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CAMBRIDGE IGCSE MATHEMATICS Additional Practice Algebra 1 Linear graphs 1. Draw the graph of y = 3x + 4 for x-values from 0 to 5 (0 x 5). 2. Draw the graph of y = 2x – 5 for 0 x 5. x 3. Draw the graph of y = + 4 for –6 x 6. 3 x x 4. a On the same axes, draw the graphs of y = – 1 and y = – 2 for 0 x 12. 3 2 b At which point do the two lines intersect? 5. Find the gradient of each of these lines. b f a d i g c j e h 6. Find the gradient of each of these lines. What is special about these lines? a b y y 6 6 x = y 4 4 2 2 x x –8 –6 –4 –2 2 4 6 –8 –6 –4 –2 2 4 6 –2 –2 –4 –4 y –6 –6 = –x –8 –8 7. Draw these lines using the gradient-intercept method. Use grids, taking x from –10 to 10 and y from –10 to 10. 1 a y = 2x + 6 b y=x+7 c y = –x – 3 4 d y=x+8 8. a Using the gradient-intercept method, draw the following lines on the same grid. Use axes with ranges –6 x 6 and –8 y 8. i y = 3x + 1 ii y = 2x + 3 b Where do the lines cross? 1 9. Give the equation of each of these lines, all of which have positive gradients. (Each square represents 1 unit.) a y b y c y 4 4 4 2 2 2 –4 –2 2 4 x –4 –2 2 4 x –4 –2 2 4 x –2 –2 –2 –4 –4 –4 d y e y f y 4 4 4 2 2 2 –4 –2 2 4 x –4 –2 2 4 x –4 –2 2 4 x –2 –2 –2 –4 –4 –4 10. Give the equation of each of these lines, all of which have negative gradients. (Each square represents 1 unit.) a y b y c y 4 4 4 2 2 2 –4 –2 2 4 x –4 –2 2 4 x –4 –2 2 4 x –2 –2 –2 –4 –4 –4 d y e y 4 4 2 2 –4 –2 2 4 x –4 –2 2 4 x –2 –2 –4 –4 2 2 Patterns and sequences 1. Lookcarefully at each number sequence below. Find the next two numbers in the sequence and try to explain the pattern. a 1, 1, 2, 3, 5, 8, 13, … b 1, 4, 9, 16, 25, 36, … c 3, 4, 7, 11, 18, 29, … 2. Triangular numbers are found as follows. 1 3 6 10 Find the next four triangular numbers. n–1 3. The first two terms of the sequence of fractions are: n+1 1–1 0 2–1 1 n = 1: = =0 n = 2: = 1+1 2 2+1 3 Work out the next five terms of the sequence. 4. A sequence is formed by the rule – × n × (n + 1) for n = 1, 2, 3, 4, … 1 2 The first term is given by n = 1: – × 1 × (1 + 1) = 1 1 2 The second term is given by n = 2: – × 2 × (2 + 1) = 3 1 2 a Work out the next five terms of this sequence. b This is a well-known sequence you have met before. What is it? 5. Findthe next two terms and the nth term in each of these linear sequences. a 3, 5, 7, 9, 11, … b 8, 13, 18, 23, 28, … c 5, 8, 11, 14, 17, … d 1, 5, 9, 13, 17, … e 2, 12, 22, 32, … 6. Find the nth term and the 50th term in each of these linear sequences. a 4, 7, 10, 13, 16, … b 3, 8, 13, 18, 23, … c 2, 10, 18, 26, … d 6, 11, 16, 21, 26, … e 21, 24, 27, 30, 33, … 7. The powers of 2 are 21, 22, 23, 24, 25, … This gives the sequence 2, 4, 8, 16, 32, … The nth term is given by 2n. a Continue the sequence for another five terms. b Give the nth term of these sequences. i 1, 3, 7, 15, 31, … ii 3, 5, 9, 17, 33, … iii 6, 12, 24, 48, 96, … 3 8. A pattern of squares is built up from matchsticks as shown. 1 2 3 a Draw the 4th diagram. b How many squares are in the nth diagram? c How many squares are in the 25th diagram? d With 200 squares, which is the biggest diagram that could be made? 9. A pattern of triangles is built up from matchsticks. a Draw the 5th set of triangles in this pattern. 1 2 3 4 b How many matchsticks are needed for the nth set of triangles? c How many matchsticks are needed to make the 60th set of triangles? d If there are only 100 matchsticks, which is the largest set of triangles that could be made? 10.A school dining hall had tables in the shape of a trapezium. Each table could seat five people, as shown on the right. When the tables were joined together as shown below, each table could not seat as many people. 