Aim for A* in GCSE Maths - Part 1

7/31/2019 Aim for A* in GCSE Maths - Part 1

1/17

GCSE Higher Tier Practice Questions 1

GCSE Higher Tier Practice Questions

Contents:

Topic Page

1 Solving Linear, Quadratic and Simultaneous Equations 2

2 Algebra: Factorising Practice and Fractions 3

3 Changing Subjects of Formulae & Solving Inequalities 4

4 Quadratic Functions: Completing the Square and Sketching 5

5 Forming and Solving Quadratic Equations 5

6 Proportionality orVariation 6

7 Number: Estimating and Factorising 6

8 Constructions, Loci and Bearings 7

9 Vectors 8

10 Probability 9

11 Calculations in Right-Angled Triangles 10

12 Calculations Needing Sine and Cosine Rules

10A

13 Upper and Lower Bounds and Standard Form 11

Answers 12

(C) E J Trickey and M E B George 2004


2/17


1 Solving Linear, Quadratic and Simultaneous Equations*Always try to CHECK your answers*

Exercise 1.1 Solve:

1. 2954 =+ x 2. 129 = x

3. 44

7 = x 4. 134315 = xx

5. ( ) 1432312 += xx 6. 612

54=

+x

7. 448

5

4

3=+

xx8. 3

10

7

5

4=

xx

Exercise 1.2 Solve the following sets ofsimultaneous equations:

1.

==+

1425

1823

ba

ba2.

==

823

1627

yx

yx

3.

==+

1523

2054

qp

qp4.

=+=+

9737

696

yx

yx

5.

==

2425

1852

ba

ba6.

=+=1536

5211

yx

yx

7.

=+

=

342

13

22

1

yx

yx

Exercise 1.3 Solve:

1. 0652 =+ xx 2. 0652 = xx

3. 02452 =+ xx 4. 024102 =+ xx

5. 024232 = xx 6. 024102 =+ xx

7. 036122 =+ xx 8. 03652 =+ xx

Exercise 1.4 Solve, to 3 sig fig, using the

formulaa

acbbx

2

42 =

1. 01192 =+ xx 2. 0753 2 =+ xx

3. 05172 2 =+ xx 4. 0935 2 =+ xx

5. 11152 2 += xx 6. 7542 = xx

7. 8213 2 = xx

Exercise 1.5 Solve by factorising or by usingthe formula

1. 865 2 =+ xx 2. 6112 2 =+ xx

3. xx 13152 2 =+ 4. yy 952 2 =

5. 2522 =+ yy

Exercise 1.6 Simultaneous equations - one linear and one quadratic:

*Expect to reach a quadratic equation in one letter which will factorise (but you may use the

quadratic formula instead of factorising if you prefer it)

*Checkyour answers in the original equations

1.

=+

=+

26

4

22 yx

yx2.

=+

=

3

113

2xxy

xy3.

=

=+

182

12

2 xyx

yx

4.

=

=+

154

132

22 yx

yx5.

=+

=

8

34

2xxy

xy6.

=

=

4494

232

22 yx

yx

7.

=+

=

182

32

22 yx

yx8.

=

=+

93

24

2 xyx

yx9.

=

=+

124

22

22 yx

yx



3/17


2 Algebra: Factorising Practice and Fractions

Exercise 2.1 Factorise fully:

1. xx 52 + 2. yy 32 2 +

3. 2915 xx 4. 22 xyyx +

5. 223 52 baba + 6. cbabc 239

7. 2452 + xx 8. 24142 + nn

9. 2422 aa 10. 48282 2 + nn

11. 1582 ++ xx 12. 45243 2 ++ xx

13. 42 x 14. 10025 2 x

15. 2142

bb 16. 1052052

bb

Exercise 2.2 Factorise fully:

