NI CHEWCH DDEFNYDDIO CYFRIFIANNELL YN Y PAPUR HWN

CALCULATORS ARE NOT TO BE USED FOR THIS PAPER

MathemategHaen GanolraddPapur 1Tachwedd 2002(2 awr)

MathematicsIntermediate TierPaper 1November 2002(2 hours)

1. (a) Ystyriwch y set ganlynol o rifau.Consider the following set of numbers.

60, 61, 62, 63, 64, 65, 66, 67, 68, Gan ddefnyddio’r rhifau yn y set yn unig, ysgrifennwchUsing only the numbers in the set, write down(i) Rhif cysefin a prime number

(ii) Rhif ciwb a cube number

(b) Darganfyddwch werth 0.3 x 0.4 Find the value of 0.3 x 0.4

= 61 neu or 67

= 64

= 0.12

2. Mae gwerslyfr yn costio £6.35. Mae ysgol yn dymuno prynu 48 o’r gwerslyfrau hyn.Cyfrifwch gyfanswm cost y 48 gwerslyfr.A text book costs £6.35. A school wishes to buy 48 of these text books.Calculate the total cost of the 48 text books.

6 . 3 5x 4 8

25400 (x 40)+ 5080 (x 8 )

30480

3. Mae côn wedi’i labelu’n A.Mae ciwboid wedi’i labelu’n B.Mae pyramid sylfaen sgwâr wedi’i labelu’n C ac mae tetrahedron wedi’i labelu’n D.A cone is labelled A.A cuboid is labelled B.A square-based pyramid is labelled C and a tetrahedron labelled D.

Cwblhewch y tabl canlynol. Mae un wedi’i wneud i chi.Complete the following table. One has been done for you.

Priodwedd y siâp

Property of the shape

Y label ar y siâp

Label on shape

Mae ganddo un wyneb crwn

It has one circular face

Mae ganddo 5 fertig yn union

It has exactly 5 vertices

Mae pob un o’i wynebau’n driongl

All its faces are triangles

Mae pob un o’i wynebau’n betryal

All its faces are rectangles

A

C

D

B

4. Mae’r diagram isod yn cynrychioli peiriant rhifau.The daigram below represents a number machine.

MEWNBWN

INPUT

Tynnu 8

Subtract 8

Rhannu â 3

Divide by 3

ALLBWN

OUTPUT

Os yw’r mewnbwn yn n, ysgrifennwch yr allbwn yn nhermau n.If the input is n, write down the output in terms of n.

= n – 8 3

5. SymleiddiwchSimplify

5x – 9 – 3x + 4

(b) Beth yw gwerth 6d – 7e pan fo d = -3 ac e = 2?What is the value of 6d – 7e when d = -3 and e = 2?

= 2x - 5

= 6 x -3 – 7 x 2

= -18 -14

= - 32

6. Mae bag du yn cynnwys pedwar disg â rhifau arnynt fel y dangosir.A black bag contains four disks numbered as shown.

1 2 5 6

81 2 4 7

Mae bag gwyrdd yn cynnwys pum disg â rhifau arnynt fel y dangosir.A green bag contains five disks numbered as shown.

Mewn gêm mae chwaraewyr yn dewis disg o’r bag du ac yna disg o’r bag gwyrdd.Mae’r rhifau ar y ddau ddisg yn cael eu lluosi â’i gilydd i gael y sgôr.In a game a player chooses a disc from the black bag and then a disc from the green bag. The numbers on the disc are multiplied together to obtain the score.

(a) Cwblhewch y tabl canlynol i ddangos pob sgôr posibl.Complete the following table to show all the possible scores.

8 8

7 7

4 4 8

3 3 6 15 18

1 1 2 5 6

1 2 5 6

Bag gwyrddGreen bag

Bag du Black bag

(b) (i) Beth yw’r tebygolrwydd y bydd chwaraewyr yn sgorio llai na 25? What is the probability that a player scores less than 25?

(ii) Beth yw’r tebygolrwydd y bydd chwaraewyr yn sgorio 25 neu fwy? What is the probability that a player scores 25 or more?