1 2 3 a In this arrangement, how many could be seated if there were: i four tables? ii n tables? iii 13 tables? b For an outside charity event, up to 200 people had to be seated. How many tables arranged like this did they need? 4 3 Substitution 1 1. Find the value of 4b + 3 when a b = 2.5, b b = –1.5, c b = –. 2 x 2. Evaluate when a x = 6, b x = 24, c x = –30. 3 12 3. Find the value of when a y = 2, b y = 4, c y = –6. y 24 1 3 4. Find the value of when a x = –5, b x = –, 2 c x = –. 4 x 5. Where A = b2 + c2, find A when a b = 2 and c = 3, b b = 5 and c = 7, c b = –1 and c = –4. 180(n – 2) 6. Where A= , find A when n+5 a n = 7, b n = 3, c n = –1. 2 y +4 7. Where Z= , find Z when 4+y a y = 4, b y = –6, c y = –1.5. 4 Simplifying expressions 1. Write down the algebraic expression for: a 2 more than x b 6 less than x c k more than x d x minus t e x added to 3 f d added to m g y taken away from b h p added to t added to w i 8 multiplied by x j h multiplied by j k x divided by 4 l 2 divided by x m y divided by t n w multiplied by t o a multiplied by a p g multiplied by itself 2. Asha,Bernice and Charu are three sisters. Bernice is x years old. Asha is three years older than Bernice. Charu is four years younger than Bernice. a How old is Asha? b How old is Charu? 3. An approximation method of converting from degrees Celsius to degrees Fahrenheit is given by this rule: Multiply by 2 and add 30. Using C to stand for degrees Celsius and F to stand for degrees Fahrenheit, complete this formula. F = …… 5 4. a Anne has three bags of marbles. Each bag contains n marbles. How many marbles does she have altogether? n n n b Beryl gives her another three marbles. How many marbles does Anne have now? c Anne puts one of her new marbles in each bag. How many marbles are there now in each bag? d Anne takes two marbles out of each bag. How many marbles are there now in each bag? 5. a I go shopping with $10 and spend $6. How much do I have left? b I go shopping with $10 and spend $x. How much do I have left? c I go shopping with $y and spend $x. How much do I have left? d I go shopping with $3x and spend $x. How much do I have left? 6. Give the total cost of: a 5 pens at 15p each b x pens at 15p each c 4 pens at Ap each d y pens at Ap each. 7. A boy went shopping with $A. He spent $B. How much has he got left? 8. Simplify the following expressions: a 2 × 3t b 2w × 4 c 2w × w d 3t × 2t 9. Joseph is given $t, John has $3 more than Joseph, Joy has $2t. a How much more money has Joy than Joseph? b How much do the three of them have altogether? 10. Write each of these expressions in a shorter form. a a+a+a+a+a b c+c+c+c+c+c c 4e + 5e d f + 2f + 3f e g+g+g+g–g f 3i + 2i – i g 5j + j – 2j h 9q – 3q – 3q i 3r – 3r j 2w + 4w – 7w k 5x2 + 6x2 – 7x2 + 2x2 l 8y2 + 5y2 – 7y2 – y2 m 2z2 – 2z2 + 3z2 – 3z2 11. Simplify each of the following expressions. a 3x + 4x b 4y + 2y c 5t – 2t d t – 4t e –2x – 3x f –k – 4k g m2 + 2m2 – m2 h 2y2 + 3y2 – 5y2 i –f 2 + 4f 2 – 2f 2 6 12. Simplify each of the following expressions. a 5x + 8 + 2x – 3 b 7 – 2x – 1 + 7x c 4p + 2t + p – 2t d 8 + x + 4x – 2 e 3 + 2t + p – t + 2 + 4 p f 5w – 2k – 2w – 3k + 5w g a+b+c+d–a–b–d h 9k – y – 5y – k + 10 13. Write each of these in a shorter form. (Be careful – some of them will not simplify.) a c+d+d+d+c b 2d + 2e + 3d c f + 3g + 4h d 3i + 2k – i + k e 4k + 5p – 2k + 4p f 3k + 2m + 5p g 4m – 5n + 3m – 2n h n + 3 p – 6p + 5 n i 5u – 4v + u + v j 2v – 5w + 5w k 2w + 4y – 7y l 5x2 + 6x2 – 7y + 2y m 8y2 + 5z – 7z – 9y2 n 2z2 – 2x2 + 3x2 – 3z2 5 Expanding and factorising 1. Expand these expressions. a 2(3 + m) b 3(2 – 4f ) c t(t + 3) d k(k2 – 5) e 5a(3a2 – 2b) 2. Simplify these expressions. a 4t + 3t b 3d + 2d + 4d c 5e – 2e d 3t – t e 2t 2 + 3t 2 f 6y2 – 2y2 g 3ab + 2ab h 7a2d – 4a2d 3. Expand and simplify. a 4(3 + 2h) – 2(5 + 3h) b 5(3g + 4) – 3(2g + 5) c 5(5k + 2) – 2(4k – 3) d 4(4e + 3) – 2(5e – 4) 4. Expand and simplify. a t(3t + 4) + 3t(3 + 2t) b 2y(3 + 4y) + y(5y – 1) c 4e(3e – 5) – 2e(e – 7) d 3k(2k + p) – 2k(3p – 4k) 5. Factorise the following expressions. a 6m + 12t b 3m2 – 3mp c 4a2 + 6a + 8 d 6ab + 9bc + 3bd e 8ab2 + 2ab – 4a2b 7 6. Expand the following expressions. a (x + 3)(x + 2) b (m + 5)(m + 1) c (x + 4)( x – 2) d ( f + 2)( f – 3) e (x – 3)(x + 4) f (y – 2)( y + 5) g (x + 3)(x – 3) h (t + 5)(t – 5) i (m + 4)(m – 4) What do you notice about your answers to g, h and i? 7. Expand the following expressions. e (2x + 3)(3x + 1) b (5m + 2)(2m – 3) c (2a – 3)(3a + 1) d (6 + 5t)(1 – 2t) e (4 – 2t)(3t + 1) 8. Expand the following squares. e (x + 5)2 b (m + 4)2 c (t – 5)2 d (3x + 1)2 e (x + y)2 9. Factorise the following. e x2 + 5x + 6 b p2 + 14p + 24 c a2 + 8a + 12 d t 2 – 5t + 6 e c2 – 18c + 32 f p2 – 8p + 15 g n2 – 3n – 18 h d2 + 2d + 1 10. Each of these is the difference of two squares. Factorise them. e x2 – 9 b t 2 – 25 c m2 – 16 d k2 – 100 e x2 – y2 f 9x2 – 1 11. Factorise the following expressions. e 2x2 + 5x + 2 b 24t2 + 19t + 2 c 6y2 + 33y – 63 d 6t 2 + 13t + 5 6 Solving equations 1. Solve these equations. f x 3y e +2=8 b + 3 = 12 c –1=8 5 8 2 t x + 10 d +3=1 e =3 5 2 2. Solve each of the following equations. Remember to check that each answer works for its original equation. e 2( x + 5) = 16 b 2(3y – 5) = 14 c 2( 3x + 1) = 11 d 9(3x – 5) = 9 3. Solve each of the following equations. a 2x + 3 = x + 5 b 7p – 5 = 3p + 3 c 2(d + 3) = d + 12 d 3(2y + 3) = 5(2y + 1) e 4(3b – 1) + 6 = 5(2b + 4) 8 Set up an equation to represent each situation described below. Then solve the equation. Remember to check each answer. 4. A man buys a daily paper from Monday to Saturday for d pence. On Sunday he buys the Observer for £1.60. His weekly paper bill is £4.90. How much is his daily paper? 5. In this rectangle, the length is 3 centimetres more than the width. The perimeter is 12 cm. a What is the value of x? (x + 3) b What is the area of the rectangle? x 6. Mary has two bags of sweets, each of which contains the same number of sweets. She eats four sweets. She then finds that she has 30 sweets left. How many sweets were in each bag to start with? 7. A boy is Y years old. His father is 25 years older than he is. The sum of their ages is 31. How old is the boy? 8. Max thought of a number. He then multiplied his number by 3. He added 4 to the answer. He then doubled that answer to get a final value of 38. What number did he start with? 7 Rearranging formulae 1. T = 3k Make k the subject. 2. A = 4r + 9 Make r the subject. m 3. g = Make m the subject. v 4. C = 2πr Make r the subject. 5. m = p2 + 2 Make p the subject. 1 6. A = πd 2, Make d the subject. 4 7. v = u2 – t a Make t the subject. b Make u the subject. 8. K = 5n2 + w a Make w the subject. b Make n the subject. 9. Make the letter in brackets the subject of each formula. a 3(x + 2y) = 2(x – y) (x) b p(a + b) = q(a – b) (b) c s(t – r) = 2(r – 3) (r) 10. When two resistors with values a and b are connected in parallel, the total resistance is given by: ab R= a+b a Make b the subject of the formula. b Write the formula when a is the subject. 9 11. a Make x the subject of this formula. x+2 y= x–2 4 b Show that the formula y = 1 + can be rearranged to give: x–2 4 x=2+ y–1 c Combine the right-hand sides of each formula in part b into single fractions and simplify as much as possible. d What do you notice? 