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Mathemateg Mathematics Haen Ganolradd Papur 1 Intermediate Tier

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					Mathemateg               Mathematics
Haen Ganolradd           Intermediate Tier
Papur 1                  Paper 1
Tachwedd 2001            November 2001
(2 awr)                  (2 hours)

         NI CHEWCH DDEFNYDDIO
      CYFRIFIANNELL YN Y PAPUR HWN

    CALCULATORS ARE NOT TO BE USED
            FOR THIS PAPER
1. Darganfyddwch werth
Find the value of
  (a) 0.2 x 0.4

     = 0.08




(b) 8.3 – 2.47
         8.30
       - 2.47
         5.83
 2. Cynilodd Sion £600. Gwariodd ⅓ o’r arian hwn ar feic a 2/5
    o’r arian hwn ar ddillad. Pa ffracsiwn o’r arian hwn sy’n
    weddill ganddo?
 John saved £600. He spent ⅓ of this money on a bike and 2/5 of
    this money on clothes. What fraction of this money has he got
    left?
                               Neu        1 x 600 = 200
 1 + 2 + x = 1                 Or         3
 3       5
                                          2 x 600 = 240
x= 1 - 1 - 2                              5
           3     5
                                        600 – 200 -240 = 160
x = 15 - 5 - 6
     15 15 15                            160 = 16 = 4
x= 4                                     600      60    15
      15
3. (a) Ysgrifennwch ddau derm nesaf y dilyniant canlynol.
       Write down the next two terms of the following sequence.
        110, 100, 80, 50, ……, ……
                               10    -40

              (Difference increases by 10
              Gwahaniaeth yn cynyddu 10)


  3. (b) Symleiddiwch
         Simplify                 6a – 3 – 2a + 8

                                  = 4a + 5
3. (c) Darganfyddwch werth 5x + 4y pan fo x = -3 ac y = -2
       Find the value of 5x + 4y when x = -3 and y = -2

              = 5 x -3 + 4 x -2
              = -15 - 8
              = -23
(ch) Mae’r diagram isod yn cynrychioli peiriant rhifau.
     The diagram below represents a number machine.
MEWNBWN             Ychwanegu 9        Rhannu a 4         ALLBWN
   INPUT                Add 9          Divide by 4        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+9
                      4
4.   Darganfyddwch faint pob un o’r onglau a ddynodir gan x ac y.
     Find the size of each of the angles marked x and y.



                      70º


                  x               y

     2x + 70 = 180
                                      55+ y = 180
     2x = 180 - 70
     2x = 110                         y = 180 - 55

       x = 55 º                       y = 125 º
 5. Mae gan Tim giwb y mae wedi’i labelu’n P, pyramid â gwaelod
    sgwâr wedi’i labelu’n Q, prism triongl wedi’i labelu’n R a
    thetrahedron wedi’i labelu’n S. Cwblhewch y tabl canlynol. Mae un
    wedi’i wneud ar eich cyfer.
 Tim has a cube, which he has labelled P, a square-based pyramid
    labelled Q, a triangular prism labelled R and a tetrahedron labelled
    S. Complete the following table. One has been done for you.
Prodweddau’r siâp                                     Y Label ar y Siâp
Properties of the shape                               Label on shape
Mae pob un o’i wynebau’n sgwâr                                  P
All its faces are square
Mae ganddo 2 wyneb trionglog a 3 wyneb petryal
It has two triangular faces and 3 rectangular faces             R

Mae pob un o’i wynebau’n drionglog
                                                                S
All its faces are triangles
Mae ganddo 5 fetrig yn union
It has exactly 5 vertices                                      Q
6.     Mae bag coch yn cynnwys pum pêl a’r rhifau arnynt yw 1,3,4,5 ac 9. Mae bag du
       yn cynnwys pedair pêl ar rhifau arnynt yw 2,3,6 ac 8
      Mewn gêm, mae chwaraewr yn cymryd un bel ar hap or bag coch ac un bêl ar hap
       or bag du. Y sgôr ar gyfer y gêm yw cyfanswm y rhifau ar y ddwy bêl.
       A red bag contains five red balls numbered 1,3,4,5 and 9 respectively. A black
       bag contains four balls numbered 2,3,6 and 8 respectively.
      In a game, a player takes one ball at random from each of the two bags. The score
       for the game is the sum of the numbers on the two balls.