1. ycybxcxb +++

2. bybxhyhx +

3. bnanbmam + 236

4. byaybxax +++ 326

5. qspsqrpr + 428

6. 352 2 ++ xx 7. 372 2 ++ xx

8. 253 2 xx 9. 152 2 xx

10. 28173 2 xx 11. 5412 2 + xx

12. 31916 2 ++ xx 13. 376 2 + yy

14. 614122 + yy 15. 6113 2 + xx

16. 494 2 m 17. 236 p

Without a calculator, evaluate:

18. 22 15.085.9

19. 22 2.78.12

20. 22 2.338.66

21. 22 5.65.65.325.3 ++

22. 22 2.02.08.928.9 ++

Exercise 2.3 Simplify:

1.3

22

15

12

mn

nm2.

2

4

3

x

xy

3.

x

yxxy 2+

4.x

x

6

189 +5.

x

xx

4

884 2++

6.x

xyx

2

28 +7.

x

xxy

9

36 +

8.xx

xx

3

22

2

+

9.32

32

2

xx

xx

10. xx

xx

102

42

2

+

11. 2

562

2

++

xx

xx

12.145

2142

2

xx

xx13.

4

1072

2

++

x

xx

14.14

22

2

+

x

xx15.

xx

xx

10

90192

2

+

Exercise 2.4 Express as a single fraction,fully simplified:

1. 22 3

2

4

3

pqqp 2.

22 5

2

3

5

ba+

3.32 8

5

4

32

aaa+ 4.

b

a

a

b

4

3

2

5

5.3

3

2

5

++

xx6.

2

3

1

7

+

+ xx

7.1

5

3

2

+ xx8.

1

4

2

3

+

xx

9.12

5

4

62

xx

x

x10.

9

4

3

22

++ x

x

x

Exercise 2.5 Solve andcheckyour solutions(all quadratics reached correctly do factorise)

1. 44

3

1

2=

xx2.

2

1

52

32

=++

m

m

m

m

3. 23

= aa

4.2

1

3

12=

+

aa



4/17


5.13

2

7

4

=

a

a



5/17


3 Changing Subjects of Formulae & Solving Inequalities

Exercise 3.1 Solve:1. In each case, makexthe subject of the

formula:

(a) cbax =+ (b) rq

p

x=

(c) mnxy = (d) tm

xl =

(e) )()( dxcbxa +=+ (f) sn

ax=

(g) cxnbax = (h) 0=++ cbyax

2. In each case, makexthe subject of the

formula:

(a) 3=+

+dcx

bax

(b) 2=

bx

axn

(c) 222 bxa =+ (d) nmax =2

(e) qp

x4

2

=

(f) ( ) 222 nlax +=

(g)g

xT 2= (h) cbax =+

(i)zyx

111+= (j) m

y

x=

4

3. rhrA 222 += Findh, ifA = 704

and r= 7 (give your answer to 2 sig. fig.)

4.3

3

4rV = Findr, ifV= 17.2

(give your answer to 2 sig. fig.)

5. 222 cbaA ++= Finda, ifA = 19.10,

b = 11 and c = 12 (give answer to 2 sig. fig.)

6.g

lT 2=

T= 6.95 and l= 12. Findg to 2 sig. fig.

7.a

c

dx

bx=

++

Findx, ifa = 7, b = 3, c = 6

and d= 5

8. In each case, make

Ror

rthe subject of

the formula:

(a) hrV2

3

1= (b) trs

121

+=

(c)3

3

4rV = (d) 24 rA =

(e) 222 42 bar += (f)100

NRAR +=

Exercise 3.2 Solve the following and showyour answer clearly on a number line:

1. 132

10 x

5. x42216 < 6. 310

11 x

7. 17234


6/17


4 Quadratic Functions: Completing

the Square and Sketching

1. 136)( 2 + xxxf

(a) Re-writef(x) in the form ( ) baxxf + 2)(

(b) Hence state the minimum value off(x) and thevalue ofx for which this minimum occurs.