12. The volume of the solid shown is given by: V = – πr 3 + πr 2h 2 3 r a Explain why it is not possible to make r the subject of this formula. b Make π the subject. h c If h = r, can the formula be rearranged to make r the subject? If so, rearrange it to make r the subject. 8 Functions 1. Given that f(x) = x + 1 and g(x) = x2 find: a f(2) b g(–2) c fg(x) d f–1(x) e fg–1(x) f gf–1(x) 2. State which values of x cannot be included in the domain of the following functions. a f : x → 1/x b g : x → √(x – 5) 10 c h:x→ (x + 1) 3. The function f(x) is defined as f(x) = x(x –1). a Find i f(3) ii f(– 3) b Given that f(x) = 6, find the values of x. 10 4. Complete the tables for the following functions and their inverses f(x) = f –1(x) = x+2 x – 10 2x 3x 1/x 3 √x sin x cos –1 x tan x = x and g(x) = 3x2 + 4 for all values of x. 1 5. f(x) 2 a Find i f(100) ii g(–1) iii fg(2) b Find an expression for gf(x) in terms of x. 2 6. The function f(x) is defined as f(x) = . (x + 2) a Given that f(x) = 5, find the value of x. b Find f –1(x). 1 7. f : x → x3 and g : x → (x – 1) a Find i fg(2) ii gf(–1) b Find i fg(x) ii gf(x) iii gg(x) giving your answer in its simplest terms c For each composite function in part b, state which values of x cannot be included in the domain. 11 9 Algebraic indices 1. Rewrite each of the following expressions in fraction form. a 5x–3 b 6t –1 c 7m–2 d 4q–4 e 10y–5 f – x–3 1 2 1 –1 g –m 2 h – t –4 3 4 i – y–3 4 5 j – x–5 7 8 2. Change each fraction to index form. 7 10 5 8 3 a b c d e x3 p t2 m5 y 3. Find the value of each of the following, where the letters have the given values. a Where x = 5 i x2 ii x–3 iii 4x–1 b Where t = 4 i t3 ii t –2 iii 5t –4 c Where m = 2 i m3 ii m–5 iii 9m–1 d Where w = 10 i w6 ii w–3 iii 25w–2 4. Simplify these and write them as single powers of a. a a2 × a b a3 × a2 c a4 × a3 d a6 ÷ a2 e a3 ÷ a f a5 ÷ a4 5. Simplify these expressions. a 2a2 × 3a3 b 3a4 × 3a–2 c (2a2)3 d –2a2 × 3a2 e –4a3 × –2a5 f –2a4 × 5a–7 6. Simplify these expressions. a 6a3 ÷ 2a2 b 12a5 ÷ 3a2 c 15a5 ÷ 5a d 18a–2 ÷ 3a–1 e 24a5 ÷ 6a–2 f 30a ÷ 6a5 7. Simplify these expressions. a 2a2b3 × 4a3b b 5a2b4 × 2ab–3 c 6a2b3 × 5a–4b–5 d 12a2b4 ÷ 6ab e 24a–3b4 ÷ 3a2b–3 8. Simplify these expressions. 6a4b3 2a2bc2 × 6abc3 3abc × 4a3b2c × 6c2 a b c 2ab 4ab2c 9a2bc 9. Rewrite the following in index form. √t 2 √ m3 √ k2 √ x3 3 4 5 a b c d 12 10 Linear programming 1. Solve the following linear inequalities. x t a x+4 7 b 2x – 3 7 c +4 7 d –2 4 2 3 2. Solve the following linear inequalities. a 4x + 1 3x – 5 b 5t – 3 2t + 5 c 3y – 12 y–4 d 2x + 3 x+1 e 5w – 7 3w + 4 f 2(4x – 1) 3(x + 4) 3. Solve the following linear inequalities. x+4 x–3 2x + 5 a 3 b 7 c 6 2 5 3 4x – 3 3t – 2 5y + 3 d 5 e 4 f 2 5 7 5 4. Write down the inequality that is represented by each diagram below. a b c 0 1 2 3 4 0 1 2 3 4 0 1 2 3 d e f –2 –1 0 1 2 –2 –1 0 1 –1 0 1 2 5. Draw diagrams to illustrate the following. a x 3 b x –2 c x 0 d x 5 e x –1 f 2 x 5 g –1 x 3 h –3 x 4 6. Solve the following inequalities and illustrate their solutions on number lines. a x+4 8 b x+5 3 c 4x – 2 12 d 2x + 5 3 x x x e 2(4x + 3) 18 f +3 2 g –2 8 h +5 3 2 5 3 7. a Draw the line x = 2 (as a solid line). b Shade the region defined by x 2. 8. a Draw the line y = –3 (as a dashed line). b Shade the region defined by y –3. 9. a Draw the line x = –2 (as a solid line). b Draw the line x = 1 (as a solid line) on the same grid. c Shade the region defined by –2 x 1. 10. a Draw the line y = –1 (as a dashed line). b Draw the line y = 4 (as a solid line) on the same grid. c Shade the region defined by –1 y 4. 