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


      Bag            8         9         11       12        13         17
      du             6         7         9        10        11         15
      Black          3         4         6         7         8         12
      bag            2         3         5         6         7         11
                               1         3         4         5         9

                             Bag coch        Red bag
6. (b) (i) Beth yw’r tebygolrwydd y                8    9   11     12   13   17
        bydd chwaraewr yn sgorio         Bag
        7?                                         6    7   9      10   11   15
                                         du
What is the probability that a = 3
        player scores 7            20    Black     3    4   6       7    8   12
                                         bag       2    3   5       6    7   11
                                                        1   3       4    5    9
(ii) Beth yw’r tebygolrwydd na fydd                              Bag coch Red bag
     chwaraewr yn sgorio 7?                = 17
     What is the probability that a player
     does not score 7                        20

    Mae chwaraewr yn ennill gwobr drwy gael sgor o 6 neu lai.
    A player wins a prize by getting a score of 6 or less.
(c) Mae Bryn yn chwarae’r gêm unwaith. Beth yw’r tebygolrwydd y bydd ef yn ennill
    gwobr?
    Brian plays the game once. What is the probability that he wins a prize?

                              = 5                 = 1
                               20                   4
  6. (ch) (i) Mae 600 o bobl yn chwarae’r gêm unwaith.
    Tua faint ohonynt y byddech chi’n disgwyl iddynt ennill gwobr?
     600 people each play the game once.
     Approximately how many would you expect to win a prize?
                = 1 x 600           = 150
                  4
(ii) Mae’n costio 30c i chwarae’r gêm unwaith. Y wobr am gael sgôr o 6
neu leiaf yw £1. Pe bai pob un o’r 600 o bobl yn chwarae’r gêm unwaith,
tua faint o elw byddech chi’n disgwyl i’r gêm ei wneud?
  It costs 30p to play the game once. The prize for getting a score of 6 or less
is £1. If the 600 people each play the game once, approximately how much
profit do you expect the game to make?
         Cost chwarae / cost of playing = 150 x £1 = £150

         Enillion / winnings = 600 x 30p = 18000p = £180

          Elw / profit = 180 - 150       = £30
7.   Mae gan Tomos flociau coch a blociau glas. Mae pob bloc glas yn pwyso x
     gram. Mae pob bloc coch yn pwyso 60 gram yn fwy na bloc glas.
Tony has some red blocks and some blue blocks. Every blue block weighs x
     grams. Every red block weighs 60 grams more than a blue block.
(a) Ysgrifennwch, yn nhermau x, bwysau un bloc coch.
     Write down, in terms of x, the weight of one red block.
                  = x + 60
(b) Mae Tomos yn darganfodbod gan 5 bloc glas yr un pwysau â 2 floc coch.
Ysgrifennwch hafaliad y mae x yn ei fodloni. Datryswch yr hafaliad.
Ysgrifennwch bwysau bloc glas a phwysau coch.
Tony finds that 5 blue blocks weigh the same as 2 red blocks.
Write down an equation that x satisfies. Solve the equation.
Write down the weight of a blue block and the weight of a red block.

