Ed Excel Algebra Questions

8/2/2019 Ed Excel Algebra Questions

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1 Use of symbols

1. Write down the algebraic expression for:

a 2 more thanx b 6 less thanx

c kmore thanx d x minus t

e x added to 3 f dadded to m

g y taken away from b h p added to tadded to w

i 8 multiplied byx j h multiplied byj

k x divided by 4 l 2 divided byx

m y divided by t n w multiplied by t

o a multiplied by a p gmultiplied by itself

2. Asha, Bernice and Charu are three sisters. Bernice is x years old. Asha is three years older thanBernice. 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 bythis rule:

Multiply by 2 and add 30.

Using Cto stand for degrees Celsius and Fto stand for degrees Fahrenheit, complete this formula.

F =

4. a Anne has three bags of marbles. Each bagcontains n marbles. How many marbles doesshe have altogether?

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 atAp each d y pens atAp each.

1

EDEXCEL IGCSE MATHEMATICS Additional Practice

Algebra


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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 $2 t.

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+gg 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 f2 + 4f2 2f2

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 + 4p f 5w 2k 2w 3k+ 5w

g a + b + c + d a b d h 9ky 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 + 3p 6p + 5n

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

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2 Algebraic indices

1. Rewrite each of the following expressions in fraction form.

a 5x3 b 6t1 c 7m2 d 4q4 e 10y5

f12x

3g

12m1

h34 t

4i

45y3

j78x

5

2. Change each fraction to index form.

a b c d e

3. Find the value of each of the following, where the letters have the given values.

a Wherex = 5

i x2 ii x3 iii 4x1

b Where t= 4

i t3 ii t2 iii 5t4

c Where m = 2i m3 ii m5 iii 9m1

d Where w = 10

i w6 ii w3 iii 25w2

4. Simplify these and write them as single powers ofa.

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 3a2 c (2a2)3

d 2a2 3a2 e 4a3 2a5 f 2a4 5a7

6. Simplify these expressions.

a 6a3 2a2 b 12a5 3a2 c 15a5 5a

d 18a2 3a1 e 24a5 6a2 f 30a 6a5


a 2a2b3 4a3b b 5a2b4 2ab3 c 6a2b3 5a4b5

d 12a2b4 6ab e 24a3b4 3a2b3


a b c

9. Rewrite the following in index form.

a3t2 b 4m3 c 5k2 d x3

3abc 4a3b2c 6c2

9a2bc2a2bc2 6abc3

4ab2c6a4b3

2ab

3y

8m5

5t2

10p

7x3

3


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3 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)


a 4t+ 3t b 3d+ 2d+ 4d c 5e 2e

d 3t t e 2t2 + 3t2 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

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 t2 5t+ 6 e c2 18c + 32 f p2 8p + 15

g n2 3n 18 h d2 + 2d+ 1

10. Factorise the following expressions.

e 2x2 + 5x + 2 b 24t2 + 19t+ 2

c 6y2 + 33y 63 d 6t2 + 13t+ 5

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11. Solve the following equations.

a + = 2 b = 1 c = 1

d = 1

12. Simplify the following expressions.

a b c

d e

4 Changing the subject

1. T= 3k Make kthe subject.

2.A = 4r + 9 Make r the subject.

3.g= Make m the subject.

4. C= 2r Make r the subject.

5. m =p2 + 2 Makep the subject.

6.A = d2, Make dthe subject.

7. v = u2 t a Make tthe 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:

R =

a Make b the subject of the formula.

b Write the formula when a is the subject.

11. a Makex the subject of this formula.

y =

b Show that the formulay = 1 + can be rearranged to give:

x = 2 +

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?

4x 2

ab

a + b

14

m

v

4x2 25

8x2 22x + 5

4x2 + x 3

4x2 7x + 3

6x2 + x 2

9x2 4

4x2 1

2x2 + 5x 3

x2 + 2x 3

2x2 + 7x + 3

43x 1

32x 1

6x + 1

2x 12

1x + 1

184x 1

5x + 2

4x + 1

5

x + 2x 2

4y 1


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12. The volume of the solid shown is given by:

V= 23r3 + r2h

a Explain why it is not possible to make r the subject of this formula.

b Make the subject.

c Ifh = r, can the formula be rearranged to make r the subject?

