values of x of x when y when y = 0 (iv) the range of values of y of y;; giving our answer in the form a y b, where a where a and b are real numbers. [7 marks] Jan 2012 8. The table below shows corresponding corresponding values values of x of x and y for the function y = x = x 2 2x 3, for integer values of x of x from -2 to 4 x -2 -2 -1 -1 0 1 2 3 4 y 5 -3 -4 0 5 (a) Copy and complete the table (b) Using (b) Using a scale of 2cm to represent 1 unit on the x-axis, x -axis, and 1cm to represent 1 unit on the y-axis y -axis,, plot the points whose x whose x and y values are recorded in your table, and draw a smooth curve through your points (c) Using your graph,estimate the value of y of y when x when x = = 3:5. Show on your graph how the values was obtained (d) Without furthur calculations, (i) write the equation of the axis of symmetry of the graph (ii) estimate the minimum value of the function y (iii) state the values of the solution of the equation: x2 2x 3 = 0 [11 marks]
Jan 2013 9. (a) The table below shows corresponding values of x of x and y for the function y = x3 ; x =0 where y where y represents the velocity of a particle after x seconds
6
x(sec) 00..25 y (m=s) m=s) 12
0. 0.5
1 3
2 1.5
3
4 0.75
5
6 0.5
(i) Copy and complete the table for the function (ii) Using a scale of 2cm to represent 1 unit on the x axis and axis and 1cm to represent 1 unit on the y the y axis, plot the points from your table, drawing a smooth curve through all points. [7 marks]
Inequalities Jan 2007 1.(a) Solve for x where x where x (i) 4 (i) 4 x 13
2Z [3 marks]
Jan 2008 2. (a) (i) Solve the inequality: 3 2x < 7 (ii) If x x is a whole number, determine the SMALLEST value that satis…es the inequality in (a) (i) above [3 marks]
Jan 2010 3.(a) (i) Solve the inequality x 3 < 3 < 3x x 7 (ii) If x x is an integer, determine the SMALLEST value of x of x that satis…es the inequality above. [4 marks]
May/Jun 2011 4. (a) (i) Solve the inequality: 5 2x < 9 (ii) If x x is an integer, determine the SMALLEST value of x of x that satis…es the inequality above.
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[3 marks]
Linear Programming Jan 2006 1. (a) A shop stocks x Sonix and y and y Zents Zents radio. radio. It has shelf shelf space for for 20 radios radios (i) Write an inequalitiy to represent this information. The owner of the shops spends $150 to purchase each Sonix radio and $300 for each Zent radio, she has $4500 to spend on the purchase of these radios (ii) Write an inequality to represent this information. The owner of the book shop decides to stock at least 6 Sonix and at least 6 Zents (iii) Write TWO inequalities to represent this information. (b) (i) Using a scale of 2cm to represent 5 Sonix radios and 2cm to represent 5 Zent radios, draw the horizontal axis for 0 x 30 and 30 and the vertical axis for 0 for 0 y 25: 25: (ii) On these axes, draw the four boundary lines for the four inequalitites written in a(i),(ii) and (iii) above. (iii) Shade the region on your graph that satis…es ALL four inequalities written in a(i),(ii) and (iii) above (iv) State the coordinates of the vertices of the shaded region.
(c) The owner of the shop sells the radios to make a pro…t of $80 on each Sonix and $100 on each Zent radio (i) Express the TOTAL pro…t in terms of x of x and y (ii) Calculate the MAXIMUM pro…t. [15 marks] May/Jun 2006 2. (a) The Owner of a parking lot wishes to park x vans and y and y cars for persons attending a function. The lot provides parking space for no more than 60 vehicles. (i) Write an inequality to represent this information To get a good bargain, he must provide parking space for at least 10 cars. (ii) Write an inequality to represnt this information. The number of cars parked must be fewer than or equal to twice the number of vans parked. (iii) Write an inequality to represent this information. (iv) (a) Using a scale of 2cm to represent 10 vans on the x axis axis and 2cm to represe represent nt 10 cars on the y the y axis, draw the graph of the lines associated with the inequality (i),(ii) and (iii) above. (b) Identify, by shading, the region which satis…es all three inequalities.
The parking fee for a van is $6 and for a car is $5. (v) Write an expression in x and y for the total fees charged for parking x vans and y and y cars. (vi) Using your graph write down the coordinates coordinates of the vertices of the shaded region. (vii) (vii) Calculate Calculate the maximum maximum fees charged. charged. [15 marks] Jan 2007 3. (a) Pam visits the stationery store where she intends to buy x pens and y and y pencils Pam must buy at least 3 pens (i) Write an inequality to represent this information The TOTAL number of pens and pencil must NOT be more than 10 (ii) Write an inequlaity to represent this information 85
EACH EACH pen cost $5.00 and EACH pencil pencil cost $2.00. More inforam inforamtion tion about the pens and pencils is represented by: 5x + 2y 2y 35 (iii) Write the information represented by this inequality as a sentence in your own words.
(b)(i) (b)(i) On the answer answer sheet provided, provided, draw the graph of TWO inequalit inequalities ies obtained obtained in a(i) and a(ii) above. (ii) Write the coordinates of the vertices of the region that satis…es the four inequalities (including y 0)
(c) Pam sells the x pens and y and y pencils and makes a pro…t of $1.50 on EACH pen and $1.00 on EACH pencil. (i) Write an expression in x and y to represent the pro…t Pam makes. (ii) Calculate the maximum pro…t Pam makes. (iii) If Pam buys 4 pens, show on your graph the graph the maximum number of pencils she can buy. [15 marks] May /Jun 2007 4. A company company manufact manufactures ures gold gold and and silve silverr stars stars to be used as party party decoratio decorations. ns. The stars stars are placed in packets so that each packet contains x gold star and y silver stars. The condition for packaging are given in the table below: Condition (1) Each packet must have at least 20 gold stars (2) Each packet must have at least 15 silver stars (3) The total number of stars in each packet must not be more than 60 (4) (a) Write down the inequality to represent conditions (2) and (3) (b) Describe in words, the condition represented by th inequality x < 2y 2 y (c) Using a scale of 2 cm to represent 10 units on both axes, draw the graphs of ALL FOUR inequalities represented in the table above. (d) Three packets packets of stars were selected selected for inspection. inspection. Their conten contentt are shown shown below Pack Packet et No. No. of gol gold d stars( stars(x x) No. of silv silver er star star((y ) A 25 20 B 35 15 C 30 25 Plot the points A,B and C on your graph. Hence determine which of the three packets satisfy ALL the conditions. [15 marks] Jan 2008 5. (a) A school buys x balls and y bats. The total number of balls and bats is no more than 30 (i) Write an inequality to represent this information. The school school budget allow allowss no more than $360 to be spent on balls balls and bats. The cost of a ball is $6 and the cost of a bat is $24. (ii) Write an inequality to represent this information (b) (i) Using a scale of 2cm on the x-axis to represent 10 balls and 2cm on the y axis to axis to represent 5 bats, bats , draw the graphs of the lines associated with the inequalities at (a)(i) and (ii) above. (ii) Shade the region which satis…es the two inequalities at a(i) and (ii) and the inequalities x inequalities x 0 and y 0 (iii) Use your graph to write the coordinates of the vertices of the shaded region.
86
Inequality x 20
x < 2y 2 y
(c) The balls and bats are sold to students students.. The school makes makes a pro…t of $1 on each each ball and $3 on on each each bat. The equatio equation n P = x + 3y 3y represents the total pro…t that may be collected from the sale of these items (i) Use the coordinates of the vertices given at (b) (iii) above to determine the pro…t for each of those combinations. (ii) Hence, state the maximum pro…t that may be made. [15 marks] May/Jun 2008 6. (a) The shaded region below shows the solution of a set of inequalities in x in x and y. y . The variabl ariablee x represents the number of boys in a cricket club and y represents the number of girls in the cricket club.
(b) Use the graph to answer the questions which follow. (i) State, State, using argument argument based on the graph, whether whether the cricket cricket club can have members: (a) 10 boys and 5 girls (b) 6 boys and 6 girls (ii) Write Write down the set of THREE inequalitit inequalitites es that de…ne the shaded shaded region region (iii) (iii) A company company sells sells uniform for the club and makes of pro…t of $3.00 on every every boy’s uniform and $5.00 on every girl’s uniform (a) Write an expression in x and y that represents the total pro…t made by the company on the sale of uniforms. (b) Calculate the minimum pro…t the company can make. [15 marks] May/Jun 2009 7. (a) The owner of a shop wishes to buy x guitars and y and y violins. violins. To satisfy satisfy the demands demands of customers, the number of violins must be less than or eqaul to the number of guitars. (i) Write an inequality to represent this information. The cost of one guitar is $150 and the cost of one violin is $300. He has $4500 to spend on the purchase of these instruments. 87
(ii) Write an inequality to represent this information. To get a good bargain the owner of the shop must buy at least 5 violins (iii) Write an inequality to represent this information. (b) (i) Using Using a scale of 2cm on the the horizontal axis horizontal axis to represent 5 guitars and 2cm on the vertical to represent 5 violins, draw the graph of the lines associated with the THREE inequalities written in a(i),(ii) and (iii) above. (ii) Shade the region on your graph that satis…es all THREE inequalities (iii) State the coordinates of the vertices of the shaded region. (c) The owner of the shop sells instrument to make a pro…t of $60 on each guitar and $100 on each violin. (i) Express the total pro…t in terms of x of x and y: y : (ii) Calculate the maximum pro…t. maximum pro…t. [15 marks] Jan 2010 8. (a) The manager of a pizza shops wishes to make x small pizzas and y large large pizzas. pizzas. His oven oven holds no more than 20 pizzas (i) Write an inequality to represent the given condition The ingredients for each small pizzas cost $15 and for each large pizzas cost $30. The manager plans to spend no more than $450 (ii) Write an inequality to represent this condition. (b) (i) Using Using a scale of 2cm on the x axis to represent 5 small pizzas and 2cm on the y axis to represent 5 large pizzas, draw the graph of the lines associated with the THREE inequalities written in a(i) and a(ii) above.