                  5x = 2(x + 60)
                   5x = 2x + 120
                   5x – 2x = 120
                        3x = 120              x = 40
8.   Ar y grid isod lluniwch helaethiad y siâp a roddir, gan ddefnyddio
     ffactor graddfa 3 a chanol A.
     Draw on the grid below, the enlargement of the given shape, using
     a scale factor of 3 and centre A




                                                                A
9.    Pan fo’n llawn, mae jwg yn dal 1 ⅓ litr. Sawl gwaith y gall y jwg
      gael ei llenwi’n llwyr o gynhwysydd 15 litr?
     When full, a jug holds 1 ⅓ litres. How many times can the jug be
      completely filled from a 15 litre container?
                = 15 ÷ 1 ⅓

               = 15 ÷ 4
                  1   3

               = 15 x 3
                  1   4

               = 45
                  4

               = 11 ¼              = 11 times
10.    Cyfesurynnau pwynt A yw (-6,7) a chyfesurynnau pwynt B yw (4,1). N
       yw gwaelod y perpendicwlar o A i’r echelin –x.
      The points A and B have coordiates (-6,7) and (4,1) respectively and N is
       the foot of the perpendicular from A onto the –x axis.

                                 Y
                                          Nid yw’r diagram wedi’i luniadu wrth raddfa
        A (-6,7)                                        Diagram not drawn to scale.

                                              B (4,1)
                                                                X
                    N                 O

Ysgrifennwch gyfesurynnau       Write down the coordinates of
(a) canolbwynt y linell AB,     the mid-point of the line AB,
                            (-1, 4)

(b) Y pwynt N The point N         ( -6, 0 )
   11. Mae rhai o’r cynhwysion agenrheidiol ar gyfer gwnwud digon o
       bastai Banoffi ar gyfer 6 phlatiad (servings) wedi cael eu rhestru
       isod:
   Some of the ingredients needed to make enough Banoffi pie for 6
      servings are listed below:
   175g o fenyn/butter         30g o sicled plaen/plain chococlate
   2 fanana /bananas          300ml o hufen dwbl/double cream
   (a) Faint o fananas y byddai eu hangen ar gyfer 18 platiad?
        How many bananas would be needed foe 18 servings?

   6 servings need 2 bananas, 18 servings need 2x3 = 6 bananas
 (b) Faint o siocled plaen y byddai ei angen i wneud digon o bastai ar gyfer 21 platiad?
 How much plain chocolate would be needed to make enough pie for 21 servings?

6 platiad angen / servings needs 30g                 Neu/or 3 servings = 15g
                                                     21 servings = 7 x 15 = 105g
1 platiad angen / serving needs 30 ÷ 6 = 5g
21 platiad angen / servings needs 21 x 5 = 105g
12.Datryswch yr hafaliad       Solve the equation.

           7x + 15 =3(x+8).
           7x + 15 = 3x + 24

           7x – 3x = 24 - 15
                4x = 9

                 x= 9
                    4
                 x=2¼
   13. Yn y tabl rhoddir cynhwysedd y peiriant (engine capacity), wedi’i fesur mewn
   centimetrau ciwbig (c.c) a’r amser, mewn eiliadau, a gymerir i bob un o 8 car
   gyflymu i fuanedd penodol.
   The engine capacity, measured in cubic centimetres (c.c) and the time, in seconds,
   taken to accelerate to a certain speed, for each of 8 cars, are given in the table.
   Cynhwysedd y peiriant (c.c)
   Engine capacity (c.c.)           1000 1100 1200 1300 1400 1600 1800                   2000
   Amser y cyflymiad (eiliadau)
   Acceleration time (s)            15.4   14.0   13.4   11.4    11.8    9.1    6.9      6.0

    (a)Ar y papur graff, lluniwch ddiagram gwasgariad yn dangos y canlyniadau hyn.
       On the graph paper, draw a scatter diagram to display these results.
      (b) Pa fath o gydberthyniad y mae eich diagram gwasgariad yn ei ddangos?           Negatif
         What type of correlation does your scatter diagram show?                        Negative
(c) Cynhwysedd cymedrig y peiriant yw 1425c.c. ac amser cymedrig y cyflymiad yw 11 eiliad.
Tynnwch linell ffit orau ar eich diagram gwasgariad.
The mean engine capacity is 1425c.c. and the mean acceleration time is 11 seconds.
Draw a line of best fit on your scatter diagram.
(d) Defnyddiwch eich llinell fit orau i amcangyfriff amser y cyflymiad ar gyfer car a
chynhwysedd peiriantt o 1750c.c.c.
Use your line of best fit to estimate the acceleration time for a car with an engine capacity of
1750c.c.                               = 7.4 eiliad / seconds
                   16