16 40 48

14 35 42

20 24

= 16 20

= 4 20

Mae chwaraewr yn ennill gwobr drwy sgorio 6 neu lai.A player wins a prize by scoring 6 or less.(c) Mae Delyth yn chwarae’r gêm unwaith. Beth yw’r tebygolrwydd y bydd hi’n ennill gwobr?Denise plays the game once. What is the probability that she wins a prize?

(ch) (i) Mae 300 o bobl yn chwarae’r gêm unwaith.Tua faint ohonynt y byddech chi’n disgwyl iddynt ennill gwobr?300 people each play the game once.Approximately how many would you expect to win a prize?

(ii) Mae’n costio £2 i chwarae’r gêm unwaith. Y wobr am ennill yw £5. Pe bai pob un o’r 300 o bobl yn chwarae’r gêm unwaith, tua faint o elw y byddech chi’n disgwyl i’r gêm ei wneud?It costs £2 to play the game once. The prize for winning is £5. If the 300 people each play the game once, approximately how much profit do you expect the game to make?

= 7 20

= 7 x 300 20

= £2 x 300 - £5 x 105

= 7 x 15 = 105

= 600 - 525

= £ 75

7. Hyd petryal yw 5 cm a’i led yw 2 cm.A rectangle has length 5 cm and width 2 cm.(a) Ysgrifennwch ddimensiynau’r petryal ar ôl i’r naill a’r llall gael eu helaethu

(enlarged) yn ôl ffactor 4. Write down the dimensions of the triangle after each has been enlarged by a factor

of 4.

(b) Sawl gwaith yn fwy nag arwynebedd y petryal gwreiddiol yw arwynebedd y petryal sydd wedi’i helaethu? How many times bigger is the area of the enlarged rectangle than the area of the original rectangle?

Hyd = 20 cm, Lled = 8cm

Length = 20cm, Width = 8cm

Arwynebedd petryal bach Area small rectangle = 5 x 2 = 10 cm²

Arwynebedd petryal mawr Area large rectangle = 20 x 8 = 160 cm²

= 160 ÷ 10

= 16 Neu = (ffactor graddfa ²) = 4² = 16Or = (scale factor ²) = 4² = 16

8. Dau borthladd yw P a Q gyda Q i’r De o P. Mae porthladd arall ar bwynt R ar gyfeiriant (bearing) o 230 ° (De 50°Gn) o P a 300° (G 60° Gn) oddi wrth Q. Trwy dynnu llinellau addas. Marciwch safle R ar eich diagram.P and Q are two ports with Q due South of P. Another port is at a point R on a bearing of 230° (S 50° W) from P and 300° (N 60° W) from Q. By drawing suitable lines, mark the position R on your diagram.

P

Q

GN

x

300°

230°

R

9. Datryswch yr hafaliadau canlynol.Solve the following equations.

(a) 6x – 8 = 10

(b) 4x – 5 = 30 – 3x

(c) 4(x + 2) = 36

6x = 10 + 86x = 18

x = 18 6

x = 3

4x + 3x = 30 + 5

7x = 35

x = 35 7

x = 5

4x+ 8 = 36

4x = 36 - 8

4x = 28

x = 28 4

x = 7

10. Dangoswch yn glir sut y byddech yn cael AMCANGYFRIF ar gyfer y cyfrifiad canlynol.Show clearly how you would obtain an ESTIMATE for the following calculation

594.3 x 7.6 38.7

= 600 x 8 40

= 600 5

= 120

11.

ChwaraewrDVD

Player £280 Sêl 35% yn llaiSale 35% Off

Pris chwaraewur DVD oedd £280. Mewn sêl cafodd ei gynnig am ostyngiad o 35%.Faint mae’n ei gostio n y sêl?A DVD player was priced at £280. In a sale it was offered at a reduction of 35%.How much does it cost in the sale?

= 35 x 280 100

= 7 x 28 2

= 7 x 14

= £98

Pris yn y sêl Sale price = 280 - 98£182

Neu or 10% = £28 5% = £14

35% = 28+ 28+ 28 + 14

= £98

Pris yn y sêl Sale price = 280 - 98

£182

12. Ddwywaith y dydd mae Carwyn yn rhoi 2/3 o fowlen o fisgedi i’w gi Carlo. Mae bag 2.5 kg o fisgedi yn cynnwys digon o fisgedi i bara 15 diwrnod i Carlo. Darganfyddwch bwysau’r bisgedi mewn powlen lawn o fisgedi, mewn gramau.Twice a day Chris gives his dog Kola 2/3 of a bowl of biscuits. A 2.5kg bag of biscuits has enough biscuits to last Kola 15 days. Find the weight, in grams, of the biscuits in a full bowl of biscuits.