(ii) Shade the region which is de…ned by the all THE following combined: - the inequalities writtin in a(i) and a(ii) above - the inequalities x 0 and y 0:
(iii) Using your graph, state the coordinates of the vertices of the shaded region. (c) The pizza can makes makes a pro…t of $8 on the sale of EACH EACH small pizza pizza and $20 on the sale of EACH large pizza. pizza. All pizzas pizzas that were made made were sold. (i) Write an expression in x and y for th eTOTAL pro…t made on the sale of the pizzas (ii) Use the coordinates of the vertices given at b(iii) to determine the MAXIMUM pro…t. [15 marks] May/Jun 2010 9. (a) A farmer supplies his neighbors with x pumpkins y pumpkins y melons daily, using the following condition First condition: y 3 Second condition: y x Third condition: condition: the total number number of pumpkins and melons must not exceed exceed 12 (i) Write an inequality to represent the third condition (ii) Using a scale of 1cm of 1cm to represent 1 pumpkin on the x the x axis and 1cm and 1cm to represent 1 melon on the y-axis, draw the graphs of the THREE lines associated with the THREE inequalities. (iii) Identify, by shading, the region which satis…es the THREE inequalities. (iv) Determine, from your graph, the minimum values minimum values of x of x and y which satisfy the conditions. [8 marks]
Jan 2012 88
10. (a) The diagram below shows the graphs of three lines and a shaded region de…ned by three inequalities associated with these lines. The inequality associated with the line 3y 3 y = x = x is is 3y 3 y x
(i) State the other TWO inequalities which de…ne the shaded region. The function p function p = = 4x + 3y 3y satis…es the solution set represented by the closed triangular region. (ii) Identify the three pairs (x; ( x; y) values for which p has a maximum or a minimum value. (iii) Which pair of ( of (x; x; y) values makes p makes p a maximum? Justify your answer. answer. [9 marks] May/Jun 2012 11. (a) A ‡orist bouquets each consisting of x of x roses and y and y orchi orchids. ds. For each each bouquet, she applies the following constraints. - the number of orchids must be at least half the number of roses - there must be at least two roses - there must be no more than 12 ‡owers (i) Write THREE inequalities for the constraints given (ii) On the answer sheet provided, shade th region of the graph which represents the solution set for the inequalities in a(i). (iii) (iii) State State the coordinates coordinates of the points points which which represen represents ts the vertices vertices of the region showing the solution set. (iv) The ‡orist sells a bouquet of ‡owers to make a pro…t of $3 on each rose and $4 on each each orchid. orchid. Determin Determinee the MAXIMUM MAXIMUM possible pro…t pro…t on the sale of a bouquet. [8 marks] May/Jun 2013 12. (a) Trish wishes to buy x oranges and y and y mangoes mangoes which which she intends intends to carry in her bag. Her bag has space for only 6 fruits. (i) Write an inequality to represent this information To get a good bargain, she must buy AT LEAST 2 mangoes. (ii) Wrtie an inequality to represent this information. More information about the number of oranges and mangoes associated with the good bargain is represented by
89
y 2x (iii) Write the information represented by this inequality as a sentence in your own words. (iv) On (iv) On the answer sheet provided, provided , draw the lines associated with the two inequalities obtained in (i) and (ii) above. (v) Shade on your graph the region which represents the solution set for the three inequalities [8 marks]
Completing the Squares Jan 2006 1.(a)(i) 1.(a)(i) Express Express 4 4x x2 12x 12x 3 in the form a form a((x + h)2 + k; where k; where a; a; h and k are real numbers. Using your answer in a(i) above or otherwise, calculate (ii) the minimum value of 4 of 4x x2 12x 12x 3 (iii) the value of x of x for which the minimum occurs (iv) the values of x for which 4x2 12x 12x 3 = 0 expressing your answers to 3 signi…cant …gures. [9 marks]
Jan 2007 2. (a) Given that f ( f (x) = 2x 2 x2 + 4x 4x
5
(i) write f write f ((x) in the form f ( f (x) = a( a(x + b)2 + c; where c; where a;b a;b;; c R (ii) state the equation of the axis of symmetry (iii) state the coordinates of the minimum point (iv) sketch the graph of f of f ((x) (v) on the graph of f of f ((x) show clearly (a) the minimum point (b) the axis of symmetry [9 marks]
2
Jan 2009 3. (a)(i) Express the function 2x 2 x2
4x 13 in 13 in the form f form f ((x)
= a( a(x + h)2 + k:
Hence, or otherwise, determine (ii) the values of x of x at which the graph cuts the x-axis x -axis (iii) the interval for which f ( f (x) 0 (iv) the minimum value of f of f ((x) (v) the value of x of x for which f ( f (x) is a minimum
[12 marks] May/Jun 2009 4.(a)(i) Express 2x 2 x2
3x + 1 in 1 in the form f form f ((x)
= a( a (x + h)2 + k where a;h; where a;h; and and k are real numbers
(b) Using your answer above or otherwise, calculate (i) the minimum value of 2 of 2x x2 3x + 1 (ii) the value of x of x for which the minimum occurs
(c) Sketch the graph of y of y = 2x2 3x + 1, 1, clearly showing - the coordinates of the minimum point - the value of the y-intercept - the values of x x where the graph cuts the x axis [9 marks]
. Jan 2010 5. (a) Given g( g (x) = 3x 3 x2 8x + 2 (i) write g write g((x) in the form a( a (x + b)2 + c, where a; where a; b and c R (ii) solve the equation g (x) = 0 writing your answer(s) correct to 2 decimal places (iii) A sketch of the graph of g (x)is shown below
2
90
Copy the sketch and state (iv) the y the y coordinate of A: A: (v) the x the x-- coordinate of C: C: (vi) the x the x and y coordinate of B B :
[10 marks] Jan 2011 6. (a) (i) Express the quadratic function 1 h and k are constants.
2
2
6x x , in the form k a(x + h) , where a; where a;
Hence state (ii) the maximum value of 1 of 1 6x x2 (iii) the equation of the axis of symmetry of the quadratic function. (iv) determine the roots of 1 of 1 6x x2 = 0 giving your answers to 2 decimal places. [8 marks]
May/Jun 2011 7. (a) (i) Express the function f ( f (x) = 4x 4 x2 8x 2 in the form a form a((x + h)2 + k; where k; where a; a; b and k are constants. State (ii) the minimum (iii) the values of x of x for which f ( f (x) is a minimum. [4 marks]
Jan 2013 8. (a) (i) Write f ( f (x) = 3x 3 x2 5x + 1 in 1 in the form a form a((x h)2 + k where a; where a; h and k are constants to be determined . (ii) Hence, or otherwise, determine the minimum value of f of f ((x) and the value of x for which f ( f (x) is a minimum. (iii) Solve the equation 3x2 5x + 1 = 0 expressing 0 expressing your answer correct to two decimal places. [8 marks]
May/Jun 2013 9. (a) (i) Write 3x 3 x2 12x 12x + 8 in 8 in the form a( a (x + h)2 + k where a; where a; h and k are constants 2 (ii) Sketch the graph of y of y = 3x 12x 12x + 8; 8; showing on your sketch (a) the intercept on the y-axis y -axis (b) the coordinates of the minimum point. [7 marks]
Transformation & Matrix Transformation Jan 2006 1. (a) The diagram below shows a quadrilateral P QRS; QRS; and and its image, quadrilateral P Q R S ; after it has been b een rotated. 0
91
0
0
0
0
(i) State the coordinates of the points R 0 and S and S (ii) Describe the rotation completely. (iii) On the answer sheet provided, draw and label quadrilateral P Q R S which is the image of P P Q R S after it has been re‡ected in the x-axis. x -axis. (iv) Describe completely the single transformation that maps quadrilateral PQRS onto onto P Q R S : [10 marks] (b) The image, ( image, (x; x; y) of any point, ( point, (x; x; y), under any transformation N is N is given by the equation x 2 0 x 2 = + 0 2 y 1 y 00
0
00
00
00
0
0
00
00
00
0
00
0
0
Calculate (i) the image (x ( x ; y ); of (3 of (3;; 1) under 1) under N N : (ii) the coordinates of the point, (x; ( x; y); which is mapped by N N onto (7 onto (7;; 4): 4): [6 marks] 0
0
May/Jun 2006 2.(a) The graph below shows the line segment AC and AC and its image A C after a transformation by p q the matrix r s 0
92
0
(i) Write in the form of a single 2 2 matrix the coordinates of (a) A and C (b) A and C (ii) Using matrices only, write an equation to represent the transformation of AC onto AC onto A A C : (iii) Determine the value of p;q;r of p;q;r and s: and s: [8 marks]
0
0
0
0
Jan 2007 3. (a) On the answer sheet provided, ABC is ABC is mapped onto A B C under a re‡ection. (i) Write down the equation of the mirror line. A B C is mapped onto A B C by a rotation of 180 0 about the point (5; (5 ; 4): 4): (ii) Determine the coordinates of the vertices of A B C (iii) State the transformation that maps ABC onto ABC onto A B C
4
4
0
0
0
4
00
00
0
0
00
00
4
4
y
0
4 00
4
00
00
00
00
10 9
C
C'
B
B'
8 7 A
6
A'
5 4 3 2 1 0 0
1
2
3
4
5
6
[6 marks]
93
7
8
9
10 10
x
May/Jun 2007 4. (a) L (a) L M N is the image of LM of LM N under N under an enlargement. 0
0
0
(i) Write on your answer sheet (a) the scale factor for the enlargement (b) the coordinates coordinates of the center of the enlargement enlargement.. L M N is the image of LM of LM N under N under a re‡ection in the line y = (ii) Draw and label the triangle L M N on your your answer answer booklet b ooklet [5 marks] 00
00
00
00
00
y
00
x
L'
8
N'
6
L
4
M' N M
2
-5
--4 4
-3
--2 2
-1
1
2
3
4
5
6
7
8
x
-2
-4
(b) The diagram below shows a parallelogram EFGH and and its images after undergoing two successive transformation.