(c) Sketch the curvey=f(x)

2. 74)( 2 + xxxf

(a) Re-writef(x) in the form ( ) dcxxf + 2)((b) Hence state the minimum value off(x) and the

value ofx for which this minimum occurs.


3. 32)( 2 ++ xxxf

(a) Re-writef(x) in the form ( ) fexxf ++ 2)((b) Hence state the minimum value off(x) and the

value ofx for which this minimum occurs.


4.2811)( xxxf

(a) Re-writef(x) in the form ( ) 2)( BxAxf +(b) Hence state the maximum value off(x) and the

value ofx for which this maximum occurs.

(c) Sketch the graph ofy=f(x)

5. 210)( xxxf (a)Re-writef(x) in the form ( ) 2)( xDCxf (b) Hence state the maximum value off(x) and the



6.248)( xxxf

(a) Re-writef(x) in the form ( ) 2)( bxaxf +(b) Hence state the maximum value off(x) and the



7. xxxf 6)( 2 (a) Sketchy=f(x)

(b) By consideringf(x) in the form

( ) baxxf 2)( , or otherwise, state thehorizontal and vertical transformations that would

map xxy 62 = onto the graph of 2xy =

5 Forming and Solving Quadratic

Equations

1. Show that the

information given

about the lengths in

this right-angled triangle satisfies the equation:0342 =+ xx . Hence findx.(Two solutions)

2.All measurements are in centimetres

Both of these

quadrilaterals

are rectangles

and the inner

one is cut out

of the outer

one. The REMAINING white area is 162cm2.

Show that this information can give the equation

165183 2 =+ xx and solve forx.

3. All measurements in cm

The shaded

right-angled

triangle is

removed from

the rectangle.

The remaining

area is 90 cm2. Show clearly that this gives the

equation 9022 2 =+xx and solve it.

4. Not drawn to scale.

The curve is 822 = xxy(a) State the coordinates of D.

(b) By completing the square

in the form baxy = 2)( orotherwise, give the coordinates

of M, the minimum point of the graph.

(c) By factorising 822 = xxy , find thecoordinates of A and B.

(d) Find the equation of the line CB and thecoordinates of C.

5. 16)( 2 + xxxf(a)Rearrangef(x) in the form dcxxf 2)()((b)Hence solve, in surd form, 0162 =+ xx(c)Now solve 0162 =+ xx using the formula

a

acbbx

2

42 = leaving your answers in

fully simplified surd form. Show that these two

answers are identical to your answers from (b).


x

y

B

C (0,4)

D M

A


7/17

6 Proportionality orVariation

Exercise 6.11.p varies directly as q.

Fill in the following table:

p 6 8 14 20

q 0.25 25 35 40 200

2. The area of a circle is proportional to the

square of the radius.

What happens if:

(a) the radius is doubled?

(b) the radius is trebled?

(c) the radius is made 10 times as big?

3. m varies as the square root ofa. When a = 4,

m = 13.6

(a) Find the equation linking m and a.

(b) Find: m when a = 16a when m = 34

m when a = 9

4. No calculators in this question!

Pis propotional to h2. When h = 15,P= 45

(a) Find Pwhen h = 20

and when h = 100

(b) Find h whenP= 180

and whenP= 0.2

5. It is known thaty is proportional tox.

Fill in the following table:

x 10 20

y 16 54 128

also find: (a) y whenx = 50

(b) x wheny = 6.75

6. If a stone is dropped from the top of a building,

the time tit takes to reach the ground is

proportional to the square root of the height h of

the building.

A stone was timed and took 4.5 seconds to drop

100 feet.

If the stone only dropped 50 feet, how long would

it take to reach the ground?

Exercise 6.21.tis inversely proportional to v.

Complete the following table:

v 10 20 50 100

t 5 4 2.5

2. It is known thaty is inversely proportional to

x2. Fill in the following table:

x 0.1 0.2

y 50 12.5 8 2

3. No calculators in this question!

His inversely proportional to the square root ofT.