13 11. a On the same grid, draw the regions defined by these inequalities. i –3 x 6 ii –4 y 5 b Are the following points in the region defined by both inequalities? i (2, 2) ii (1, 5) iii (–2, –4) 12. a Draw the line y = 3x – 4 (as a solid line). b Draw the line x + y = 10 (as a solid line) on the same diagram. c Shade the diagram so that the region defined by y 3x – 4 is left unshaded. d Shade the diagram so that the region defined by x + y 10 is left unshaded. e Are the following points in the region defined by both inequalities? i (2, 1) ii (2, 2) iii (2, 3) 13. Pens cost 45p each and pencils cost 25p each. Jane has £2.00 with which to buy pens and pencils. She buys x pens and y pencils. a Write down an inequality that must be true. b She must have at least two more pencils than pens. Write down an inequality that must be true. 14. Mushtaq has to buy some apples and some pears. He has £3.00 to spend. Apples cost 30p each and pears cost 40p each. He must buy at least two apples and at least three pears, and at least seven fruits altogether. He buys x apples and y pears. a Explain each of these inequalities. i 3x + 4y 30 ii x 2 iii y 3 iv x+y 7 b Using a suitable scale draw four lines to show the inequalities in part a. Shade the unwanted regions. c What is the maximum number of apples and pears which Mushtaq can buy? 15. A shop decides to stock only sofas and beds. A sofa takes up 4 m2 of floor area and is worth $300. A bed takes up 3 m2 of floor area and is worth $500. The shop has 48 m2 of floor space for stock. The insurance policy will allow a total of only $6000 of stock to be in the shop at any one time. The shop stocks x sofas and y beds. a Explain each of these inequalities. i 4x + 3y 48 ii 3x + 5y 60 b Using a suitable scale draw two lines to show the inequalities in part a. Shade the unwanted regions. c What is the maximum number of sofas and beds which can be stocked? 14 16. The 300 pupils in Year 7 are to go on a trip to Adern Towers theme park. The local bus company has six 40-seat coaches and five 50-seat coaches. The school hires x 40-seat coaches and y 50-seat coaches. a Explain each of these inequalities. i 4x + 5y 30 ii x 6 iii y 5 b Using a suitable scale draw three lines to show the inequalities in part a. Shade the unwanted regions. c The cost of hiring each coach is $100 for a 40-seater and $120 for a 50-seater. Which combination of 40-seater and 50-seater coaches gives the cheapest option? 11 Direct and inverse proportion In each case, first find k, the constant of proportionality, and then the formula connecting the variables. 1. T is directly proportional to M. If T = 20 when M = 4, find the following. a T when M = 3 b M when T = 10 2. W is directly proportional to F. If W = 45 when F = 3, find the following. a W when F = 5 b F when W = 90 3. Q varies directly with P. If Q = 100 when P = 2, find the following. a Q when P = 3 b P when Q = 300 4. X varies directly with Y. If X = 17.5 when Y = 7, find the following. a X when Y = 9 b Y when X = 30 5. The distance covered by a train is directly proportional to the time taken. The train travels 105 miles in 3 hours. a What distance will the train cover in 5 hours? b What time will it take for the train to cover 280 miles? 6. T is directly proportional to x2. If T = 36 when x = 3, find the following. a T when x = 5 b x when T = 400 7. W is directly proportional to M 2. If W = 12 when M = 2, find the following. a W when M = 3 b M when W = 75 8. « E varies directly with √« . If E = 40 when C = 25, find the following. C a E when C = 49 b C when E = 10.4 9. « X is directly proportional to √« . If X = 128 when Y = 16, find the following. Y a X when Y = 36 b Y when X = 48 10. P is directly proportional to f 3. If P = 400 when f = 10, find the following. a P when f = 4 b f when P = 50 15 11. In an experiment, the temperature, in °C, varied directly with the square of the pressure, in atmospheres. The temperature was 20 °C when the pressure was 5 atm. a What will the temperature be at 2 atm? b What will the pressure be at 80 °C? 12. The weight, in grams, of ball bearings varies directly with the cube of the radius measured in millimetres. A ball bearing of radius 4 mm has a weight of 115.2 g. a What will a ball bearing of radius 6 mm weigh? b A ball bearing has a weight of 48.6 g. What is its radius? In each case, first find the formula connecting the variables. 