(i) Describe in words, the geometric transformations (a) J (a) J which which maps EFGH onto E onto E F G H onto E (b) K (b) K which which maps E maps E F G H H onto E F G H (ii) Write the matrix which represents the transformation described above as 0
0
0
0
0
00
0
0
00
00
94
00
(a) J (a) J (b) K (b) K (iii) The point P (6 P (6;; 2) is 2) is mapped onto P onto P by the transformation J: State the coordinates of P P (iv) The point Q point Q(5 (5;; 4) is 4) is mapped onto Q by the transformation K: State K: State the coordinates of Q Q [8 marks] 0
0
0
0
Jan 2008 5. The diagram diagram below shows a pattern made of congruent congruent right-ang right-angled led triangles. triangles. In each each triangle, triangle, the sides meeting meeting at a right angle are 1 unit and 2 units long.
(i) Describe FULLY the single transformation that will map triangle BCL, onto triangle FHL. (ii) Describe FULLY the single transformation that will map triangle BCL onto triangle HFG. [6 marks] May/Jun 2008 6. (a)(i) The …gure …gure labelle labelled d P , P , undergoes a transformation, such that its image is Q: Describe this transformation completely.
95
(ii) On the answer sheet provided, provided , draw and label. (a) the line y line y = x = x (b) S (b) S the the image of P under P under a re‡ection in the line y = x = x [6 marks] (b) The vertices of triangle DEF D EF are D(5; (5; 12); 12); E (2; (2; 7) and 7) and F (8 F (8;; 4) (i) The triangle DEF D EF underogoes underogoes an enlargement with center, O; O ; and scalefactor, k, its iamge is D is D E F where D(5; (5; 12) D (7: (7:5; 18): 18): (a) Determine the value of k of k (b) Hence write down the coordinates of E of E and F and F 0
!
0
0
0
0
0
0
0
0
(ii) D E F undergoes a clockwise rotation of 90 0 about the origin. (a) Determine the 2 2 matrix that represents a clockwise rotation of 90 0 about the origin. (b) Determine the coordinates of D of D E F the image of D D E F under this rotation (c) Determine the 2 2 matrix that maps triangle DEF D EF onto onto triangle D E F [9 marks]
00
00
00
0
0
0
00
00
00
Jan 2009 7. (a) The diagram diagram below shows shows three triangles triangles labelled labelled L;M:and L;M:and N N L is mapped onto M by M by a translation.
(i) State, in the form
x
; the vectors which represents the translation. y (ii) L is mapped onto N by N by an enlargement. (a) Find and label on label on your answer sheets the sheets the point G; the center of the enlargement. Show Show your method clearly
. (b) State the coordinates of G: of G: (c) State the scalefactor of the enlargement. [6 marks] (b) The diagram diagram belo b elow, w, not not drawn to scale, shows a triangle, ABC AB C whose coordinates are stated. 96
Triangle AB Triangle ABC C undergoes undergoes two successive transformation, V followed V followed by W by W ,, where 2 0 1 0 V = and W and W = 0 2 0 1 (i) State the e¤ect of V of V on triangle ABC: (ii) Determine the 2 2 matrix that represents the combined transformation of V of V followed by W: by W: (iii) Using your matrix in b(ii), determine the coordinates of the image of triangle ABC under ABC under this combined transformation. [8 marks]
May/Jun 2009 8. (a) The diagram below shows triangles A; B and D and D.. The The line line y y = x = x is is also shown.
(i) Describe FULLY the single transformation which maps triangle A onto (a) triangle D (b) triangle B (ii) State the coordinates coordinates of the vertices vertices of triangle C; the image of triangle A after a re‡ection in the line y = x: = x: 97
[10 marks]
1 0 (b) The transformation R is represented by the matrix 00 11 The transformation S transformation S is is represented by the matrix a b 1 0 (i) Write a single matrix in the form
c
d
to represent the combined
transformation S transformation S followed followed by R: (ii) Calculate the image of the point (5; (5 ; 2) under 2) under the combined transformation in b(i) above. [5 marks]
Jan 2010 9. (a) (i) Using scale of 1cm to represent 1 unit on both axes , draw the trinagle ABC AB C with vertices A(2 A (2;; 3); 3); B (5; (5; 3) and 3) and C (3 C (3::6): 6): (ii) On the same axis used in (a)(i), draw and label the line y = 2: (iii) Draw the triangle ABC AB C under under a re‡ection in the line y = 2. Label Label the the . image A image A B C : 0
0
0
00
00
00
00
00
(iv) Draw a new triangle A B C with vertices with A ( 7; 4), 4), B ( 4; 4) and 4) and C ( 6; 7): 7): 00
(v) Name and describe the single transforma transformation tion that maps triangle triangle ABC AB C onto onto triangle A triangle A B C : [9 marks] 00
00
00
(b) A geometric transformation, R; maps the point (2; (2 ; 1) to 1) to ( 1; 2): 2): 0 p Given that R that R = , calculate the values of p of p and q q 0 r (c) A translation T = maps the point (5; (5 ; 3) onto 3) onto (1 (1;; 1); 1); Determine the values of r of r and s s
(d) Determine the coordinates of the image of (8 of (8;; 5) under 5) under the combined transformation. R followed by T : [10 marks] May/Jun 2010 10. The diagram below shows triangle triangle LMN, and its image, triangle triangle L 0 M N , after undergoing a rotation. 0
98
0
(i) Describe the rotation FULLY by stating (a) the center (b) the angle (c) the direction (ii) State TWO geometric relationships between triangle LM N and N and its image, triangle L M N 1 (iii) Triangle LM riangle LM N is N is translated by the vector 2 Determine the coordinates of the image of the poin L under this transformation. [7 marks] 0
0
0
Jan 2011 11.(a) The diagram below shows triangle RS T and T and its image R S T after a transformation. 0
0
0
0
(i) Wrtite down the coordinates of R of R and R (ii) Describe completely the transformation which maps triangle RS T onto T onto triangle R S T (iii) RS T undergoes T undergoes an enlargement, center, (0; (0 ; 4); 4); scale factor 3 (a) On your answer sheet, draw triangle R S T , the image of triangle RS T T under the enlargement. (b) Given that the area of triangle RS T is T is 4 square units, calculate the area of triangle R S T : (c) State TWO geometrical relationship between triangles RS T and R and R S T : [12 marks] 0
00
00
00
00
00
00
00
(b) Under a matrix transformation, M = 0
a b c
d
00
00
; the point, V and W and W are are mapped onto
0
V and W and W such that: V (3 V (3;; 5) W (7 W (7;; 2)
0
! V (5; (5; 3) ! W (2; (2; 7) 0
(i) Determine the values of a;b of a;b;; c and d: (ii) State the coordinates coordinates of Z of Z such such that Z (x; y ) Z (5; (5; 1) under 1) under the transformation M: (iii) Describe FULLY the geometric transformation M: M : [7 marks]
!
99
0
0
0
May/Jun 2011 12. (a) The transformation, M =
0 a
0
0
; maps the point R point R and T T onto R onto R and R and R such that:
b 0 R(7; (7; 2) R (2; (2; 7) and 7) and T ( T ( 5; 4) T (4; (4; 5) (i) Determine the values of a of a and b (ii) Describe fully the transformation, M :
! !
0
0
[5 marks] Jan 2012 13. (a) The diagram below shows a triangle, P QR, QR, and its image P Q R : 0
0
0
(i) State the coordinates of P of P and Q: and Q: (ii) Describe fully the transformation that maps triangle P QR onto QR onto triangle P Q R: (iii) Write down the coordinates of the images of P of P and Q and Q under the translation 3 : 6 0
0
May/Jun 2012 14. (a) The graph below shows triangle LM N and N and its image P QR after QR after an enlargement. On the answer sheet provided
100
(i) Locate the center of enlargement, showing clearly your method clearly. (ii) State the scale factor and the coordinates of the center of enlargement. (iii) Determine the value of of Are Area a of P QR Area of LMN (iv) Draw and label the triangle ABC with coordinates ( 4; 4); 4); ( 1; 4); 4); and ( 1; 2) respectively. 2) respectively. (v) Describe fully the single transformation which maps triangle LM N on N on to triangle ABC: [11 marks]
Jan 2013 15. (a) The matrix matrix J J =
0 1
represents a single transformation. 1 0 The image of the point P under P under transformation J is (5 is (5;; 4): 4): Determine the coordinates of P of P (b) (i) Write down a matrix, H; H ; of size 2 2 which represents an enlargement of scale factor 3 about the origin. (ii) Determine the coordinates of the point (5; (5 ; 7) under 7) under the combined transformation H followed H followed by J [6 marks]
May/Jun 2013 16. (a) The graph below shows triangle LM N and N and its image L M N after undergoing a single transformation. 0
101
0
0
y
8 7 N
6
N'
5 L
4
M
M'
L'
3 2 1 0 0
1
2
3
4
5
6
(i) Describe fully the transformation that maps
7
8
9
10
0
11
0
12
13
14 14
x
0
4LM N N onto 4L M N : 00
00
00
(ii) On the answer sheet provided, provided , draw triangle L M N the image of triangle 0 LMN; after LMN; after a translation by the vector 3:
(iii) (iii) Name and describe describe a com combinat bination ion of TWO transformat transformation ion which may be used to map L M N onto L M N : [7 marks]
4
00
00
00
4
0
0
0
Sequence and Patterns May/Jun 2006 1.(a) The drawing below shows a sequence of square made from toothpicks.