When T= 36,H= 2.5

(a) Find: Hwhen T= 100

and when T= 0.01

(b) Find: TwhenH= 3

and whenH= 0.3

4. The resistanceR in a fixed length of wire

varies inversely as the square of the diameterdof

the wire. If the diameter is 5mm, the resistance is

0.04 ohms. Find the resistance if the diameter is

4mm.

5. (Give your answers to 3 sig. fig.)

The air pressure available from a bicycle pump is

inversely proportional to the square of the

diameter of the pump. If a pressure of 16 units is

available from a diameter of 25mm,

(a) Find the pressure available from a pump with

a diameter of 12mm.

(b) Find the diameter of a pump whose pressure

is 20 units.

7 Number: Estimating and Factorising(NO calculators to be used!)

1. Given that 28000 x 0.0032 = 89.6

Find: (a) 28 x 320 (b)00032.0

896.0

(c)000280

8960

2. hrV2

3

1=

EstimateVwhen r= 19.8 and h = 10.1(give your answer to 1 sig. fig.)

3.g

lT 2= Estimate Twheng= 9.8

and l= 155. (try 2sf for this estimate)

4. Express the following numbers as products of

their prime factors: {e.g. 12 = 22 x 3}

(a) 192 (b) 264 (c) 168

Hence find their highest common factor and their

lowest common multiple infactor form.

5. Find an estimate to 1 sig. fig. for:

08.249.17

8.2698.33.189

++


8/17

8 Constructions, Loci and BearingsIn questions 1, 2 and 6 use only a pencil, a ruler and a pair of compasses on PLAIN paper. Do NOT erase

any construction lines or arcs.

1. Construct the sketched triangle accurately:

AB = 8cm

angle B = 90

angle A = 30

Measure AC to the nearest mm.

2. (a) Construct

rectangle ABCD.

(b) Construct the

locus of all points

equidistant fromAD and DC.

(c) Construct the locus of all points 6cm from A,

inside ABCD.

(d) Shade the region inside ABCD where the

points are nearer DC than AD but less than 6cm

from A.

(e) Calculate the area of this shaded region, to 3

sig. fig.

3. {not drawn to scale CALCULATE your answers!}

A ship sails

from port P on a

bearing of 062 to Q.

The distance PQ is 11km.

At Q the ship changes direction

and sails to Ron a bearing of 175.

QR= 19km.

Find the direct distance PRand the bearing ofR

from P, giving your answers to 3 sig. fig.

4. not to scale

Bs bearing from A is 076.

Bs bearing from C is 340.

AB = BC.

Find the bearing ofA fromC.

5.

The bearing ofA from B is 122.

The bearing ofC from B is 231.

BA = 8cm CA = 14cm.

Find the bearing ofC from A and the distance BC

(each to 3 sig. fig.)

6.

A B

Draw a line AB 8cm long and construct

accurately the locus of all points 4cm from any

point on AB.

Find the area within this locus to the nearest cm2.

7.

(a) Evaluate the area of the quadrilateral ABCD.

(b) Find the lengths ofAB, BC, CD and AD to 3

sig. fig.

(c) Calculate the angle A to 3 sig. fig.

2

4

6

8

2 4 6 8 10 120 x

y

A

B

C

D


9/17

9 Vectors

Exercise 9.11.

(a) Write a, b, c, p, q, r, s, and t each as column

vectors.(b) Give, in simplified surd form,

|a| |b| |c| |q| |t|

(c) As you can see, p = 2a

Write q, r, s, and t each as a multiple

ofa orb orc.

2. {Tip: When you have a vector question in an

exam or test, NEVER attempt to solve it without a

DIAGRAM even if there is no grid drawn for

you!}

O is the origin and Sis the point (0,3).

=

2

6SP . Tis the midpoint ofSP.

