Edexcel-IGCSE-algebr..

26 Pages·2007·0.2 MB·English

by  Sally Boothroyd

EDEXCEL IGCSE MATHEMATICS Additional Practice
Algebra
1 Use of symbols
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 = ……
4.a Anne has three bags of marbles. Each bag
contains n marbles. How many marbles does
she 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 at Ap each d y pens at Ap each.
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7. Aboy 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 –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 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 + 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 5x–3 b 6t–1 c 7m–2 d 4q–4 e 10y–5
f –1x–3 g –1m–1 h –3t–4 i –4y–3 j –7x–5
2 2 4 5 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.
a √3t2 b √4m3 c √5k2 d √x3
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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)
2. Simplify these expressions.
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.
4 5 18 1 2x – 1 6
a + = 2 b – = 1 c – = 1
x + 1 x + 2 4x – 1 x + 1 2 x + 1
3 4
d – = 1
2x – 1 3x – 1
12.Simplify the following expressions.
x2 + 2x – 3 4x2 – 1 6x2 + x – 2
a b c
2x2 + 7x + 3 2x2 + 5x – 3 9x2 – 4
4x2 + x – 3 4x2 – 25
d e
4x2 – 7x + 3 8x2 – 22x + 5
4 Changing the subject
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.
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6.A = πd2, Make d the subject.
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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.
11.a Make x the subject of this formula.
x + 2
x – 2
y =
4
b Show that the formula y = 1 + can be rearranged to give:
4 x – 2
y – 1
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?
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12.The volume of the solid shown is given by:
V = –2πr3 + πr2h
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a Explain why it is not possible to make r the subject of this formula.
b Make π the subject.
c If h = 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 = –1.
2
x
2.Evaluate when a x = 6, b x = 24, c x = –30.
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3.Find the value of when a y = 2, b y = 4, c y = –6.
24
4.Find the value of when a x = –5, b x = –1, c x = –3.
x 2 4
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.
y2 + 4
7.Where Z = , find Z when
4 + y
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.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
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7. W is directly proportional to M2. If W = 12 when M = 2, find the following.
a W when M = 3 b M when W = 75
8. E varies directly with √«C«. If E = 40 when C = 25, find the following.
a E when C = 49 b C when E = 10.4
9. X is directly proportional to √«Y«. If X = 128 when Y = 16, find the following.
a X when Y = 36 b Y when X = 48
10.P is directly proportional to f3. If P = 400 when f = 10, find the following.
a P when f = 4 b f when P = 50
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. Aball bearing of radius 4 mm has a weight of 115.2 g.
a What will a ball bearing of radius 6 mm weigh?
b Aball 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 t2. 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 √«T«. If W = 6 when T = 16, find the following.
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?
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20.Anew 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?
7 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 x + 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)
Set up an equation to represent each situation described below. Then solve the equation. Remember
to check each answer.
4.Aman 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?
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 four
sweets. She then finds that she has 30 sweets left. How many sweets were in each bag to start with?
7.Aboy 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?
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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 = 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. Ataxi firm charges a fixed amount plus so much per mile. Ajourney of 6 miles costs £3.70.
Ajourney 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?
9. Solve these pairs of simultaneous equations.
a xy = 2 b xy = –4
y = x + 1 2y = x + 6
10.Solve these pairs of simultaneous equations.
a x2 + y2 = 25 b x2 + y2 = 9 c x2 + y2 = 13
x + y = 7 y = x + 3 5y + x = 13
11.Solve these pairs of simultaneous equations.
a y = x2 + 2x – 3 b y = x2 – 2x – 5 c y = x2 – 2x
y = 2x + 1 y = x – 1 y = 2x – 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 = 0
5.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 of
the rectangle.
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7.Solve the equation x + = 7. Give your answers correct to 2 decimal places.
x
8.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.
9.Atrain 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.
10 Inequalities and graphs
1.Solve the following linear inequalities.
x t
a x + 4 (cid:2) 7 b 2x – 3 (cid:2) 7 c + 4 (cid:2) 7 d – 2 (cid:3) 4
2 3
2.Solve the following linear inequalities.
a 4x + 1 (cid:3) 3x – 5 b 5t – 3 (cid:4) 2t + 5 c 3y – 12 (cid:4) y – 4
d 2x + 3 (cid:3) x + 1 e 5w – 7 (cid:4) 3w + 4 f 2(4x – 1) (cid:4) 3(x + 4)
3.Solve the following linear inequalities.
x + 4 x – 3 2x + 5
a (cid:4) 3 b (cid:5) 7 c (cid:2) 6
2 5 3
4x – 3 3t – 2 5y + 3
d (cid:3) 5 e (cid:5) 4 f (cid:4) 2
5 7 5
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