13. T is inversely proportional to m. If T = 6 when m = 2, find the following. a T when m = 4 b m when T = 4.8 14. W is inversely proportional to x. If W = 5 when x = 12, find the following. a W when x = 3 b x when W = 10 15. Q varies inversely with (5 – t ). If Q = 8 when t = 3, find the following. a Q when t = 10 b t when Q = 16 16. M varies inversely with t 2. If M = 9 when t = 2, find the following. a M when t = 3 b t when M = 1.44 17. « W is inversely proportional to √« . If W = 6 when T = 16, find the following. T a W when T = 25 b T when W = 2.4 18. The grant available to a section of society was inversely proportional to the number of people needing the grant. When 30 people needed a grant, they received $60 each. a What would the grant have been if 120 people had needed one? b If the grant had been $50 each, how many people would have received it? 19. While doing underwater tests in one part of an ocean, a team of scientists noticed that the temperature in °C was inversely proportional to the depth in kilometres. When the temperature was 6 °C, the scientists were at a depth of 4 km. a What would the temperature have been at a depth of 8 km? b To what depth would they have had to go to find the temperature at 2 °C? 20. A new engine was being tested, but it had serious problems. The distance it went, in km, without breaking down was inversely proportional to the square of its speed in m/s. When the speed was 12 m/s, the engine lasted 3 km. a Find the distance covered before a breakdown, when the speed is 15 m/s. b On one test, the engine broke down after 6.75 km. What was the speed? 21. The amount of waste which a firm produces, measured in tonnes per hour, is inversely proportional to the square root of the size of the filter beds, measured in m2. At the moment, the firm produces 1.25 tonnes per hour of waste, with filter beds of size 0.16 m2. a The filter beds used to be only 0.01 m2. How much waste did the firm produce then? b How much waste could be produced if the filter beds were 0.75 m2? 16 12 Simultaneous equations 1. By drawing their graphs, find the solution of each of these pairs of simultaneous equations. a x + 4y = 8 b y=x x–y=3 x + y = 10 c y=x+8 d 3x + 2y = 18 x+y=4 y = 3x x e y= +1 3 x + y = 11 2. Solve these simultaneous equations. a 4x + y = 17 b 3x + 2y = 11 c 2x + 5y = 37 2x + y = 9 2x – 2y = 14 y = 11 – 2x 3. Solve these simultaneous equations. a 5x + 2y = 4 b 2x + 3y = 19 c 5x – 2y = 4 4x – y = 11 6x + 2y = 22 3x – 6y = 6 4. Solve these simultaneous equations. a 2x + 5y = 15 b 3x – 2y = 15 c 2x + y = 4 3x – 2y = 13 2x – 3y = 5 x–y=5 d 3x + 2y = 2 e 3x – y = 5 2x + 6y = 13 x + 3y = –20 Read each situation carefully, then make a pair of simultaneous equations in order to solve the problem. 5. Amul and Kim have $10.70 between them. Amul has $3.70 more than Kim. Let x be the amount Amul has and y be the amount Kim has. Set up a pair of simultaneous equations. How much does each have? 6. Three chews and four bubblies cost 72p. Five chews and two bubblies cost 64p. What would three chews and five bubblies cost? 7. A taxi firm charges a fixed amount plus so much per mile. A journey of 6 miles costs £3.70. A journey of 10 miles costs £5.10. What would be the cost of a journey of 8 miles? 