2.5 kg = 2500g

1 diwrnod / 1 day = 2500 g 15

2 x powlen / bowl = 2500 g3 15

Powlen / bowl = 2500 x 315 2

= 250g

13. Mae Arwyn, Beti a Ceri yn rhannu £3600 yn ôl y gymhareb 4:5:9. Faint fydd pob un yn ei gael?Arwyn, Betty and Clive share out £3600 in the ratio of 4:5:9. How much do they each get?

Nifer o rannau Number of parts = 4+5+9 = 18

1 rhan part = £3600 ÷ 18

=£200

Arwyn yn cael Arwyn gets 4 x 200 = £800

Beti yn cael Betty gets 5 x 200 = £1000

Mae Ceri yn cael Clive gets 9 x 200 = £1800

14. Mewn pedrochr PQRS, mae’r llinell PQ yn baralel i SR, gyda PQ = 16cm ac SR yn 18cm. Y pellter perpendicwlar rhwng PQ ac SR yw 8cm.Cyfrifwch arwynebedd y pedrochr PQRS.In quadrilateral PQRS, the line PQ is parallel to SR, with PQ = 16cm and SR = 18cm. The perpendicular distance between PQ and SR is 8cm.Calculate the area of the quadrilateral PQRS.

S

P Q

R18cm

16cm

Nid yw’r diagram wedi’i luniadu wrth raddfa.Diagram not drawn to scale.

8cm

Arwynebedd trapesiwm = (a + b) x u 2

Area of a trapesium = (a + b) x h2

= (18 + 16) x 8 2

= 34 x 8 2= 34 x 4

= 136cm²

15. Cafodd masau 8 person a aeth ar ddiet eu mesur cyn ac ar ôl y diet. Roedd y canlyniadau fel y dangosir yn y tabl canlynol.The masses of 8 people who went on a diet were measured before and after the diet. The results were as shown in the following table.

Màs cyn y diet (kg)

Mass before the diet (kg)

130 50 75 93 112 68 61 83

Màs ar ôl y diet (kg)

Mass after the diet (kg)

112 51 61 83 92 60 49 68

(a) Ar papur graff gyferbyn, lluniwch ddiagram gwasgariad i ddangos y canlyniadau hyn.On the graph paper opposite, draw a scatter diagram to display these results.

(b) Pa fath o gydberthynaid y mae eich diagram gwasgariad yn ei ddangos?What type of correlation does your scatter diagram show?

(c) Màs cymedrig yr 8 person cyn y diet oedd 84kg ac ar ôl y diet eu màs cymedrig oedd 72kg. Defnyddiwch y wybodaeth hon i dynnu llinell ffit orau ar eich diagram gwasgariad.The mean mass of the 8 people before the diet was 84kg and after the diet it was 72kg. Use this information to draw a line of best fit on your scatter diagram.

Positif Positive

40

60

80

100

120

140

50 60 70 80 90 100 110 130120

Màs ar ôl y diet (kg)Mass after the diet (kg)

Màs cyn y diet (kg) Mass before the diet (kg)

MM

(ch) Defnyddiwch eich llinell ffit orau i amcangyfrif y màs ar ôl y diet ar gyfer person a oedd â’i fàs yn 95kg cyn mynd ar y diet.Use your line of best fit to estimate the mass after the diet for a person whose mass was 95kg before going on the diet.

80kg

16. Mae’r tabl yn dangos rhai o werthoedd y = 3x² - 2x – 6 ar gyfer gwerthoedd x o -3 i 3.The table shows some of the values of y = 3x² - 2x – 6 for values of x from -3 to 3.(a) Cwblhewch y tabl drwy ddarganfod gwerth y pan fo x = -2. Complete the table by finding the value of y when x = -2.

x -3 -2 -1 0 1 2 3

y = 3x2 - 2x – 6 27 -1 -6 -5 2 15

(b) Ar y papur graff gyferbyn, lluniwch graff y = 3x² - 2x – 6 ar gyfer gwerthoedd x rhwng -3 a 3.On the graph paper opposite, draw the graph of y = 3x² - 2x – 6 for values of x between -3 and 3.