102
Column 1 Column 2 Length, n, of Pattern for one side of calcula calculati ting ng the square Number Number of tooth tooth picks picks In squa square re
Column 3 Total number of toothp toothpick ickss In squa square re
1
1x2x2
4
2
2x3x2
12
3
3x4x2
24
4 7
n s
r
220
(i) draw the next shape in the sequence (ii) insert appropriate values in column 2 and 3 when (a) n (a) n = 4 (a) n (a) n = 7 (b) Complete the table by inserting appropriate values at (i) r (i) r (ii) s (ii) s [10 marks] Jan 2007 2. (a) A large large equilatera equilaterall triangle triangle is subdivided subdivided into a set of smaller smaller equilateral equilateral triangle triangle by the following procedure: The midpoint of the sides of each equilateral triangle are joined to form a new set of smaller triangles. The procedure is repeated repeated many many times. times. The table shows the results when the above procedure has been repeated twice, that is when n when n = 2
103
(i) Calculate the number of triangles formed when n = 3: (ii) Determine the number of triangles formed when n = n = 6 A shape has 65536 triangles. (iii) Calculate the value of n of n (iv) Determine the number of small triangles in a shape after carrying out the procedure m times. [10 marks] May/Jun 2007 3. Rectangle W Rectangle W XY X Y Z below below represents one whole unit which has been divided into seven smaller parts. parts. These parts are labelled labelled A;B;C A;B;C;D;E; ;D;E; F and G: and G:
(a) Copy and complete the following table, stating what fraction of the rectangle each part represents. Part Part Fractio raction n A B 1 C 24 D E F 1 G 18 (b) Write the parts in order to the size of their perimeters. (c) The area of G is 2 square units. units. E,F and G are rearranged rearranged to form a trapezium trapezium (i) What is the area of the trapezium in square units? (ii) Sketch the trapezium clearly showing the outline of each of the three parts. [10 marks] Jan 2008 4. (a) The table below shows the work done by a student in calculating the sum of the …rst n natural numbers. Informati Information on is missing missing from some rows of the table. Study the pattern pattern and complete, complete, in your answer booklet, booklet , the rows marked (i) and (ii)
104
(b) After doing additional calculation, the student stated that: 13 + 23 + 33 = 36 = 62 and 13 + 23 + 33 + 43 = 100 = 10 2 Using the pattern observed in these two statements, determine the sum of the series (i) 13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 (ii) 13 + 23 + 33 + :::n 3 (c) Hence, or otherwise, determine the EXACT value of the sum of the series 13 + 23 + 33 + 43 + ::: + 123 [10 marks] Jan 2009 5. The …gure below below shows the …rst three diagrams diagrams in a sequence. sequence. Each Each diagram is made up of dots connected connected by line line segments. segments. In each diagram, diagram, there there are d dots and l and l line segments.
On the answer sheet provided (a) Draw the fourth diagram in the sequence (b) Complete the table by inserting the missing values at the rows marked (i) and (ii)
105
(c) (i) How How many dots are in the sixth diagram diagram of the sequence? sequence? (ii) How many line segments are in the seventh diagram of the sequence? (iii) Write the rule which shows how l is related to d: [10 marks] May/Jun 2009 6. The drawing drawing show the …rst three three diagram diagram in a sequence. sequence. Each diagram diagram in the sequence sequence is obtained by drawing a 1-unit square on each side that forms the perimeter of the previous diagram . For example, Diagram 2 is obtained by drawing a 1-unit square on each of the four sides of Diagram 1.
(i) Draw diagram 4 in the sequence (ii) Complete the table by inserting the appropriate values at the rows marked (i),(ii) and (iii). [10 marks] Jan 2010 7. Bianca Bianca makes hexagons hexagons using using sticks of equal length. length. She then creates creates pattern pattern s by joining joining the hexagon hexagon together. together. Pattern Pattern 1,2 and 3 are shown shown below: below:
106
The table below shows the number of hexagons in EACH pattern created and the number of sticks used to make each pattern.
(a) Determine the values of (i) x (i) x (ii) y (iii) z (b) Write down an expression for S in S in terms of n, n , where S where S represents represents the number of sticks used to make a pattern of n n hexagons. (c) Bianca uses 76 sticks to make a pattern of h of h hexagon. hexagon. Determine Determine the the value value of h: of h: May/Jun 2010 8.The diagram diagram below below shows the …rst three …gures …gures in a sequence sequence of …gures. …gures. Each Each …gure is made up of square of side 1cm
107
(a) On your answer sheet draw the FOURTH …gure (Fig 4) in the sequence. (b) Study the pattern in the table shown below, and on the answer sheet provided, complete the rows numbered (i), (ii), (iii) and (iv).
[10 marks] Jan 2011 9. The answer sheet shows a rectangle A; of area 20 square units and perimeter 24 square units.
Use the information below to complete the table on the answer sheet, which shows the length, width, area and perimeter of rectangles B;C;D and E: (a) On your answer sheets (i) Draw and label (a) rectangle rectangle B B of area 27 square units and perimeter 24 units (b) rectangle C of C of area 32 square units and perimeter 24 square units. (ii) Complete Complete the table table to show show the dimensions dimensions of rectangle rectangle B and C; and C; drawn at (a) (i) above. (b) Rectangle D Rectangle D has a perimeter of 24cm, with length, l and width, w width, w.. If the the area area of the the rectangle D rectangle D is to be as large as possible, determine the values of l and w: w : In the table write the values of l of l;; w and the area of rectangle D: D : (c) Rectangle E has E has a perimeter of 36cm, with length, l and width, w width, w.. If the the are area a of the the rectangle E rectangle E is is to be as large as possible, determine the values of l of l and w and w:: In the table 108
write the values of l of l;; w and the area of rectangle E
[10:marks [10:marks]] May/Jun 2011 10. The …gure …gure below shows shows the the …rst …rst three diagram diagram in in a sequence. sequence. Each Each diagram diagram is made up of sticks sticks joined at the ends by thumb thumb tacks. tacks. The sticks sticks are represented represented by lines and the thumb thumb tacks tacks by dots. dots. In each each diagram, diagram, there there are t thumbs tacks and s sticks.
On the answer sheet provided: (a) Draw the FOURTH diagram in the sequence (b) (b) (i) (i) How How ma many ny stic sticks ks are are in in the the SIXT SIXTH H dia diagr gram am?? (ii) How many thumb tacks are in the SEVENTH diagram? (c) Complete the table by inserting the missing values at the rows marked (i) and (ii) .
109
(d) Write the rule, in terms of s and t, to show how t is related to s:
Jan 2012 11. Sarah Sarah is making making a pattern of squares squares using straws. straws. She uses four straws straws for the sides and two longer longer straws straws for the diagonals. diagonals. The …rst three …gures …gures in her sequence sequence of shapes are shown below:
(a) On your answer sheet, draw Figure 4, the FOURTH shape in the pattern (b) On your answer sheet, complete the TWO rows in the table for (i) Figure 4 (ii) Figure 10
110
(c) Which …gure in the sequence uses 106 straws? (d) Obtain an expression in n, n , for the total number of straws used in the n th pattern. [10 marks] May/Jun 2012 12. The diagram diagram below below shows the the …rst three three …gures …gures in a sequence sequence of …gures. …gures. Each Each …gure is an isosceles isosceles triangle triangle made of rubber streche streched d around pins on a geo-board. geo-board. The pins are arranged in rows and columns, one unit apart.
(a) On the answer sheet provided, draw the fourth …gure (Figure 4) in the sequence. (b) Study the patterns in the table below and on your answer sheet, complete the rows numbered numbered ( i) ,(ii), (iii) (iii) and (iv). The breaks in columns columns are to indicate indicate that the rows do not follow one after the other.
111
[10 marks] Jan 2013 13. The …rst three diagrams in a sequence are shown below
(a) In your answer booklet, draw the FOURTH diagram in the sequence. (b) The table below shows the number of squares in EACH diagram.
112
Determine the values of (i) a (i) a (ii) b (iii) c (c) Write down, in terms of n; of n; the number of squares in the n th diagram of the sequence. [10 marks] May/Jun 2013 14. The drawings below show the …rst three diagrams in a sequence.
Each diagram is made up of wires of equal length which are joined at the ends by balls of plasticine. Diagram Diagram 1 is made of 12 wires and 8 balls. Each Each new diagram in the sequence sequence is formed by …tting …tting the frame shown below to the right of the previous diagram.
113
Thus, Diagram 2 has 8 more wires and 4 more balls than Diagram 1. On the answer sheet provided: (a) Draw a sketch of Diagram 4, the fourth diagram in the sequence. (b) Complete the table by inserting the missing values at the rows marked (i) and (ii).