If so, rearrange it to make r the subject.

5 Substitution

1. Find the value of 4b + 3 when a b = 2.5, b b = 1.5, c b = 12.

2. Evaluate when a x = 6, b x = 24, c x = 30.

3. Find the value of when a y = 2, b y = 4, c y = 6.

4. Find the value of when a x = 5, b x = 12, c x =34.

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.

6. Where A = , find A when

a n = 7, b n = 3, c n = 1.

7. Where Z = , find Z when

a y = 4, b y = 6, c y = 1.5.

6 Proportion

In each case, first find k, the constant of proportionality, and then the formula connecting the variables.

1. Tis directly proportional toM. IfT= 20 whenM= 4, find the following.

a TwhenM= 3 b Mwhen T= 10

2. Wis directly proportional toF. IfW= 45 whenF= 3, find the following.

a WwhenF= 5 b Fwhen W= 90

3. Q varies directly withP. IfQ = 100 whenP = 2, find the following.

a Q whenP = 3 b P when Q = 300

4.Xvaries directly with Y. IfX= 17.5 when Y= 7, find the following.

a Xwhen Y= 9 b YwhenX= 30

5. The distance covered by a train is directly proportional to the time taken. The train travels 105 milesin 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. Tis directly proportional tox2. IfT= 36 whenx = 3, find the following.

a Twhenx = 5 b x when T= 400

y2 + 44 +y

180(n 2)n + 5

24x

x

3

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7. Wis directly proportional toM2. IfW= 12 whenM= 2, find the following.

a WwhenM= 3 b Mwhen W= 75

8. Evaries directly with C. IfE= 40 when C= 25, find the following.

a Ewhen C= 49 b CwhenE= 10.4

9. Xis directly proportional to Y. IfX= 128 when Y= 16, find the following.

a Xwhen Y= 36 b YwhenX= 48

10. P is directly proportional tof3. IfP = 400 whenf= 10, find the following.

a P whenf= 4 b fwhenP = 50

11. In an experiment, the temperature, in C, varied directly with the square of the pressure, inatmospheres. 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 inmillimetres. 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. Tis inversely proportional to m. IfT= 6 when m = 2, find the following.

a Twhen m = 4 b m when T= 4.8

14. Wis inversely proportional tox. IfW= 5 whenx = 12, find the following.

a Wwhenx = 3 b x when W= 1015. Q varies inversely with (5 t). IfQ = 8 when t= 3, find the following.

a Q when t= 10 b twhen Q = 16

16. Mvaries inversely with t2. IfM= 9 when t= 2, find the following.

a Mwhen t= 3 b twhenM= 1.44

17. Wis inversely proportional to T. IfW= 6 when T= 16, find the following.

a Wwhen T= 25 b Twhen W= 2.4

18.

The grant available to a section of society was inversely proportional to the number of peopleneeding 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 thetemperature in C was inversely proportional to the depth in kilometres. When the temperature was6 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?

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20. A new engine was being tested, but it had serious problems. The distance it went, in km, withoutbreaking down was inversely proportional to the square of its speed in m/s. When the speed was12 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 produces1.25 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?

7 Solving equations

1. Solve these equations.

e + 2 = 8 b + 3 = 12 c 1 = 8

d + 3 = 1 e = 3

2. Solve each of the following equations. Remember to check that each answer works for itsoriginal equation.

e 2(x + 5) = 16 b 2(3y 5) = 14

c x + 1) = 11 d 9(3x 5) = 9

3. Solve each of the following equations.

a

2x + 3 =x + 5b

7p 5 = 3p + 3c

2(d+ 3) = d+ 12d 3(2y + 3) = 5(2y + 1) e 4(3b 1) + 6 = 5(2b + 4)

Set up an equation to represent each situation described below. Then solve the equation. Rememberto check each answer.

4. A man buys a daily paper from Monday to Saturday for dpence. On Sunday he buys the Observer for1.60. 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 ofx?b What is the area of the rectangle?

6. Mary has two bags of sweets, each of which contains the same number of sweets. She eats foursweets. 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 isthe boy?