                   14
Time (seconds)




                   12

                   10
Amser (eiliadau)




                    8


                    6

                    4


                    2

                    0
                        1000 1100 1200     1300     1400 1500    1600 1700 1800    1900   2000
                           Cynhwysedd y peiriant (c.c.)   Engine capacity (c.c.)
     14. (a) Cwblhewch y tabl sy’n rhoi gwerthoedd y = 2x² + 4x – 5 ar gyfer gwerthoedd x
     o – 4 i 3.
     Complete the table which gives the values of y = 2x² + 4x – 5 for values of x ranging
     from – 4 to 3.

                 x              -4       -3      -2      -1      0       1       2      3

        Y = 2x² + 4x – 5        11       1      -5       -7      -5      1      11     25

  (b) Ar y papur graff lluniwch graff y = 2x² + 4x – 5 ar gyfer gwerthoedd x o -4 i 3.
  On the graph paper draw the graph of y – 2x² + 4x – 5 for values of x ranging from -4 to 3.

 (c) Tynnwch y llinell y = 8 ar yr un papur graff ac ysgrifennwch werthoedd x ar gyfer y
 pwyntisu lle mae eich dau graff yn croestorri.
 Draw the line y = 8 on the same graph paper and write down the x-values of the points where
 the two graphs intersect.
                                                                 -3.7 ac / and 1.7
(d) Ysgrifennwch yr hafaliad yn x y mae’r gwerthoedd x a gawsoch yn (c) yn ddatrysiadau iddo.
Write down the equation in x whose solutions are the x-values you found in (c).
          2x² + 4 x – 5 = 8
         2x² + 4 x – 13 = 0
                       y
                      30

                      25

                      20


                      15

                      10
          y=8

                       5

                                        x
-4   -3     -2   -1    0    1   2   3

                       -5


                      -10
  15. Mae Elin a Huw yn cuddio blwch yn eu gardd. Maent yn gwneud map o’r ardd, gan ddefnyddio’r
  raddfa 1cm yn cynrychioli 1m. Maent yn rhoi’r map i rai o’u ffrindiau ynghyd â’r Cliwiau canlynol.
              Mae’r blwch yn agosach at ben A i’r berth nag yw at y pen C.
              Mae’r blwch yn llai na 6m i ffwrdd o’r goeden a ddynodir gan T.
              Mae’r blwch yn agosach at wal yr ardd AB nag yw at y berth AC.
  Ar y map a ddangosir isod, dangoswch ym mha ardal o’r ardd y mae’r blwch wedi’i guddio drwy
  liwio’r ardal hon yn dywyll.
  Enid and George hide a box in their garden. They make a map of the garden, using a scale of 1cm to
  represent 1m. They give the map to some friends together with the following clues.
  The box is nearer the end A of the hedge than the end C.
  The box is less than 6m away from the tree marked T.
  The box is nearer the garden wall AB than the hedge AC.
  On the map shown below, shade the region of the garden in which the box is hidden.
                          Perth      Hedge
     A                                                                              C
                                                                                        Wal y tŷ
                                                               T
 Wal
                                                                                        House wall
  yr
 ardd
                                                                                   Graddfa: 1cm = 1m
Garden                                                                             Scale: 1cm = 1m
 wall
        B
16. Mewn pac bach o naw cerdyn, y rhifau ar y cardiau yw 1,2,3,4,5,6,7,8 a 9.
Ar wynebau dis ciwbigol teg mae’r rhifau 1,2,3,4,5 a 6.
Mae Arwel yn tynnu cerdyn ar hap o’r pac ac yn rholio’r dis.
Cyfrifwch y tebygolrwydd y bydd y rhif ar y cerdyn yn eilrif ac y bydd y dis yn dangos 5.

In a small pack of nine cards, the cards are numbered 1,2,3,4,5,6,7,8 and 9 respectively.
A fair cubical dice has faces numbered 1,2,3,4,5 and 6 respectively.
Terry draws a card at random from the pack and rolls the dice.
Calculate the probability that the number on the card is even and that the dice shows 5.