(c) Write the rules which may be used to …nd the values of W of W and and of B B where N is N is known. (i) W (i) W = _____________ (ii) H = H = _____________ [10 marks]
Trigonometrical Graphs Jan 2009 1.(a)(i) Copy and complete the table for the graph of y = 21 tan x for 100 x 600
x y
100 0.13
200
300 0.29
400
500 0.60
600 0.87
(ii) Using a scale of 2cm to represent 10 0 (10 degrees) on the horizontal axis and 10 cm to represent 1 unit on the vertical axis, plot the values from the table and draw a smooth curve throught your points. (iii) Use your graph to estimate the value of x of x when y when y = 0:7 [6 marks]
114
1.(a)(i) 18 1.(a)(i) 18
Fraction/Decimal/Signi…cant raction/Decimal/Signi…cant Figures/Standard Form (ii) 14.3 (ii) 14.3
2.(a)(i) 151.208 2.(a)(i) 151.208
(ii) 150 (ii) 150
3.(a)(i) 12.2092 3.(a)(i) 12.2092
(ii) 2.1 (ii) 2.1
4.(a)(i) 8.89 4.(a)(i) 8.89 11 14
5(a)(i)
(ii) 0.4 (ii) 0.4
6.(a)(i) 1.873 6.(a)(i) 1.873 7.(a)(i)
5 2
8.(a)(i)
29 48
(ii)
(ii) 3.2
34 15
2
10
9. (a)(i) 79.14 (a)(i) 79.14 10.(a)(i)
1 3
(ii) 6.1 (ii) 6.1
11.(a)(i) 86.65 11.(a)(i) 86.65 12.(a)(i)
3 4
13.(a)(i)
49 96
14.(a)
(ii)
2 21
(ii) (ii) 4.46 (ii) 4.46
1
10
19 21
15.(a) 9.257 15.(a) 9.257 16.(a)(i)
11 18
(ii) 1.3524 (ii) 1.3524
Substitution 1.(a) 0 1.(a) 0 2.(a)(i) 6 2.(a)(i) 6
(ii) (ii) - 12
3.(a)(i) 30 3.(a)(i) 30 (ii) 45 (ii) 45 4.(a)(i) 4 4.(a)(i) 4 (ii) (ii) - 2 5.(a) 21 5.(a) 21 6.(a) 4 13 or 4.333 : 7.(a)(i)
2
(ii)
5
8.(a) 25 8.(a) 25
Changing the Subject of a Formula 1.(a) u 1.(a) u = = 2. (a) t (a) t = =
2s
t
v
gp 2 4
r 115
q 3.(a)(i) v 3.(a)(i) v =
2E
m
4.(a) p 4.(a) p = =
(ii) 2.63 (ii) 2.63
p qt + r
h 5.(a)(i) r 5.(a)(i) r = 1 (ii) r (ii) r = h
6.(a)(i) C 6.(a)(i) C = = 95 (F
32)
v h
p
(ii) 45 (ii) 45
Simplifying Expressions 1.(a)(i)
2x 19 15
2.(a)(i) 7 2.(a)(i) 7x x2 3.(a)(i)
19x 19x 36
5 12 p
4.(a)(i) 4( 4.(a)(i) 4(x x + 5) (ii) ab (ii) ab + 16 (b)(i) x (b)(i) x b(ii) a b(ii) a 2 b4 5.(a)(i)
m 3n
6.(a)(i) 7 6.(a)(i) 7x x + 5y 5y (ii) 2 (ii) 2x x3 7.(a)(i) 7( 7.(a)(i) 7(x x + y ) (ii) y (ii) y((y + 1) x
(ii) 14 (ii) 14 p p7 q 4
8.(a)(i)
15
9.(a)(i)
7x 5 12
Ratio:-Proportion-Direct-Inverse-Variation Ratio-Proportion 1.(a) $210 1.(a) $210 b(i) $17.35 b(i) $17.35 b(ii) 14 b(ii) 14 2.(a)(i) 40 (ii) 720 (ii) 720 (iii)
3 25
3.(a)(i) 3 3.(a)(i) 3 (ii) 30 (ii) 30 Ratio-Direct/Inverse Variations 1.(a) y 1.(a) y =
k x2
2.(a)(i) V 2.(a)(i) V = 3.(a)(i)
1 2
b(i) 18 b(i) 18 b(ii) 5.6 b(iii) 1.5 k P
(ii) 6400 (ii) 6400 (iii) 13 13
(ii) 450 (ii) 450
4.(a) a 4.(a) a = = 3; b = 8 5.(a)(i) A 5.(a)(i) A = kR = kR 2
(ii) k (ii) k = 4 (iii) A = A = 100; 100; R = 7
116
Factorising-Simple-Grouping-Quadratic actorising-Simple-Grouping-Quadratic 1.(a)(i) (2 1.(a)(i) (2x x + 5)(2x 5)(2x
5)
(ii)
(3s (3s 2)(3 p + 2 p 2 p)) or (3 p (3 p + 2q 2q )(2 )(2 3s)
(iii) ( (iii) (x x + 2)(3x 2)(3x
2.(a)(i) x 2.(a)(i) x((x
5) (ii) ( (ii) (x x 9)(x 9)(x + 9) 3.(a)(i) (2 3.(a)(i) (2 p p 1)( p 3) (ii) ( (ii) ( p p q + + 5)( p 5)( p + q ) 4.(a)(i) x 4.(a)(i) x((x y ) (ii) ( (ii) (a a 1)(a 1)(a + 1) (iii) ( p 2)( p q ) or ( p ( p q )(2 )(2 p) p) 5.(a)(i) 6 5.(a)(i) 6a a b(2a (2a + b ) (ii) ( (ii) (m m + 5)(2m 5)(2m 1) 6.(a)(i) ( 6.(a)(i) (x x + 3)(x 3)(x 2y ) 7.(a)(i) (2 7.(a)(i) (2x x + 3y 3y)(a )(a b) (ii) 5( (ii) 5(x x 2)(x 2)(x + 2) (iii) (x ( x + 3)(3x 3)(3x 5) 8.(a)(i) (2 8.(a)(i) (2yy z )(2y )(2y + z ) (ii) ( (ii) (x x y )(2a )(2a b) (iii) ( (iii) (x x + 4)(3x 4)(3x 2) 2
2
2
9.(a)(i) ab 9.(a)(i) ab((a + 2) 10(a)(i) xy 10(a)(i) xy((x + y 2 ) (ii) ( (ii) (m m
n)(2h )(2h 3k) 11.(a)(i) ( 11.(a)(i) (x x + 4)(x 4)(x 4) (ii) ( (ii) (x x + 4)(2x 4)(2x 3) 12.(a)(i) 2 12.(a)(i) 2x x y (x + 3y 3y) (ii) (3 (ii) (3x x 2)(3x 2)(3x + 2) (iii) (4x (4 x y )(x )(x + 2y 2y ) 13.(a)(i) (5 13.(a)(i) (5m m + 1)(5m 1)(5m 1) (ii) (2 (ii) (2n n + 5)(n 5)(n 4) 14.(a)(i) 2 14.(a)(i) 2x x(x 2)(x 2)(x + 2) (ii) (3 (ii) (3x x + 1)(x 1)(x 2) 2
Simple Equations-Grouping-Brack Equations-Grouping-Bracket-F et-Fraction-W raction-Worded orded Problems 1.(a)(i) x 1.(a)(i) x = 6 b(i)(a) 5 b(i)(a) 5m m b(i)(b) 4 b(i)(b) 4m m b(ii) 5 b(ii) 5m m + 4m 4m = 31: 31:50 2.(a)(i) 2( 2.(a)(i) 2(k k + 5) (ii) 2( (ii) 2(k k + 5) + 10k 10 k + 4(2k 4(2 k ) (iii) k (iii) k = 6:5 3.(a)(i) x 3.(a)(i) x = 4.(a)(i) x; 4.(a)(i) x; x
13 10
3; 2x
(ii) x (ii) x + x
3 + 2x 2x
(iii) x = x = 6
5.(a)(i) 8 5.(a)(i) 8x x + 5(2x 5(2x + 3) (ii) a (ii) a = = 10; 10; b = 23 6.(a)(i)(a) (28 6.(a)(i)(a) (28 7.(a)(i)
x)
(a)(i)(b) 30x 30 x (a)(i)(c) 15(28 (a)(i)(c) 15(28
x
(a)(i)(b) 6 (a)(i)(b) 6x x + 10(500
x)
a(iii) x a(iii) x = 16
39 7
8.(a)(i) 32 9.(a)(i)(a) 500 9.(a)(i)(a) 500
x)
a(ii) x a(ii) x = = 2277
Simultaneous Equations 1.(a)(i) x 1.(a)(i) x = 5; y = 2 2.(a)(i) x 2.(a)(i) x = $20 117
2)
3.(a)(i) 5 3.(a)(i) 5a a + 3b 3b = 105 (ii) a (ii) a = = 12; 12; b = 15 4a + b = 63 4.(a)(i) x 4.(a)(i) x + 2y 2y = 8 (ii) x (ii) x = $2; $2; y = $3 3x + y = 9 5.(a)(i) x 5.(a)(i) x = 3; y = 1 6.(a)(i) x 6.(a)(i) x = 3; y = 2 7.(a)(i) x 7.(a)(i) x = 4; y = 1 8.(a)(i) 5 8.(a)(i) 5x x + 12y 12y = 61 (ii) x (ii) x = 5; y = 3 10x 10x + 13y 13y = 89
Quadratic Equations 1.(a)(i) 4 1.(a)(i) 4x x + 2l 2l + 6 (iv) x (iv) x = = 3:5orx = orx = 2:5 2.(a)(i) 2 2.(a)(i) 2x x2 + 5x 5x 3.(a)(i) ( 3.(a)(i) (a a
2
7)
3
(ii) x (ii) x = = 11 (iii) length = 21cm, width = 14cm
+ a2 = (a ( a + 1)2
(ii) a (ii) a = 12 or 12 or a = a = 4 (iii) Sides are 5, 12 and 13cm
4.(a)(i) x 4.(a)(i) x = 1:43; 43; or x = x = 0:23 Linear and Non-Linear Simultaneous Equations 1.(a)(i) ( 1.(a)(i) ( 32 ; 25 ) , (3; (3 ; 1) 3 2.(a)(i) ( 2.(a)(i) ( 1; 1) , 1) , (2; (2 ; 4)
3.(a)(i) (5 3.(a)(i) (5;; 7) , 7) , (2; (2; 19) 4.(a)(i) ( 4.(a)(i) ( 32 ; 1) , 1) , ( 1; 6)
5.(a)(i) (1 5.(a)(i) (1;; 3) , 3) , ( 3; 15) 6.(a)(i) (4; 12) (ii) The line is a tangent to the curve because it touches the curve at one point
only.