8. When you book Bingham Hall for a conference, you pay a fixed booking fee plus a charge for each delegate at the conference. The total charge for a conference with 65 delegates was $192.50. The total charge for a conference with 40 delegates was $180. What will be the charge for a conference with 70 delegates? 13 Quadratic equations 1. Solve these equations. a (x + 2)(x + 5) = 0 b (x + 3)(x – 2) = 0 c (x – 1)(x + 2) = 0 d (x – 3)(x – 2) = 0 2. First factorise, then solve these equations. a x2 + 5x + 4 = 0 b x2 – 8x + 15 = 0 c t 2 + 4t – 12 = 0 d x2 + 4x + 4 = 0 e t 2 + 8t + 12 = 0 17 3. First rearrange these equations, then solve them. a x2 + 10x = –24 b x2 + 2x = 24 4. Write an equivalent expression in the form (x ± a)2 – b. a x2 + 4x b x2 – 4x c x2 + 10x d x2 + 2x 5. Write an equivalent expression in the form (x ± a)2 – b. a x2 + 4x – 1 b x2 – 4x – 1 c x2 + 8x – 6 d x2 + 2x – 9 6. Solve by completing the square. Give your answers to two decimal places. a x2 + 2x – 5 = 0 b x2 – 4x – 7 = 0 c x2 + 2x – 9 = 0 7. Solve the following equations using the quadratic formula. Give your answers to 2 decimal places. a 2x2 + x – 8 = 0 b x2 – x – 10 = 0 c 7x2 + 12x + 2 = 0 d 4x2 + 9x + 3 = 0 e 3x2 – 7x + 1 = 0 f 4x2 – 9x + 4 = 0 8. The sides of a right-angled triangle are x, (x + 2) and (2x – 2). The hypotenuse is length (2x – 2). Find the actual dimensions of the triangle. 9. The length of a rectangle is 5 m more than its width. Its area is 300 m2. Find the actual dimensions of the rectangle. 3 10. Solve the equation x + = 7. Give your answers correct to 2 decimal places. x 11. On a journey of 400 km, the driver of a train calculates that if he were to increase his average speed by 2 km/h, he would take 20 minutes less. Find his average speed. 12. A train has a scheduled time for its journey. If the train averages 50 km/h, it arrives 12 minutes early. If the train averages 45 km/h, it arrives 20 minutes late. Find how long the train should take for the journey. 14 Algebraic fractions 1. Simplify each of these. x x x y 2x – 1 3x – 1 x – 4 2x – 3 a + b + c + d + 2 3 2 3 2 4 5 2 2. Simplify each of these. x x x y 2x + 1 3x + 3 x – 4 2x – 3 a – b – c – d – 2 3 2 3 2 4 5 2 3. Solve the following equations. x+1 x+2 2x – 1 3x + 1 3x + 1 5x – 1 a + =3 b + =7 c – =0 2 5 3 4 5 7 4. Simplify each of these. x x 4y2 3x2 2x + 1 3x + 1 x–5 5 a × b × c × d × 2 2 3 9x 2y 2 4 10 x – 5x 18 5. Simplify each of these. x x 4y2 2y 2x + 1 4x + 2 x – 5 x2 – 5x a ÷ b ÷ 2 c ÷ d ÷ 2 3 9x 3x 2 4 10 5 6. Simplify each of these. Factorise and cancel where appropriate. 3x x 3x x 3x + 1 x – 2 2x2 2y2 a + b ÷ c × d – 4 4 4 4 2 5 9 3 7. Solve the following equations. 4 5 3 4 a + =2 b – =1 x+1 x+2 2x – 1 3x – 1 8. Simplify the following expressions. x2 + 2x – 3 4x2 + x – 3 a b 2x2 + 7x + 3 4x2 – 7x + 3 15 Quadratic and cubic graphs 1. a Copy and complete the table for the graph of y = 3x2 for values of x from –3 to 3. x –3 –2 –1 0 1 2 3 y 27 3 12 b Use your graph to find the value of y when x = –1.5. c Use your graph to find the values of x that give a y-value of 10. 2. a Copy and complete the table for the graph of y = x2 + 2 for values of x from –5 to 5. x –5 –4 –3 –2 –1 0 1 2 3 4 5 y = x2 + 2 27 11 6 b Use your graph to find the value of y when x = –2.5. c Use your graph to find the values of x that give a y-value of 14. 3. a Copy and complete the table for the graph of y = x2 – 2x – 8 for values of x from –5 to 5. x –5 –4 –3 –2 –1 0 1 2 3 4 5 x2 25 9 4 –2x 10 –4 –8 –8 –8 y 27 –8 b Use your graph to find the value of y when x = 0.5. c Use your graph to find the values of x that give a y-value of –3. 