(a) Find, as a column vector, TO

(b) Find OT

(c)R is on the liney = 0. If OR = OT , find

the possible coordinates ofR.

3.In each set of vectors below, find the one

vector that is notparallel to all the othervectors in its row:

(a)

5

2,

10

4,

25

4,

2

12

1

,

2

15

1

(b)

3

2,

6

4,

30

20,

9

4,

12

8

4. Given that in each case p and q AREparallel, find k:

(a) p = 6a + 18b q = ka + 6b

(b) p = 12a + kb q = 9a 21b

Exercise 9.21. O is the centre of a regular hexagon,

ABCDEF.

x=ABn=BC

Find, in terms ofxor

n or both, the following vectors:

AC , AD , CD , FA , FB , AE

2. This diagram shows a rough sketch of 2

quadrilaterals,ABEFandBCDE.

(a)Ifa = 2b, what can you conclude aboutA,B

and C?

(b)Ifa = b = e = d, what type of figure is

ABCDEF?

(c)Ifg= 2c, what type of figure isEBCD?

(d) Ifd+ c = e +g, name the four points that

are vertices of a parallelogram.

3.ABCD is a parallelogram.

s3=ABt3=AD

(a) Find, in terms ofs ortor both:

(i)BD (ii) BP (iii) BQ (iv) AQ (v)QC


10/17

(b) Show clearly, giving reasons, thatAPCQ is

a parallelogram.

10 ProbabilityIf you are asked to find any probability, remember that the answer:

1Pr0..

obabilityeiNEGATIVEBECANNOT

ONEEXCEEDCANNOT

1. Three coins are tossed at the same time.

List all the possible outcomes. Find the

probability of obtaining:

(a) 3 Heads (b) 2 Heads and one Tail

(c) no Heads (d) at least one Head.

2. Cards with the numbers 2 to 101 are placed in

a hat. Find the probability of selecting:

(a) an even number

(b) a number less than 14

(c) a square number(d) a prime number less than 40

(e) a prime number greater than 90.

3. Two dice, one red and one blue, are thrown

simultaneously. Show all the possible outcomes

on a sample space (probability space). Find the

probability of obtaining:

(a) a total of 10

(b) a total of 12

(c) a total of less than 6

(d) the same number on both dice

(e) a total more than 9.(f) Which is the most likely total?

4.In each of the following, state whether the

events A and B are mutually exclusive:

(a) Two coins are tossed

(i)

TailHeadBEvent

HeadsAEvent

1,1:

2:

(ii)

HeadleastatBEvent

HeadsAEvent

1:

2:

(b) A card is drawn from a pack

aceanBEvent

spadeaAEvent

:

:

(c) A counter is drawn from a box of red, blue

and white counters

counterblueanotBEvent

counterredaAEvent

:

:

5. In a European car factory, 85% of the cars

manufactured are left-hand drive, the rest

(obviously) are right-hand drive.

The probability that a car needs its steering

adjusted before leaving the factory is 0.2. The

percentage of cars that are left-hand drive and do

NOT need adjustment to their steering is 75.

Show that steering adjustment and right / left

hand drive are NOT independent.

6. What is the probability of selecting a King

from a full pack of cards?What is the probability of selecting a red card

from a full pack?

Show that the event selecting a King and the

event selecting a red card are independent.

7. A bag contains 6 red marbles and 4 blue

marbles. A marble is drawn at random and not

replaced. Two further draws are made, again

without replacement.

Find the probability of drawing:

(a) 3 red marbles

(b) 3 blue marbles(c) no red marbles

(d) at least one red marble.

8. Sally goes to school in the mornings. The

probability that her alarm clock works is 0.9. If

her alarm clock has worked the probability that

she is on time for registration is 0.85. If her alarm

clock has failed the probability that she is late for

registration is 0.95.

Draw a tree diagram to help you work out the

probability that:

(a) Her alarm clock worked and she is on time

(b) She is on time, regardless of her clock.