Consumer Arithmetic 1.(a)(i) $1680 1.(a)(i) $1680 (ii) $13680 (iii) $5880 (iv) $823.20 2.(a)(i) p 2.(a)(i) p = $35200 , $35200 , q = q = 15 (ii) $30976 (ii) $30976 3.(a)(i) $354 3.(a)(i) $354 (ii) $34.05 (iii) 10.64% (iii) 10.64% 4.(a)(i) $18125.00 4.(a)(i) $18125.00 (ii) $163.79 (b)(i) $52.20 (b)(i) $52.20 (c)(i) 5-$4 (c)(i) 5-$4 , 1-$2.50 , 1-$1.20 , 2-$1.00 (c)(ii) 6-$1.00 (c)(ii) 6-$1.00 , 6-$1.20 , 5 -$2.50 d(i) 17 d(i) 17 stamps (d)(ii) 6-$1.00 , 6-$1.20 , 5-$2.50 5.(a)(i) EC 5.(a)(i) EC $1.35 (a)(ii) BDS (a)(ii) BDS $320
(b) $27933.60 (b) $27933.60
6.(a)(i) $14 (ii) $210 (iii) $875 7.(a)(i) $37680 7.(a)(i) $37680 (ii) $14400 (ii) $14400 (iii) $52080 118
8.(a)(i) $12.80 8.(a)(i) $12.80 (ii) $2985 (ii) $2985 (iii) $1065 (iv) 55% (iv) 55% 9.(a)(i) $380 9.(a)(i) $380 (ii) $85.50 (ii) $85.50 (iii) $684 (iii) $684 (iv) 48 (iv) 48 hours 10.(a)(i) W 10.(a)(i) W = = $15: $15:60
X = = $13: $13:20
Y = 6
Z = = 12%
11.(a)(i) $900 11.(a)(i) $900 (ii) $33.75 (ii) $33.75 (iii) $405 (iii) $405 (iv) 16 (iv) 16 hours 12.(a)(i) 20% 12.(a)(i) 20% (a)(ii) $80 (b)(i) TT (b)(i) TT $2.50 (b)(ii) EC $216 (b)(iii) US (b)(iii) US $96 13.(a)(i) US 13.(a)(i) US $647 (ii) US (ii) US $596.30 (iii) Angie (iii) Angie since $596.50 < $647 (iv) $1400 (iv) $1400 14.(a)(i) 450 14.(a)(i) 450 ml since, for the dollar ;
450 5:13
>
350 4:20
(b)(i) $768 (b)(ii) $6000 (b)(ii) $6000 (b)(iii) $480 (b)(iii) $480
Mensuration/Measurements 1.(a)(i) B 1.(a)(i) B C = = 6cm (a)(ii) 1200km (a)(ii) 1200km 2 (b)(iii) 207.24cm (b)(iii) 207.24cm
(a)(iii) 550
2.(a)(i) 4320cm 2.(a)(i) 4320cm 3
(b)(i) 11.78cm (b)(i) 11.78cm (b)(ii) 41.78cm (b)(ii) 41.78cm (b)(iii) 88.31cm2
(a)(ii) 1728cm (a)(ii) 1728cm2
3.(a)(i) 120m (a)(ii) 232m (a)(iii) 1600m2 4.(a)(i) 38.5cm 4.(a)(i) 38.5cm2
(ii) 12.25cm (ii) 12.25cm2
(b)(i) 150.72cm (b)(i) 150.72cm 2
(a)(iv) 41600m 2
(b)(i) AB = 8cm (b)(ii) 608cm2
(iii) 2.625cm (iii) 2.625cm 2
5.(a)(i) RS 5.(a)(i) RS = = 6m (ii) x (ii) x = = 4m (iii) 40m (iii) 40m (iv) 74m (iv) 74m2 2 (b)(iii) 41.54cm (b)(iii) 41.54cm (b)(iv) (b)(iv) 36.03cm 6.(a)(i) 7200cm 6.(a)(i) 7200cm 3
(b)(ii) (b)(ii) 56.52cm2
(b) 2776cm (b) 2776cm2
(c)(i) 1200cm (c)(i) 1200cm3
(v) 250 (v) 250 (b)(i) (b)(i) = = 1170
(b)(ii) 32.19cm (b)(ii) 32.19cm2
(c)(ii) 60cm (c)(ii) 60cm2
7.(a)(i) 5.5cm 7.(a)(i) 5.5cm (ii) 2.75km (ii) 2.75km (iii) 9cm 8.(a)(i) 6.5cm 8.(a)(i) 6.5cm (ii) 81.25m (ii) 81.25m (iii)(a) 8.36m/s (iii)(a) 8.36m/s (iii)(b) 30.1km/h (iii)(b) 30.1km/h 9.(a)(i) E 9.(a)(i) E F = 3cm 3 cm (ii) D (ii) DF F = 4cm 4 cm (iii) 42cm2 10(a)(i) 240cm 10(a)(i) 240cm3
(ii) 13 (ii) 13 cartons (iii) 12.2cm
11.(a)(i) 11cm 11.(a)(i) 11cm (a)(ii) 44cm (a)(ii) 44cm 12.(a)(i) 4h 12.(a)(i) 4h cm2
(b)(i) 7cm (b)(ii) 154cm2
(ii) 52h (ii) 52h cm3
(iii) 5.5cm (iii) 5.5cm
13.(a)(i) 16.5cm 13.(a)(i) 16.5cm (ii) 23.5cm (ii) 23.5cm (iii) 28.875cm 2
(b)(i) 577.5cm (b)(i) 577.5cm3
(b)(ii) 4216 (b)(ii) 4216 kg
14.(a)(i) 6 14.(a)(i) 6 cm (ii) 37.68cm (ii) 37.68cm b(i)a b(i)a = 37: 37:68cm 68cm , , b = 8cm (b)(iii) 301.44cm 2
Functions(1) 1(a)(i)
1 6
(ii) -10 (ii) -10
2.(a)(i) 53
(ii)
3.(a)(i) 1 3.(a)(i) 1 (ii)
3 2
2x+9 5
(iii) 1
(iii)
5x 1 2
(iii) 2 (iii) 2
4.(a)(i) 3 4.(a)(i) 3 (ii) x (ii) x + 3 5.(a)(i) -9 5.(a)(i) -9 (ii) -22 (ii) -22 (iii) 4 (iii) 4 119
(c) h (c) h = 4:4cm
x+5
6.(a)(i) 3 6.(a)(i) 3 (ii) 12 (iii)
2
7.(a)(i) k 7.(a)(i) k = 5 (ii) 4 (iii) x (iii) x = = 10 8 (ii) 2 (ii) 2x x + 2 (iii) x 6
8.(a)(i) 21 9.(a)(i) x 9.(a)(i) x =
4y +3 y 2
(ii)
4x+3 x 2
(c) x (c) x = = 23
10.(a)(i) 7 10.(a)(i) 7 (ii) 4 11.(a)(i) 9 11.(a)(i) 9 (ii) 4 (ii) 4
Geometry(Construction,Polygons, Similar & Congruent Shapes) 1.(a)(i) 40 1.(a)(i) 40 0
(ii) 40 (ii) 40 0
4KLN KLN is isosceles
3.(a)(i) 450
(ii) 12cm2
(iii) 110 0 isosceles triangle
(iii) 101.6cm (iii) 101.6cm2
4.(a)(i) 90 4.(a)(i) 90 0 - Co-interior angles (ii) 48 (ii) 48 0 9.(a)(i) x 9.(a)(i) x = 540 alternate angles, angles, y = 65 Co-interior 65 Co-interior angles 11.(c)(i) scalefactor 11.(c)(i) scalefactor = = 2 (ii) 10cm (ii) 10cm
12.(b)(i) 72 12.(b)(i) 72 0
(iv) 86 (iv) 86 0
(ii) 22 0
(iii) 72 (iii) 72 0
(iii) 20cm (c)(i) 600
(c)(ii) 65cm (c)(ii) 65cm2
14.(a)(i) 72 14.(a)(i) 72 0 (ii) 54 0 (iii) Similar (iii) Similar since corresponding are equal and not congruent because length of corresponding sides are unequal. unequal . 15.(a)(i) 6m (ii) angle A is the same size and lies in both triangles (b)(i) 5cm (b)(i) 5cm (b)/(ii) 34 (b)/(ii) 34 0 (b)(iii) 18cm (b)(iv) 12cm (b)(iv) 12cm2
(iii) 3.6m (iii) 3.6m
Coordinate Geometry 1.(a)(i) 4 (ii) y (ii) y = 4x 2.(a)(i)
3 2
14
(ii) (4, (ii) (4, 7)
3.(a)(i) 7 (ii) 27
(iii) (1 , 3.5)