19 4. a Copy and complete the table for the graph of y = x2 + 2x – 1 for values of x from –3 to 3. x –3 –2 –1 0 1 2 3 x2 9 1 4 +2x –6 –2 4 –1 –1 –1 –1 y 2 7 b Use your graph to find the y-value when x = –2.5. c Use your graph to find the values of x that give a y-value of 1. x d On the same axes, draw the graph of y = + 2. 2 x e Where do the graphs y = x2 + 2x – 1 and y = + 2 cross? 2 5. a Copy and complete the table for the graph of y = 2x2 – 5x – 3 for values of x from –2 to 4. x –2 –1.5 –1 –0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 y 15 9 –3 –5 –3 9 b Where does the graph cross the x-axis? 6. a Copy and complete the table to draw the graph of y = x2 + 4x for –5 x 2. x –5 –4 –3 –2 –1 0 1 2 x2 25 4 1 +4x –20 –8 4 y 5 –4 5 b Use your graph to find the roots of the equation x2 + 4x = 0. 7. a Copy and complete the table to draw the graph of y = x2 – 6x for –2 x 8. x –2 –1 0 1 2 3 4 5 6 7 8 x2 4 1 16 –6x 12 –6 –24 y 16 –5 –8 b Use your graph to find the roots of the equation x2 – 6x = 0. 8. a Copy and complete the table to draw the graph of y = x2 – 4x + 4 for –1 x 5. x –1 0 1 2 3 4 5 y 9 1 b Use your graph to find the roots of the equation x2 – 4x + 4 = 0. c What happens with the roots? 20 2 9. a Copy and complete the table to draw the graph of y = for –4 x 4. x x 0.2 0.4 0.5 0.8 1 1.5 2 3 4 y 10 4 2.5 1 0.5 b Use your graph to find the following. i the y-value when x = 2.5 the x-value when y = –1.25 ii 1 10. a Copy and complete the table to draw the graph of y = for –5 x 5. x x 0.1 0.2 0.4 0.5 1 2 2.5 4 5 y 10 2.5 1 0.2 b On the same axes, draw the line x + y = 5. c Use your graph to find the x-values of the points where the graphs cross. 5 11. a Copy and complete the table to draw the graph of y = for –20 x 20. x x 0.2 0.4 0.5 1 2 5 10 15 20 y 25 10 0.25 b On the same axes, draw the line y = x + 10. c Use your graph to find the x-values of the points where the graphs cross. 12. a Copy and complete the table to draw the graph of y = x3 + 3 for –3 x 3. x –3 –2.5 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2 2.5 3 y –24.00 –12.63 2.00 3.00 3.13 11.00 30.00 b Use your graph to find the y-value for an x-value of 1.2. 13. a Copy and complete the table to draw the graph of y = –x3 for –3 x 3. x –3 –2.5 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2 2.5 3 y 27.00 8.00 3.38 0.00 –0.13 –8.00 –15.63 b Use your graph to find the y-value for an x-value of –0.6. 14. a Copy and complete the table to draw the graph of y = x3 – 2x + 5 for –3 x 3. x –3 –2.5 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2 2.5 3 y –16.00 1.00 4.63 5.00 4.13 15.63 b On the same axes, draw the graph of y = x + 6. c Use your graph to find the x-values of the points where the graphs cross. 21 16 Gradients and tangents 1. Draw a graph of the curve y = x2 + 1. Use your graph to find the gradient of the curve at the following points: a x=5 b x=1 c x = –2 d x = –5 2. Draw a graph of the curve y = x(x – 3). Use your graph to find the gradient of the curve at the following points: a x=5 b x=3 c x=0 d x = 1.5 3. Draw a graph of the curve y = x3 – 3. Use your graph to find the gradient of the curve at the following points: a x=2 b x = –1 c x = –2 4. y 5 4 3 2 1 x –3 –2 –1 0 1 2 3 –1 –2 –3 –4 –5 a Use the graph to find the gradient of the curve when x = 2. b Write down the coordinates of the points where the gradient is zero. 22 5. Draw a graph of the curve y = sinx for 0° < x < 360 °. Use your graph to find the gradient of the curve at the following points: a x = 60° b x = 90° c x = 240° 6. a Draw a graph of the curve y = x2. b Use your graph to complete the following table: x –3 –2 –1 0 1 2 3 gradient at x c Use your table to find a formula for the gradient of the curve y = x2. d What can you say about the formula for the gradient of the curve y = x2 + 1 and y = x2 – 1. 23