                 = P(eilrif) a P(5)
                   P (even) and P(5)
                 = 4x1
                   9 6

                = 4
                 54
                = 2
                  27
17. Lluniwch ddelwedd y siap A ar ol 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.
                                            y

                                            5


                                            3
                            B
                                            1
                                                                        x
                       -5       -3     -1        1     3       5
                                            -1

                                      A



                                            -5
18. Mae sachau’n cael eu llenwi â 50kg o dywod wedi’u mesur yn gywir i’r kg agosaf.
Sacks are filled with 50kg of sand measured correct to the nearest kg.
(a) Ysgrifennwch y maint lleiaf a’r maint mwyaf o dywod a allai fod yn y sach.
Write down the least and greatest amounts of sand there could be in the sack.
                49.5 kg                                        50.5 kg
Lleiaf Least ………………….                      Mwyaf Greatest ………………….




(b) Mae person yn prynu 20 sach o dywod.
Ysgrifennwch y maint lleiaf a’r maint mwyaf o dywod y gallai eu derbyn.
A person buys 20 sacks of sand.
Write down the last and greatest amounts of sand he could receive.


               = 20 x 49.5                                    = 20 x 50.5
Lleiaf Least = 990               kg           Mwyaf Greatest = 1010         kg
19. Datryswch yr hafaliadau cydamserol drwy ddull algebraidd (nid graffigol).
Dangoswch eich holl waith cyfrifo.

Solve the simultaneous equations by an algebraic (not graphical) method.
Show all your working.
             4x – 3y = 20   1
             6x – 5y = 22
                            2                Amnewid y = 16 yn hafaliad 1
Lluosi 1 x 3 a 2 x 2
Multiply eqn 1 x 3 and eqn 2 x 2             Substitute y = 16 in equation 1
                                             4x – 3y = 20
1x3         12x – 9y = 60              3       4x – 3 x 16 = 20
2x2       12x – 10y = 44               4        4x – 48 = 20
           -9y - -10y = 60 - 44
3 - 4                                           4x = 20 + 48
                  y = 16
                                                4x = 68
                                                 x = 68              x = 17
                                                      4
20. Mae gan bob un o’r meintiau canlynol nifer penodol o ddimensiynau. Rhowch nifer y
dimensiynau ar gyfer pob maint. Mae’r un cyntaf wedi’i wneud ar eich cyfer.
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
    Cyfaint côn
    The volume of a cone
                                                                           3
    Perimedr polygon
    The perimeter of a polygon
                                                                           1
    Cynhwysedd bwced
                                                                           3
    The capacity of a bucket
    Y pellter y bydd lloeren yn ei deithio mewn un
    orbit o’r Ddaear                                                       1
    How far a satellite travels in one orbit of the Earth
    Arwynebedd trawsdoriad prism
                                                                           2
    The area of the cross-section of a prism
21. (a) Gan roi rhesymau, dangoswch NAD yw’r trionglau ABC ac XYZ isod yn gyflun.
Mae’n rhaid i chi ddangos eich holl resymau.
Show, giving reasons, that the triangles ABC and XYZ below are not similar.
You must show all your reasoning.                                           Y
        B
                     16cm                               12cm
 8cm
                                                                        6cm
                               C
  A
               12cm                         X
                                                                          Z
      Nid yw’r diagramaaau wedi’i lluniadu wrth raddfa.   8cm
      Diagrams not drawn to scale.
Os yn gyflun BA = AC = CB
If similar then YZ ZX XY
    BA = 8 = 4
    YZ 6        3
    AC = 12 = 3
    ZX       8   2
    CB = 16 = 4           2 ≠ 3 felly ddim yn gyflun
    XY       12           3 4 therefore shapes not similar
                   3
(b) Mae pob sgwâr yn gyflun â phob sgwâr arall. Enwch ffigur geometregol
arall sydd â’r briodwedd hon.
Every square is similar to every other square. Name another geometrical figure
that has this property.