4.(a)(i) P(0 4.(a)(i) P(0 , 3) Q(-2 , 0) 5.(a)(i) y 5.(a)(i) y = 53 x 6.(a)(i)
2 3
7 5
or 5 or 5yy = 3x
(b)(i)
(b)(ii) y (b)(ii) y = 23 x + 3
(ii) y (ii) y = 23 x + 13
4 3
3 2
(c) (2 (c) (2 , 0)
7
7.(a)(i) A 7.(a)(i) A(( 2; 3) , 3) , B (4; (4; 6) (ii) 8.(a)(i)
(b) 14 (b) 14
1 2
(iii) y (iii) y = 21 x + 4
(ii) y (ii) y = 34 x
9.(a)(i) 23 10.(a)(i) 21
2 (iii) (3 , 2) (ii) 3 (ii) 3yy = 2x 5 or y = x 2 3
(ii) (1 , 3)
(iv) 10 (iv) 10
5 3
(iii) y = 2x + 1 Sets
1.(a)(i) 120
(c) t (c) t =
9
(d) y (d) y = 32 x + 6
U=300
B
C
2x
x
70
18-x
x
15-x
80
(ii) 3x 3 x + 150 (iii) 100 (iii) 100 2.(a)(i)
3x
(ii) 2x 2 x + 33 (iii) 3 (iii) 3
4
6
8
10
Q
P 3 2
5
1
9
7
0
0
0
3.(a)(i) A B (ii) (A B ) or A B (iii) (A B ) A (c)(i) 17 (c)(i) 17 (c)(ii) 4 (c)(ii) 4 (c)(iii) 21 (c)(iv) 25 (c)(iv) 25
[
[
\
[ \
4.(a)(i) Tennis and Hockey (ii) Squash and Tennis (iii) Hockey (iv) Members (iv) Members who play squash and tennis only 5.(a)(i) U
T S L
m
K p
n
q
r
121
(ii) {k, l, m, p, q} (iii) {n, q, r} 6.(a)(i)
(ii) {15, 21, 25} 7(a)(i) U
6
M
P
54 - x
x
42 – x
(iii) x (iii) x = 12 8.(a)(i) U
D C
23 – x
x
18 – x
2x
(ii) 41 + x (iii) x = x = 9 9 9.(a)(i) 122
E T 12 3
9
6
10 2 4 8
1
(ii) {6 , 12 } (iii) {1} (iii) {1} 10.(a)(i)
(ii) 58
2x
(iii) x (iii) x = 9
11.(a)(i) {5, 11.(a)(i) {5, 7, 9, 11} (ii) {2, (ii) {2, 3, 5, 7, 11}
12.(a)(i) 4 (ii) x (ii) x = 9 (iii) 14 (iii) 14 13.(a)(i) A 13.(a)(i) A = 51; 51; 53; 53; 55; 55; 57; 57; 59
f
g
(ii) B (ii) B = 53; 53; 59
f
g
123
14.(a)(i) y 14.(a)(i) y = 30
9x ,
z = 4 (ii) x (ii) x + 34 (iii) x (iii) x = = 2
15.(a)(i)
(ii) x = x = 10 16.(a)(i) U
2
M
C
4x
x
20 – x
(ii) 4x 4 x + 22 (iii) 8 (iii) 8
Statistics 1.(a)(i) 17 1.(a)(i) 17 - 19 - 6; 23 - 25 - 5 (ii) 13.5 (ii) 13.5 (iii) 3 (iii) 3 (iv) 17 (iv) 17 - 19
124
(v)
3 5
(vi) 22cm (vi) 22cm
(vii) 20.4 (vii) 20.4
2.(a)(i) 22, 2.(a)(i) 22, 27, 32 (ii) 17.85 kg (iv)
32 100
or
8 25
3.(a)(i) 3.(a)(i) Cumulative Frequencies [5, 23, 46, 68, 89, 100] 4.(a)(i) Frequencies 4.(a)(i) Frequencies [3, 7 ,4, 5]
(iv)
(iii) 36 (iii) 36 (iv)
32 100
or
8 25
7 32
5.(a)(i)
(b) 36 (b) 36
(c) 2 (c) 2
6.(a)(i) t 6.(a)(i) t = 105
(d) 76 (d) 76
(e) 2.111 (e) 2.111
(f)
11 36
(ii) 800 ; 630 ; 350 ; 720 ; 1100
7.(a)(i) Cummulative 7.(a)(i) Cummulative Frequencies [30, 46, 58, 66, 70] or 1 16 8 (c) 30 (c) 30 (d) 70 or 35
(b)(ii) No (b)(ii) No student got a mark of less than 0.5
8(a)(i) Cumm 8(a)(i) Cummulati ulative ve Frequen Frequencies cies [4, 11, 22, 30. 52, 62, 67, 70] 9(a)(i) Frequencies 9(a)(i) Frequencies [1, 2, 4, 6, 7, 3, 2, 1]
(c)
6 26
3 or 13
10(a)(i)
(b) 19.5 (b) 19.5 (d)(i) 8 (d)(i) 8 (d)(ii)
1 3
11.(a)(i)
(ii) March and April (iii) $25000 (iv) $25000 (iv) $25000 12.(a)(i)
125
(d) Mean (d) Mean
(c)(i) 24 kg (c)(ii)
80 100
or 45
13(a)(i)
(b)(i) 11 (b)(i) 11 - 20 (ii) 100 (ii) 100 (iii) 24.2 (iv) 14.(a) 35 14.(a) 35 and 47 (c)(i) 64 (c)(i) 64 yrs (c)(ii) 15.(a)(i) modal 15.(a)(i) modal class 20 - 29
34 100
or 17 50
43 60
(ii) 19.4 (ii) 19.4 lies in the class of 10 -19
(b)(i)
(b)(ii) 31.4 (b)(ii) 31.4 (c) (c) The The calculated calculated mean is an estimate estimate because because the assumption assumption was made that 32 8 the scores in all the intervals are the midpt values. (d) 100 or 25 16(a)(i) 16(a)(i) Cummulative Frequencies [3, 10, 19, 30, 38, 40]
Vectors 1.(a)(i)
1 5 6 2
4
; ; ; (ii) (b) 4 (b) 4 units 2 2 4 4 0 (c)(i) AB is parallel and equal to DC (c)(ii) AB and DC are DC are opposite to each other and are equal and parallel to each other 3:5 (d) OG = OG = ; G = (3: (3:5; 3) 3
!
!
2.(b)(i)
6 2
!
!
!
(b)(ii)
0 4
(b)(iii)
6 2
(c)(i) (3 (c)(i) (3 , 3) 126
3.(b)(i) r 3.(b)(i) r + + s (b)(ii) 31 s 4.(b)(i) k 4.(b)(i) k
m
(b)(ii) m (b)(ii) m
2 3
r (b)(iii) 65 r
1 2
1 3
k (b)(iii)
7 12
s mk
3 2
5.(a)(ii)a) 2 5.(a)(ii)a) 2x x + 3y 3y (a)(ii)b) x (a)(ii)b) x + y (b)(i)a) 6.(b)(ii) b 6.(b)(ii) b
1 3
a;
a; 21 b
7.(a)(i)
3
8.(a)(i)
6 2
2
1 6
a 1 (ii)
(d)
9 6
9.(a)(i)a)
4
(a)(i)b)
10.(a)(ii)a) 3 10.(a)(ii)a) 3u u (a)(ii)b) 11.(a)(i)
2 7
12.(a)(i)
(a)(ii)
b + a 1 13.(a)(i) 3
14.(a)(i)
3 2
15.(a)(i)
4 4
4
1 2
k
4 2
(b)(ii) F (4 F (4;; 1)
(c)(i)
a + 1 b
3
(c)(ii) a (c)(ii) a = = 2; b = 5
(b)(ii) 23 a + 21 b (b)(iii) 21 a + 21 b (b)(iv)
2a + 2b 2b
(d) They (d) They are co-linear
(b)(i)a) 2 (b)(i)a) 2u u (b)(i)b) v (b)(i)b) v + 3u 3u (b)(i)c) 2v 2 v + u
u + v
6
(b)(i)b)
6
p (b)(iii) 74
; (b)(i) 3a + b 3 1 DX is ! is parallel to DC !; DX ! =4DC ! (c) DX
1
2
p 20units 20units
(b)(i)
3
(b)(iv) 31 m
(a)(ii)c)
(b)(i)
3u + 3v 3v
(iii) KL is parallel to MN; MN ; K L = M = M N
8 6
(ii) 31 ( b + a) (iii) 31 (b + 2a 2a)
(ii)
(ii)
u + v 16.(a)(i) 2a + 2b 2b
5 1
9 6
(iii)
6 2
!
(b)(i)
3 1
(b)(ii)
2 2
!