(c) Tynnwch y llinell y = 5 ar eich papuer graff ac ysgrifennwch werthoedd x ar gyfer y pwyntiau lle mae eich dau graff yn croestorri.Draw the line y = 5 on your graph paper and write down the x-values of the points where your two graphs intersect.

(ch) Ysgrifennwch yr hafaliad yn x y mae’r gwerthoedd x a gawsoch yn (c) yn ddatrysiadau iddo.Write down the equations in x whose solutions are the x-values you found in (c).

10

x = -1.7 neu / 0r x = 2.2

3x² - 2x – 6 = 5 Neu / or 3x² - 2x – 11 = 0

-10

10

20

30

1 2 3 4 x-1-2-3-4

y

y = 5

17. Tywyllwch ranbarth y pwyntiau y tu mewn i’r triongl ABC sy’n bodloni y naill a’r llall o’r amodau canlynol.Shade in the region of points inside the triangle ABC which satisfy both of the following conditions.(i) Mae’r pwyntiau’n agosach at y pwynt A nag at y pwynt B. The points are nearer the point A than the point B

A (ii) mae’r pwyntiau’n bellach o B na’r pellter BC.And (ii) the points are further from B than the distance BC.

A B

C

1 2 3 4 5

-1

-2

-3

-4

-5

0

1

2

3

4

5

-1-2-3-4-5 x

A

18. (a) Lluniwch ddelwedd y siâp A ar ôl trawsfudiad (translation) o 3 uned i’r cyfeiriad x a -5 i’r cyfeiriad y. Labelwch y ddelwedd B.

Draw the image of the shape A after a translation of 3 units in the x direction and -5 in the y direction. Label the image B.

B

1 2 3 4 5

-1

-2

-3

-4

-5

0

1

2

3

4

5

-1-2-3-4-5 x

C

(b) Cylchdrowch y siâp C trwy 90° yn glocwedd o amgylch y pwynt (1, -2).Labelwch y ddelwedd D.

Rotate the shape C through 90° clockwise about the point (1, -2). Label the image D.

D

19. Ysgrifennir llythrennau’r gair MAESTEG ar saith cerdyn, un llythyren am bob cerdyn ac fe’u rhoddir mewn blwch. Yn yr un modd, ysgrifenir deg llythyren CAERNARFON ar ddeg cerdyn a’u rhoi mewn blwch arall.Mae person yn dewis un cerdyn ar hap o’r naill flwch a’r llall. Beth yw’r tebygolrwydd bod gan y person y llythyren E ar y ddau cerdyn?The letters of the word MAESTEG are written on several cards, one letter per card and placed in a box. Similarly, the ten letters of CAERNARFON are written on ten cards and placed in another box.A person selects one card at random from each of the two boxes. What is the probability that the person has the letter E on both cards?

= P (E in MAESTEG) x P (E in CAERNARFON)

= 2 x 1 7 10

= 2 70

= 1 35

= P (E yn MAESTEG) x P (E yn CAERNARFON)

20. Datryswch yr hafaliadau cydamserol canlynol drwy dull algebraidd (nid graffigol).Dangoswch eich holl waith cyfrifo.Solve the following simultaneous equations by an algebraic (not graphical) method.Show all your working.

5 x + 2 y = 102 x + 3 y = - 7

Multiply equation 1 x 2 and equation 2 x 5

10x + 4y = 20

10x + 15y = - 35

1

2

3

4

4 - 3 15y – 4y = - 35 - 20

y = - 55 11

y = - 5

Amnewid y = -5 yn hafaliad 1Substitute y = -5 in equation 1

5 x + 2 y = 10

5 x + 2 x -5 = 10

5 x - 10 = 10

5x = 10 + 10

5x = 20

1 x 2

2 x 5

11y = - 55

x = 20 5x = 4

21. (a) Ysgrifennwch bob un o’r rhifau canlynol yn y ffurf safonol.Write each of the following numbers in standard form. (a) 0.0000086

(ii) 62400 000

(b) Darganfyddwch, yn y ffurf safonol, werthFind, in standard form, the value of

(i) (5 x 10-8) x (3.2 x 10-4)(ii) (2 x 10-5) ÷ (5x107)