   Cylch                           Circle
   Triongl hafalochrog            Equilateral triangle

   Pentagon rheolaidd             Regular pentagon

  Hecsagon rheolaidd              Regular hexagon

  Polygon rheolaidd               Regular polygon
22. (a) Symleiddiwch     Simplify         (2a4c) x (5a³c²).
                                          =10a7c3

(b) Ehangwch y mynegiad canlynol, gan symleiddio eich ateb gymaint ag y sy’n bosibl.
Expand the following expression, simplifying your answer as far as possible.
                            (x – 2 ) ( x – 6 )

                          = x² -2x -6x +12                    CAMO      FOIL
                          = x² -8x +12
(c) Gwnewch r yn destyn y fformiwla
Make r the subject of the formula

                            3t+7=5(t–2r)
                            3t + 7 = 5t – 10r
                            10r = 5t -3t -7
                             10r = 2t - 7
                             r = 2t – 7
                                   10
  23. Dau fath o fylbiau golau trydan yw Gleini a Serolau. Mesurwyd a chofnodwyd hyd oes
  (lifetimes), mewn wythnosau cyflawn, 80 bwlb o’r naill fath a’r llall. Mae’r canlyniadau ar
  gyfer bylbiau Gleini wedi’u crynhoi yn y tabl canlynol.
  Glomo and Staybrite are two types of electric light bulbs. The lifetimes, in complete weeks,
  of eighty bulbs of each type were measured and recorded. The results for the Glomo bulbs
  are summarised in the following table.

 Hyd oes mewn           0-9    10-19 20-29 30-39 40-49 50-59           60-69    70-79 80-89
 wythnosau cyflawn
 Lifetime in
 complete weeks
 Amlder                 2      3       4       11       31       15    8        5       1
 Frequency
(a) Cwblhewch y tabl amlder cronus canlynol ar gyfer y bylbiau Gleini
    Complete the following cumulative frequency table for the Glomo bulbs.
  Hyd oes mewn wythnosau cyflawn(llai         10 20 30 40 50 60                 70 80 90
  na)
  Lifetime in complete weeks (less than)
  Amlder cronus                               2     5        9   20 51 66 74 79 80
  Cumulative frequency
     (b) Mae’r gaff isod yn dangos y diagram amlder cronus ar gyfer yr 80 bwlb Serolau.
     Gan ddefnyddio’r un papur graff, lluniwch ddiagram amlder cronus ar gyfer y bylbiau Gleini.
     The graph below shows the cumulative frequency diagram for the 80 Staybrite bulbs.
     Using the same graph paper, draw a cumulative frequency diagram for the Glomo bulbs.
                                                            (c) Defnyddiwch eich diaqgram amlder
Amlder cronus Cumulative frequency




                                                            cronus i ddarganfod y canolrif a’r
         80                Gleini / Glomo                   amrediad rhyngchwartel ar gyfer y
                                                            bylbiau Gleini.
             Chwartel Uchaf
         60 Upper Quartile                                  Use your cumulative frequency diagram
                                        Serolau / Staybrite to find the median and interquartile
              Canolrif                                      range for the Glomo bulbs.
         40 Median
                                                                           Canolrif / Median = 48
                                             Chwartel Isaf
                                     20      Lower Quartile
                                                                          Amrediad rhyngchwartel
                                     0                                    Interquartile range = 56 – 40
                20       40      60      80
                                         0      100                                              16
             Hyd oes mewn wythnosau cyflawn
             Lifetime in complete weeks
(d) Mae Dafydd yn dymuno cael bwlb a fydd yn para am 75 wythnos o leiaf. Os nad yw
cost yn ffactor, pa fath y dylai ei brynu? Rhowch rheswm dros eich dewis.
David wants a bulb that will last at least 75 weeks. If cost is not a factor, which type should
he buy? Give a reason for your choice.        Staybrite – only 54 bulbs have blown whereas
                                                              77 Glomo bulbs have blown in 75 weeks

				
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