(iii) BC = 3 BA
(ii) 32 u + 31 v (iii) 31 u + 32 v (ii) b (ii) b (iii) OB is parallel to PQ; OB = 2PQ
Circle Theorem 1.(a)(i) 65 1.(a)(i) 65 0
(ii) 30 0
(iii) 65 0
2.(a)(i) 90 2.(a)(i) 90 0
(ii) 70 0
(iii) 140 0
(c)(i) 10.92 (c)(i) 10.92 cm (c)(ii) 22.7 cm 2 (iv) 40 (iv) 40 0
3.(a)(i) PTQ 3.(a)(i) PTQ is a straight line because T is a common point of cantact of two circles and a common tangent drawn through T will be perpendicular to both PT and QT (ii) 7 cm (iii) PS is parallel because PSR = QRS = 90 0 . The corresponding angles are equal. (b)(i) 3cm (b)(i) 3cm (b)(ii) 40 (c)(i) 550 (c)(ii) 35 (c)(ii) 35 0
p
4.(a)(i) 90 4.(a)(i) 90 0 (Angles in a semicircle) (b) 90 (b) 90 0 ( Angles between tangent and radius) 0 (c) 23 (c) 23 ( Angles between tangent and chord) (d) 113 (d) 113 0 (Opposite angles in a cyclic Quadrilateral) 5.(a)(i) 100 5.(a)(i) 100 0 (Angle at the center of a circle is twice the angle at circumference) (ii) 40 0 (WOY isosceles) 6.(a)(i) 64 6.(a)(i) 64 0
(ii) 38 0
7.(a)(i) 90 7.(a)(i) 90 0
(ii) 132 0
(iii) 66 (iii) 66 0
(iv) 114 (iv) 114 0 127
8.(a)(i) 48 8.(a)(i) 48 0
(ii) 96 0
(iii) 64 0
(iv) 64 (iv) 64 0
9.(a)(i) 440
(ii) 54 0
(iii) 54 (iii) 54 0
(b)(i) 7.99cm (b)(i) 7.99cm2
10.(a)(i) 46 10.(a)(i) 46 0
(ii) 56 0
11.(a)(i) 31 11.(a)(i) 31 0
(ii) 56 0
12.(a)(i) 116 12.(a)(i) 116 0
(vi) 22 (vi) 22 0 (b)(ii) 12.28cm (b)(ii) 12.28cm2
(b)(iii) 4.29cm 2
(iii) 124 (iii) 124 0
(ii) 23 0
13.(a) 350
(v) 42 (v) 42 0
(b) 550
(iii) 26 (iii) 26 0 (c) 20 (c) 20 0
14.(a)(i) 700
(ii) 20 (ii) 20 0
15.(a)(i) 55 15.(a)(i) 55 0
(ii) 100 0
(d)(i)
4Y OK
(d)(ii)
4ZU X
(iii) 630 (iii) 50 (iii) 50 0
(iv) 15 (iv) 15 0
Trigonometry(Trig rigonometry(Trig Ratios/Sin-Cos Rule/Bearings) 1.(a)(iii) 5km 1.(a)(iii) 5km (iv) 067.40
(b)(ii) 15cm (b)(ii) 15cm (c)(i) 6.03cm (c)(i) 6.03cm (c)(ii) 23.68cm (c)(ii) 23.68cm 2
2.(b)(i) 18.7km 2.(b)(i) 18.7km (ii) 50.8 (ii) 50.8 (iii) 290.8 (iii) 290.80 3.(a 3.(a)( )(i) i) 5.6k 5.6km m
(a)( (a)(ii ii)) 108k 108km m
(b)( (b)(i) i)
1 2
(b)(ii)
p 3
(b)(iii)
1 2
4.(a)(i) 28.6m 4.(a)(i) 28.6m (ii) 12.9m (ii) 12.9m 5.(b)(i) 19.2m 5.(b)(i) 19.2m (b)(ii) 4.4m (d)(i) 101 (d)(i) 101 0 6.(a)(i) 5m
7.(a)(ii) 1250
(d)(ii) 47.14 (d)(ii) 47.140
(a)(i) 7m (a)(i) 7m (b)(i) 9.17m (b)(i) 9.17m (b)(ii) 32.40
(d)(iii) 260 0
(e)(ii) 73.9km (e)(ii) 73.9km
(b)(iii) 34.6m (b)(iii) 34.6m2
(ii) 985km (ii) 985km (iii) 060.6 0
8.(a)(i) 10cm 8.(a)(i) 10cm (a)(ii) 30 0
(b)(ii) 128m (b)(ii) 128m (c)(i) 26m (c)(i) 26m (c)(ii) 14.1m
9.(a)(i) 9.(a)(i) 8.57m (ii) 12.2m (ii) 12.2m (iii) 46.90 10.(a)(v) 1440
(a)(vi) 174.15km (a)(vi) 174.15km (a)(vii) 245.7 (a)(vii) 245.70
11.(a)(i) 3.26m 11.(a)(i) 3.26m (a)(ii) 67.2 (a)(ii) 67.20
(a)(iii) 32.40
(b)(i) 76 (b)(i) 76 0
(b)(ii) 299km (b)(ii) 299km (b)(iii) 218.50
12.(b)(i) 7.98m 12.(b)(i) 7.98m (b)(ii) 28.8m (b)(iii) 23 (b)(iii) 23 0 13.(a)(ii) 25km 13.(a)(ii) 25km (a)(iii) 53 (a)(iii) 53 0
(b)(i) 6cm (b)(i) 6cm (b)(ii) 44 (b)(ii) 44 0
14.(a)(i) 5.8 14.(a)(i) 5.8 cm (a)(ii) 65 0 (c)(ii) 3849.5m 2
(a)(iii) 174m (a)(iii) 174m (b)(i) 10cm (b)(i) 10cm (b)(iii) NRM (b)(iii) NRM (c)(i) 137.1 (c)(i) 137.10
15. (a)(iii) 65.5m (a)(iv) 30 (a)(iv) 30 0
Matrices 1.(a)(i) x 1.(a)(i) x 2
4
(a)(ii) x = x = 2 (b)(i)
2.(a)(i) x 2.(a)(i) x = 3 (a)(ii)
1 3 1 9
1 3 2 9
3 8 1 8
1 16 3 16
(b)(ii) x = x = 5; y =
128
3
2 5 x 6 3. (a) p (a) p = = (b)(i) = (b)( (b)(ii ii)( )(a) a) -7 (b)( (b)(ii ii)( )(c) c) x x = = ; y = 3 4 y 8 3a 3b 2 3 3a + 2 3b 3 4.(a)(i) (ii) (iv) a = 4; b = 1; c = 2; d = 0 3c 3d 3 5 (iii) 3c 3 3d + 5 (iv) a 5.(a) x 5.(a) x = = 4; y = 3 (b)(ii) (b)(iv) x (b)(iv) x = = 1; y = 2 8 1 6.(a) 2 8 15 39 11 6 x 6 7.(a)(i) (b)(i) = (b)(ii) x = x = 12; y = 23 12 33 9 5 y 7 2 1 8.(a) (b) x (b) x = = 6 1 0 1 0 1 2 x 1 2 2 9.(a)(i) (ii) (iii) = (iv) x (iv) x = 8; y = 19 0 1 2 5 y 2 5 3 2 10.(a) 1 1 8 3 13 1 11.(a)(i) (a)(ii) x = 1; y = 2 1 8 1 11 (b)(i) 2 (b)(ii) x = 2 12.(a)(i) a 12.(a)(i) a = = 3; b = 2 (ii) (iii) x = 4; y = 1 1 (iii) x 02 0 1 02 0 1 13.(a)(i) @5 6 A (ii) 40 55 120 (iii) 40 55 120 @5 6 A 3 10 3 10 2 3
16 7
3 7 1 7
2 7
1 7 2 7
5 2 3 2
1 2 1 2
3 2 1 2
3 2
14.(a)(i)
2
1 2
(iii) r = 1; s = 2; t = 0; u =
1
3
Speed-Distance-Time 1(.(a)(i) 6 1(.(a)(i) 6 hrs : 50 mins (a)(ii) 60 km/h 2.(a)(i) 20 2.(a)(i) 20 min (ii) 150 (ii) 150 km/h (iii) 50 (iii) 50 min 3.(a)(i) 126 3.(a)(i) 126 km (ii) 630 (ii) 630 km/h 4.(a)(i) 4.(a)(i) 12m/s (ii) 4 (ii) 4 sec (iii) 102 m
(iv) 0 (iv) 0 - 6 sec
(v) 10 (v) 10 - 13 sec (vi) 6 (vi) 6 - 10 sec
5.(a)(i) x = 20sec (ii) Gradient is 0, the car traveled at a constant speed of 12m/s for 5 seconds (iii) 210 (iii) 210 m 6.(a)(i) 10 6.(a)(i) 10 min (ii) 55 (ii) 55 min (iii) 49.5 km 7.(a)(i) 15 7.(a)(i) 15 m/s (ii) 300 (ii) 300 m
Graphs-Curves 1.(a)(i)
t h
0 0.0
0.5 8.8
1 15
1.5 18.8
2 20
2.5 18.8
3 15
3.5 8.8
(a)(ii) 129
4 0.0
h
20
15
10
5
0 0
1
2
3
4
t
(c)(i) 20m (c)(i) 20m (c)(ii) 2.5 sec (c)(iii) 2 sec 2.(a) a 2.(a) a = =
2; b = 4
3.(a)(i)
x y
-1 -1 5
(b) x (b) x =
0 0
1 -3
2 -4
1; x = 3 3 -3
4 0
(c) (1, (c) (1, -4)
(d) x (d) x = = 0; 1; 2; 3
f
(e) 2 (e) 2
g
5 5
(ii)
y
4
2
-1
1
2
3
4
5
x
-2
-4
(c)(ii) x (c)(ii) x = =
0:4; 4:4
(c)(iii) x 2
4x = 2
4.(a) K 80 70 60 50 40 30 0
(b)(i) 500 5.(a)(i)
x y
-4 5
10
20
30
(b)(ii) 0.66 0 C= min -3 0
(a)(ii) x (a)(ii) x 2 + 2x 2x
-2 -3
-1 -4
0 -3
1 0
2 5
3 130
40
50
60
t
y
4
2
-4
-3
-2 -2
-1 -1
1
2
x
-2
-4
6.(a)(i) f ( f (x) = 3 (ii) x = 0:5 (iii) (-1.3, 6.1) (iv) x = 0:5
7.(a)(i) 2c 7.(a)(i) 2cm m : 1unit 1unit (ii) y (ii) y = 8.(a)(i)
x y
-2 -2 5
--1 1 0
0 -3
1 -4
3:75
2 -3
3 0
(iii) x = x =
3; x = 1
1:3
(iv)
(v) x =
2; 0:5
4 y 5
4 5
(b)
y
4
2
-2
-1
1
2
3
4
x
-2
-4
(c) y (c) y = 2:25 (d)(i) x (d)(i) x = 1 (d)(ii) y (d)(ii) y = 9.(a)(i)
x(sec) y(m/s)
0.25 12
0. 0.5 6
1 3
2 1.5
3 1
4 4 0.75
(d)(iii) x = x = 5 0.6
1; x = 3
6 0.5
(a)(ii)
y
12 10 8 6 4 2 1
2
3
131
4
5
6
x
(vi)
2 < x <
Inequalities 1.(a) x 1.(a) x
9 2.(a)(i) x 2.(a)(i) x 2
(ii) Smallest (ii) Smallest whole number is 0
3.(a)(i) x 3.(a)(i) x > 2 (ii) Smallest (ii) Smallest whole number is 3 4.(a)(i) x 4.(a)(i) x >
2
(ii) Smallest (ii) Smallest integer is -1
132