9. A box containsxmilk chocolates andy plain

chocolates. Two are selected one after the other,

WITHOUT replacement.

Find, in terms ofxandy, the probability of

choosing:

(a) a milk chocolate on the first choice

(b) milk chocolates on both the first and second

choices

(c) one of each sort of chocolate(d) two plain chocolates.


11/17

11 Calculations in Right-Angled Triangles

Exercise 11.1 Find each angle x to 3 sig.fig.1. 2.

3. 4.

5. 6.

7.

Exercise 11.2 Find each sidexto 3 sig.fig.1. 2.

3. 4.

5. 6.

7.

12 Calculations Needing Sine and Cosine Rules

Remember:SinC

c

SinB

b

SinA

a== and

bc

acbCosACosAbccba

22

222222 +=+= or

In each case, find the side lettered x, or the angle markedx, to 3 sig. fig. (Triangles NOT drawn to scale)

1. 2. 3.

4. {two possible answers!} 5. 6.

7. 8. 9.


12/17


13/17


14/17

ANSWERS

1 Solving EquationsExercise 1.11 5=x 2 4=x3 12=x 4 4=x

5 2=x 6 4=x7 32=x 8 30=x

Exercise 1.2

1

==

3

4

b

a2

==

1

2

y

x

3

==

0

5

q

p4

==

9

10

y

x

5

==

2

4

b

a6

==

3

1

y

x

7=

=14

9

y

x

Exercise 1.3

1 32 orx =2 16 = orx3 83 = orx4 64 orx =5 241 orx =6 122 = orx

7 6=x (repeated)8 94 = orx

Exercise 1.4

1 46.154.7 orx =2 57.2907.0 = orx3 305.019.8 orx =4 67.107.1 = orx5 673.017.8 = orx6 89.69.10 = orx

7 566.007.7 = orxExercise 1.5

1 8.02 orx =

22

16 orx =

32

115 orx =

4 52

1ory =

5 221 = ory

Exercise 1.6

1

==

==

5

1

1

5

y

xor

y

x

2

==

==

5

1

1

5

y

xor

y

x

3=

==

=5

1

1

5

y

xor

y

x

4

==

==

5

1

1

5

y

xor

y

x

5

==

==

5

1

1

5

y

xor

y

x

6

==

==

5

1

1

5

y

xor

y

x

7=

==

=5

11

5yxor

yx

8

==

==

5

1

1

5

y

xor

y

x

9

==

==

5

1

1

5

y

xor

y

x

2 Algebra Practice

Exercise 2.11 )5( +xx 2 )32( +yy3 )35(3 xx 4 )( yxxy +5 )52(2 baba +6 )3(3 babc 7 )3)(8( + xx8 )2)(12( nn9 )6)(4( + aa10 )2)(12(2 nn11 )5)(3( ++ xx12 )5)(3(3 ++ xx13 )2)(2( + xx14 )2)(2(25 + xx15 )7)(3( + bb16 )7)(3(5 + bb

Exercise 2.2

1 ))(( cbyx ++2 ))(( bhyx

3 )2)(3( banm 4 )2)(3( yxba ++5 )4)(2( qpsr

6 )1)(32( ++ xx7 )3)(12( ++ xx8 )2)(13( + xx9 )3)(52( + xx10 )7)(43( + xx

11 )12)(56( + xx12 )1)(316( ++ xx13 )13)(32( + yy14 )13)(32(2 + yy15 )3)(23( xx16 )72)(72( + mm17 )6)(6( pp +18 97 19 112