= 8.6 x 10-6

= 6.24 x 107

= 5 x 3.2 x 10 (-8 + -4)

= 16.0 x 10 -12

= 1.6 x 10 -11

= 2 x 10 (-5 - 7)

5

= 0.4 x 10 -12

= 4 x 10 -13

22. Safodd grŵp o 200 o ddisgyblion arholiad. Mae’r tabl yn rhoi dosraniad amlder grŵp o’u marciau yn yr arholiad.A group of 200 pupils sat an examination. The table gives a grouped frequency distribution of their marks in the examination.

Marc

Mark

0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79

Amlder

Frequency

4 14 46 52 46 30 6 2

(a) Cwblhewch y tabl amlder cronnus acnlynol. Complete the following cumulative frequency table.

Marc (llai na)

Mark (less than)

9.5 19.5 29.5 39.5 49.5 59.5 69.5 79.5

Amlder cronnus

Cumulative Frequency

(b) Ar y papur cyferbyn, lluniwch ddiagram amlder cronnus i ddangos y wybodaeth hon.On the graph paper opposite, draw a cumulative frequency diagram to show this information.

4 18 64 116 162 192 198 200

(c) Defnyddiwch eich diagram amlder cronnus i ddarganfod y canolrif.Use your cumulative frequency diagram to find the median.

(d) Y marc isaf ar gyfer y radd uchaf A oedd 58. Defnyddiwch eich diagram amlder cronnus i amcangyfrif faint o ddisgyblion a gafodd radd A.The minimum mark for the top grade A was 58. Use your cumulative frequency diagram to estimate how many pupils achieved grade A.

20 40 60 80

40

80

120

160

200

Marciau Marks

Am

lder

cro

nnus

Cum

ulat

ive

freq

uenc

y

Canolrif Median

58

= 2009 - 192 = 8 disgybl / pupils

= 36 Marc

23. Mae gan bob un o’r meintiau canlynol nifer penodol o ddimensiyna8uyghgbtu. Rhowch nifer y dimensiynau ar gyfer pob maint. Mae’r un cyntaf wedi’i wneud i chi.Each of the following quantities has a particular number of dimensions. Give the number of dimensions of each quantity. The first one has been done for you.

Maint

Quantity

Nifer y dimensiynau

Number of dimensions

Arwynebedd cylch

The area of a circle

2

Y pellter y bydd athletwr yn taflu gwaywffon

The distance a javelin is thrown by an athlete.

Cynhwysedd jwg

The capacity of a jug

Perimedr cylch

The perimeter of a circle

Arwynebedd trawstoriad prism

The cross-sectional area of a prism

1

3

1

2

24. (a) Gan roi rhesymau, dangoswch pam NAD yw’r trionglau ABC a PQR isod yn gyflun. Mae’n rhaid i chi roi eich holl resymu.Show giving reasons, why the triangles ABC and PQR below are NOT similar. You must give all your reasoning.

18cm9cm

10cm

12cm

8cm

6cm

A

B

C

P

Q

R

Nid yw’r diagram wedi’i luniadu wrth raddfa.Diagram not drawn to scale.

Os yn gyflun AB = AC = BCIf similar QR PR PQ

6 = 8 = 129 10 18

2 = 4 = 23 5 3

AC is not in the same ratio.PRTydi AC ddim yn yr un gymhareb. PR

(b) Mae pob ciwb yn gyflun â phob ciwb arall. Enwch wrthrych 3 dimensiwn arall sydd â’r briodwedd hon.Every cube is similar to every other cube. Name another 3 dimensional object that has this property.

Sffêr / Sphere

Regular Tetrahedron Rheolaidd

Regular Octahedron Rheoliadd

Regular Dodecahedron Rheolaidd

25. Datryswch yr hafaliad canlynol.Solve the following equation.

2x + 6 - 4x – 1 = 1 3 2 2

6 (2x + 6) - 6(4x – 1) = 6 x 1 3 2 2

2(2x + 6) - 3(4x – 1) = 3

4x + 12 – 12x + 3 = 3

-8x + 15 = 3

-8x = 3 - 15

-8x = -12

x = -12 -8

x = 1 ½ neu / or 1.5