8. Max thought of a number. He then multiplied his number by 3. He added 4 to the answer. He thendoubled that answer to get a final value of 38. What number did he start with?

x + 102

t

5

3y2

x

8f

5

8


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8 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 = 3xe y = + 1

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


a 5x + 2y = 4 b 2x + 3y = 19 c 5x 2y = 4

4x y = 11 6x + 2y = 22 3x 6y = 6


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. Letx be the amountAmul has andy be the amount Kim has. Set up a pair of simultaneous equations. How much doeseach have?

6. Three chews and four bubblies cost 72p. Five chews and two bubblies cost 64p. What would threechews 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 eachdelegate at the conference. The total charge for a conference with 65 delegates was $192.50. Thetotal charge for a conference with 40 delegates was $180. What will be the charge for a conference

with 70 delegates?

9. Solve these pairs of simultaneous equations.

a xy = 2 b xy = 4y = x + 1 2y = x + 6


a x2 + y2 = 25 b x2 + y2 = 9 c x2 + y2 = 13x + y = 7 y = x + 3 5y + x = 13


a y = x2 + 2x 3 b y = x2 2x 5 c y = x2 2xy = 2x + 1 y = x 1 y = 2x 3

x

3

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9 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 t2 + 4t 12 = 0

d x2 + 4x + 4 = 0 e t2 + 8t+ 12 = 0

3. First rearrange these equations, then solve them.

a x2 + 10x = 24 b x2 + 2x = 24

4. 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 = 05. 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.

6. The length of a rectangle is 5 m more than its width. Its area is 300 m2. Find the actual dimensions ofthe rectangle.

7. Solve the equationx + = 7. Give your answers correct to 2 decimal places.

8. On a journey of 400 km, the driver of a train calculates that if he were to increase his average speedby 2 km/h, he would take 20 minutes less. Find his average speed.

9. 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 thejourney.

10 Inequalities and graphs

1. Solve the following linear inequalities.

a x + 4 7 b 2x 3 7 c + 4 7 d 2 4


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)


a 3 b 7 c 6

d 5 e 4 f 25y + 3

53t 2

74x 3

5

2x + 53

x 35

x + 42

t

3x

2

3x

10


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4. Write down the inequality that is represented by each diagram below.

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

e 2(4x + 3) 18 f + 3 2 g 2 8 h + 5 3

7. a Draw the linex = 2 (as a solid line).

b Shade the region defined byx 2.

8. a Draw the liney = 3 (as a dashed line).

b Shade the region defined byy 3.

9. a Draw the linex = 2 (as a solid line).

b Draw the linex = 1 (as a solid line) on the same grid.

c Shade the region defined by 2 x 1.

10. a Draw the liney = 1 (as a dashed line).

b Draw the liney = 4 (as a solid line) on the same grid.

c Shade the region defined by 1 y 4.

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 liney = 3x 4 (as a solid line).

b Draw the linex +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)

x3

x5

x2

11


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13. Pens cost 45p each and pencils cost 25p each. Jane has 2.00 with which to buy pens and pencils.She buysx pens andy 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 andpears cost 40p each. He must buy at least two apples and at least three pears, and at least seven

fruits altogether. He buysx apples andy 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 m

2

of floor area and is worth $500. The shop has 48 m

2

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 stocksx sofas andy beds.


i 4x + 3y 48 ii 3x + 5y 60

b Using a suitable scale draw two lines to show the inequalities in part a.


c What is the maximum number of sofas and beds which can be stocked?

16. The 300 pupils in Year 7 are to go on a trip to Adern Towers theme park. The local bus companyhas six 40-seat coaches and five 50-seat coaches. The school hires x 40-seat coaches andy 50-seatcoaches.


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.


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?

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11Patterns and sequences

1. Look carefully at each number sequence below. Find the next twonumbers 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.

Find the next four triangular numbers.

3. The first two terms of the sequence of fractions are:

n = 1: = = 0 n = 2: =

Work out the next five terms of the sequence.

4. A sequence is formed by the rule 12 n (n + 1) for n = 1, 2, 3, 4,

The first term is given by n = 1: 12 1 (1 + 1) = 1

The second term is given by n = 2: 12 2 (2 + 1) = 3

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. Find the next two terms and the nth term in each of these linearsequences.

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,

13

2 12 + 1

02

1 11 + 1

n 1n + 1

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8. A pattern of squares is built up from matchsticks as shown.

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 upfrom matchsticks.

a Draw the 5th set of triangles inthis pattern.