20 3360 21 100

22 100

Exercise 2.3

1n

m

5

42

x

y

4

3

3 xyy + 4x

x

2

63+

5x

xx 2221 ++6 y+4

73

12 +y8

3

2

+

x

x

91+x

x10

)5(24+

xx

112

5

+

x

x12

2

3

++

x

x

132

5

+

x

x14

12 xx

15x

x 9

Exercise 2.4

1 2212

89

qp

pq

222

22

15

625

ba

ab +

33

2

8

5616

a

aa +

4ab

ab

4

310 22

5 )3)(2(

98

++xx

x


15/17


6)2)(1(

114

+++xx

x

7)1)(3(

173

+xx

x

8

)1)(2(

11

+

+

xx

x

912

182

+xx

x

10)3)(3(

)1(6

+xx

x

Exercise 2.5

14

313 orx =

2 4

1

=m3 13 = ora4 34 += ora

53

25 = ora

3 Changing Subjects &

InequalitiesExercise 3.11

(a)a

bcx = (b) )( qrpx +=

(c)n

myx

= (d) )( tlmx =

(e)ca

abcdx

= (f) ansx +=

(g)ca

bnx

++

= (h)a

bycx

=

Remember:

ac

cdab

ca

abcd

2

(a)ca

bdx

3

3

= (b)ab

nx

+=

2

(c) 22 abx =

(d)a

nmx

=

(e) qpx 2=

(f)a

nlx

22 +=

(g) xgT

=

2

2

(h)a

bcx

=

2

(i) yz

yzx

+=

(j)16

2ymx =

3 )2(0.9 sfh =4 )2(6.1 sfr=5 )2(10 sfa =6 )2(8.9 sfg= 7 9=x

8

(a)h

Vr

3= (b)

st

str

=

2

(c) 34

3

Vr= (d)

4

Ar =

(e)2

4 22 bar

+=

(f)N

AR

=

100

100

Exercise 3.2

1 6


16/17


5 (a) ( ) 2525)( xxf (b) max is 25 whenx = 5

(c)

6 (a) ( ) 224)( + xxf(b) max is -4 whenx = -2

(c)

7

( ) 936 22 xxx , so for

this curve to map onto2xy = ,

it needs to move

9

3or 3 to

the left and 9 up.

5 Quadratic Equations1By Pythagoras:

( ) 222 )14()2)4( +=++ xxx18164416 222 ++=+++ xxxxx

0342 =+ xx 0)3)(1( = xx

==

3

1

xor

x

2

162)2)(3()3)(14( =++ xxxx16)65()3134( 22 =+++ xxxx

1623183 2 =+ xx 0165183 2 =+ xx

0)556(3

2

=+ xx 0)5)(11(3 =+ xx511 = xx

3

90)12)(2(2

1)4)(6( += xxxx

90224 22 += xxx 9022 2 =+xx

09022 2 =+ xx0)2)(4522( =+ xx

2, =xsonegativebecannotx4 (a) D is at (0,-8), (b)

9)1( 2 = xyso M is at (1,-9)

(c)A is at(-2,0) and B is at (4,0)

(d) CB has equation

4+= xy and C is at (-3,7)5 (a)

8)3(16 22 + xxx(b) 223=x(c)

223

2

246

2

326

2

2366

=

+=

+=

+=

x

x

x

x

6 Proportionality or

VariationExercise 6.11

Missingp values: 0.1,10,16,80

Missing q vlaues: 15,20,50

2 (a) becomes 4 x as big

(b) becomes 9 x as big

(c) becomes 100 x as big

3 (a) am 8.6=

(b)m = 27.2, a = 25, m = 20.4

4 (a)P= 80, P= 2000

(b) 1,30 == hh5

Missingxvalues: 30,40

Missingy value: 2

(a)y = 250

(b)x = 15

6 time = 3.18 seconds (3sf)

Exercise 6.21

Missing vvalues: 40, 80

Missing tvalues: 20, 10, 2

2

Missingxvalues: 0.4, 0.5,

Missingy value: 200

3 (a)H= 1.5, H= 150

(b) T= 25, T= 2500

4 Resistance = ohms161 or

0.0625 ohms

5(a) pressure = 69.4 units (3sf)