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 couldseat five people, as shown on the right. When the tables were joined together asshown below, each table could not seat as many people.

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 likethis did they need?

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12 Interpreting graphs

1. This graph is a conversion graph between C and F.

a How many F are equivalent to a temperatureof 0 C?

b What is the gradient of the line?

c From your answers to parts a and b,write downa rule which can be used to convert Cto F.

2. This graph illustrates charges for fuel.

a What is the gradient of the line?

b The standing charge is the basic chargebefore the cost per unit is added. What isthe standing charge?

c Write down the rule used to work out thetotal charge for different amounts of unitsused.

3. This graph shows the hire charge for heatersover so many days.

a Calculate the gradient of the line.

b What is the basic charge before the daily hirecharge is added on?

c Write down the rule used to work out the totalhire charge.

4. This graph shows the length of a springwhen different weights are attached to it.

a Calculate the gradient of the line.

b How long is the spring when noweight is attached to it?

c By how much does the springextend per kilogram?

d Write down the rule for finding the

length of the spring for differentweights.

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5. Paul was travelling in his car to a meeting. He set off from home at 7:00 am, and stopped on the wayfor a break. This distancetime graph illustrates his journey.

a At what time did he:

i stop for his break?

ii set off after his break?

iii get to his meeting place?

b At what average speed was he travelling:

i over the first hour?

ii over the second hour?

iii for the last part of his journey?

6.James was travelling to Cornwall on his holidays. This distancetime graph illustrates his journey.

a His fastest speed was on the motorway.

i How much motorway did he use?

ii What was his average speed on the motorway?

b i When did he travel the slowest?

ii What was his slowest average speed?

7. Richard and Paul had a 5000 m race. The distance covered is illustrated below.

a Paul ran a steady race. What is

his average speed in:

i metres per minute?

ii km/h?

b Richard ran in spurts. What washis quickest average speed?

c Who won the race and byhow much?

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8. Three friends, Patrick, Araf and Sean, ran a 1000 metres race. The race is illustrated on thedistancetime graph below.

a Describe the race of each friend.

b i What is the average speed of Araf in m/s?

ii What is this speed in km/h?

9. Calculate the average speed of the journey represented by each line in the following diagrams.

13 Linear graphs and coordinates

1. Draw the graph ofy = 3x + 4 forx-values from 0 to 5 (0 x 5).

2. Draw the graph ofy = 2x 5 for 0 x 5.

3. Draw the graph ofy = + 4 for 6 x 6.

4. a On the same axes, draw the graphs ofy = 1 andy = 2 for 0 x 12.

b At which point do the two lines intersect?

5. Find the gradient of each of these lines.

x

3

17

x

3x

2


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6. Find the gradient of each of these lines. What is special about these lines?

7. Draw these lines using the gradient-intercept method. Use grids, takingx from 10 to 10 and y from10 to 10.

a y = 2x + 6 b y =x + 7 c y = 14x 3 d y =x + 8

8. a Using the gradient-intercept method, draw the following lines on the same grid. Use axes withranges 6 x 6 and 8 y 8.

i y = 3x + 1 ii y = 2x + 3

b Where do the lines cross?

9. Give the equation of each of these lines, all of which have positive gradients. (Each squarerepresents 1 unit.)

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10. Give the equation of each of these lines, all of which have negative gradients. (Each squarerepresents 1 unit.)

14 Quadratic

1. a Copy and complete the table for the graph ofy = 3x2 for values ofx from 3 to 3.

b Use your graph to find the value ofy whenx = 1.5.

c Use your graph to find the values ofx that give ay-value of 10.

2. a Copy and complete the table for the graph ofy =x2 + 2 for values ofx from 5 to 5.



3. a Copy and complete the table for the graph ofy =x2 2x 8 for values ofx from 5 to 5.



54324

4

8

8

101239

4525

10

8

27

x

x2

2x

8

y

5432

6

10123

11

45

27

x

y = x2 + 2

32

12

101

3

23

27

x

y

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4. a Copy and complete the table for the graph ofy =x2 + 2x 1 for values ofx from 3 to 3.

b Use your graph to find they-value whenx = 2.5.


d On the same axes, draw the graph ofy = + 2.

e Where do the graphsy =x2 + 2x 1 andy = + 2 cross?