(b) diameter = 22.4 mm (3sf)

7 Number: Estimating

and Factorising1 (a) 8960 (b) 2800 (c) 0.032

2 4000V 3 24T4(a) 326

(b) 11323

(c)

7323 HCF = 24323 =LCM = 117326 {=14784 is unnecessary}

5 approx. 20

8 Constructions, Loci

and Bearings1 AC = 9.2 cm

2

Area = 10.3cm2

3PR = 17.9km, bearing = 140

4 bearing = 298

5 bearing = 264, BC = 9.18cm

x

y

0

(5,25)

x

y

(0,8)

(2,4)

x

y

0

(3,9)

(6,0)


17/17


6

Area = 114 cm2

7 (a) Area = 332

1square units

(b)AB = 5.83 units, BC = 5

CD = 7.21 AD = 5.39

(c) 99.2

9 VectorsExercise 9.1

1 (a) a =

1

2b =

2

2

c =

3

2p =

2

4q =

1

1

r =

3

2s =

3

6t =

44

(b) |a| = 5 |b| = 22

|c| = 13 |q| = 2 |t| = 24

(c) q =2

1 b r = - c

s = -3a t = 2b

2 (a)

=4

3TO

(b) 5=OT(c) R is either (-5,0) or (+5,0)

3 (a)

25

4(b)

9

4

4 (a) k= 2 (b) k= -28

Exercise 9.2

1 nx+=AC n2=AD

x-n=CD nx=FA

nx= 2FB x-n2=AE

2 (a)A,B and Cwould be

collinear.

(b) It would be all one

parallelogram.

(c) It would be a trapezium.(d)Vertices would beE, C, B, F

3 (a) (i) st 33 =BD(ii) st=BP(iii) st 22 =BQ

(iv) st+= 2AQ

(v) ts += 2QC

(b) ts += 2AP

QCAP =

QCtoparallelisAPand

ramlogparalleaisAPCQ10 Probability

1 (a)8

1 (b)

8

3

(c)8

1 (d)

8

7

2 (a)2

1 (b)

25

3

(c)100

9 (d)

25

3

(e)50

1

3 (a)12

1 (b)

36

1

(c)18

5 (d)

6

1 (e)

6

1

(f) 7 is the most likely total

4 (a) (i) mutually exclusive

(ii) not mutually exclusive

(b) not mutually exclusive

(c) not mutually exclusive

5 P(LHD) x P(no adjustment)

75.0

68.08.085.0

==

6 P(King) =13

1

52

4=

P(Red) =2

1

P(Red) x P(King) =26

1

P(Red King) =261

522 =

7 (a)6

1 (b)

30

1

(c)30

1 (d)

30

29

8

(a)P(alarm ok & on time) = 0.765

(b) P(on time) = 0.77

9 (a)

P(Milk 1st) =yx

x

+(b)

P(M,M)=)1)((

)1(

++

yxyx

xx

(c)

P(one of each)= )1)((

2

++ yxyx

xy

(d) P(Plain,Plain)=

)1)((

)1(

++

yxyx

yy

11 Calculations in

Right-Angled TrianglesExercise 11.11 x = 47.7 2 x = 67.4

3 x = 55.8 4 x = 26.5

5 x = 71.6 6 x = 65.97 x = 120 {all to 3 s.f.}

Exercise 11.21 x= 5.30 cm 2 x= 7.68 cm

3 x= 6.43 cm 4 x= 10.4 cm

5 x= 34.9 cm 6 x= 30.8 cm

7 x= 16.9 cm

12 Sine and Cosine

Rules

1 x= 7.45 cm 2x= 14.2 cm3 x = 60.9

4 x = 54.4 or126

5 x= 8.13 cm 6 x= 5.06 cm

7 x = 32.5 8 x = 47.9

9 x= 6.68 cm

Aim for A* in GCSE Maths - Part 1

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