5. a Copy and complete the table for the graph ofy = 2x2 5x 3 for values ofx from 2 to 4.

b Where does the graph cross thex-axis?

6. a Copy and complete the table to draw the graph ofy =x2 + 4x for 5 x 2.

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 ofy =x2 6x for 2 x 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 ofy =x2 4x + 4 for 1 x 5.

b Use your graph to find the roots of the equation x2 4x + 4 = 0.

c What happens with the roots?

543

1

2101

9

x

y

87654

16

24

8

321

1

6

5

012

4

12

16

x

x2

6x

y

21

1

4

5

012

4

8

4

345

25

20

5

x

x2

+4x

y

49

3.532.53

21.510.55

03

0.511.59

215

x

y

x

2

x

2

32

4

4

1

7

1

1

01

2

2

1

3

9

6

1

2

x

x2

+2x

1

y

20


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15Gradients and tangents

1. Draw a graph of the curvey =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 curvey =x(x 3).


a x = 5

b x = 3

c x = 0

d x = 1.5

3. Draw a graph of the curvey =x3 3.


a x = 2

b x = 1

c x = 2

4.

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.

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5. Draw a graph of the curvey = sinx for 0


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4. Complete the tables for the following functions and their inverses

5. f(x) =x and g(x) = 3x2 + 4 for all values ofx.

a Find i f(100)

ii g(1)

iii fg(2)

b Find an expression for gf(x) in terms ofx.

6. The function f(x) is defined as f(x) =2

.(x + 2)

.

a Given that f(x) = 5, find the value ofx.

b Find f1(x).

7.f:x x3 andg:x 1

(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 ofx cannot be included in the domain.

23

f(x) = f1(x) =

x + 2

x 10

2x

3x

1/x3x

sinx

cos 1x

tanx


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17 Calculus

1. Differentiate:

a 3x 7

b 5x + 11

c x2 8x + 5

d 3x2 2x + 121

e (x 1)(x + 5)

f (x 3)2

g x3 +x2 +x 1

h 5x3 + 4x2 8x 17

i x(x + 1)(x 4)

j 1/x + 2/x2

k 10/x + 4/x2 + 3/x3 7/x4

2. Find the gradient of the tangent at the given point on the following curves:

a y =x2 7x + 2 at the point wherex = 5

b y =x2 4x 21 at the point (2, 9)

c y = 3x2 + 5x 2 at the point (1, 4)

d y = (2x 1)(x + 1) at the point (0, 1)

3. A curve has the equationy =x2 6x + 11.

a Find dydx

.

b Find the gradient of the curve at the point with coordinates (4, 3).

c Find the coordinates of the turning point.

4. A curve has the equationy =x2 2x 8.

a Finddydx

.

b Find the coordinates of the turning point.

c Is this turning point a maximum or a minimum?

Give a reason for your answer.

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5. a Differentiate with respect tox:

y =x3 3x2 + 8

b Hence find the coordinates of the minimum point on the curve.

6. The graph shows the curvey = 2x3 15x2 + 24x + 20.

a Finddydx

.

b Find the gradient of the curve at A.

c Find the coordinates of the turning points at B and C.

7. A body is moving in a straight line which passes through a fixed point O.

The displacement,s metres, of the body from O at time tseconds is given by the formula:

s = t(t 1)(t 2)a Find an expression for the velocity, v m/s, at time tseconds.

b Find the acceleration after 3 seconds.

8. The temperature Tof liquid after tseconds is given by the formula T= t2 11t+ 28.

a Find the rate of change of the temperature after 3 seconds.

b Find the time when the temperature is at its maximum.

9. A car is moving along a straight road. It passes a point O.

After tseconds its distance,s (in metres) from O is given by the formulas = 10(t t2) where 0 t 10.

a Find the time when the car passes through the origin again.

b Finddsdt

.

c Find the maximum distance of the car from O.

d Find the speed of the car 4 seconds after passing O.

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10. A farmer wishes to make a rectangular pen for keeping sheep.

She has 30 metres of fencing which she builds against the wall AB.

She wants to make the area of the pen as large as possible.

a If the width of the pen isx metres, show that the areay, of the pen is given by the formula:

y = 30x 2x2

b Find the value ofx for which the area is a maximum.

c Hence find the largest possible area of the sheep pen.

Ed Excel Algebra Questions

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