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SQA National 5 Mathematics

2017 Paper

Worked Solutions

Papers One & Two

2017 SQA N5 Past Paper Worked Solutions

SQA Past Papers & Specimen Papers

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The contents of these worked solutions have not been checked or

approved by the Scottish Qualifications Authority. They reflect the authors’

opinions of good answers to exam questions and where possible have

been checked against publicly available marking instructions.

FOR OFFICIAL USE

l\ati*naI*r"xal{ftcati*ns?*t 7

x747 t75t01

FRIDAY, 5 MAY

1:00 PM - 2:00 PM

MathematicsPaper 1

(Non-Calculator)

I il]t il]t ffit ilil tilr lll] tilil ilil il] fftr.X74775O1?k

Fitt in these boxes and read what is printed betow.

Futt name of centre

Surname

Date of birthDay Month Year Scott"ish candidate number

Total marks - 40

Atfempt ALL questions.

You may NOT use a calculator.

Fut[ credit wit[ be given onty to solutions which contain appropriate working.

State the units for your answer where appropriate. .

Write your answers ctearty in the spaces provided in this booktet. Additionat space for answers isprovided at the end of this booktet. lf you use this space you must ctearty identify the questionnumber you are attempting.

Use blue or btack ink.

Before [eaving the examination room you must give this book to the lnvigitator; if you do not,you may lose atlthe marks for this paper.

I ilil il]t tffi lilililil ff] tflil ilIililt l]] il] ll]

mrum

I

Town

Forename(s) Number of seat

tX74775O1O1,k

Xtot

FORMULAE LIST

The roots of

Sine rute:

Votume of a pyramid:

Standard deviation:

ax'+br*c=0d[er= -bt[g -+*1

a _b c

sinA sinB sinC

tZ ) 2D +C -ACosine rute:

Area of a triangle: A= )absinC

Volume of a sphere: 1,, =tnr'

Votume of a cone: v =trcr,h

2 t) ) ;ta' = b'+ c- -'Lbc cos I or cos ,-l =

r = t,ttt

5'--2 ( Ex\'

nn-1

, where zz is the sampte size.

I ll|il il]t lllil ]]t flil ffit Ililt illll il]t ilfl illl ilr<X74775O1O2't

Page 02

2(.r-r)',1

i::::111

6:)+(-l) = Fsl'+ 3i ?3 tS7 tO

Total marks - 40

Attempt ALL questions

1. Given that /(x) = x' +3x, evatuat" f (-S).

2. The number of calts received by the potice was recorded over 10 days.

The results are shown betow.

1g8 216 z1B z3o ,rr. l. ,0, 248tl

Find the semi-interquartite runj" of thjs data.I

&t

2v'

^f.t\XL -- cpj-Ql

250 265 267I

I

I

(,3tr5c>

2tu *Li?')*

=lu^

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Poge 03

No16O-.gs;Tt(utE

s* tuL-{@,*,f ,-, =,f)

e5J'= -S r-f=2<

[Turn over

=r!..tj.x.:'i:lr.l.:E:: ::::,rr

:::::

ii;.i

,o*orJF

L

r3. Evaruate ,Z-1.

Give your answer in its simplest form'

a)ft\\j-,,r

G+. \-a

Jl, >< sXqe 3

r-bm-5-

O iYfrLe16€ lffrn\q

cLtY#f, + 4 ><

3 tnr Q-,()

n1/-L

=-(AI

.+A1

4. Expand and simptifv (2x + s)("t' - +x + t)'

)xW - LpN *' )'r3VL * r;ttt,)

)f -u*&ur 3t'ruLrs

2t - 5x" - /ca -t-3

I ll]l lilll llll lill lil lllll lllil llill lill llill il lllr<X74775O1O4't

Page 04

L J

r5. The diagram shows a square-based pyramid ptaced on top of a cube, retative

to the coordinate axes.

The height of the pyramid is lfqg,_ol-99_[e1g!it,of the cube.

A is the point (6,0,0).

The point C js djrectty above the centre of the base.

Write down the coordinates of B and C.

IT

I

L cx,\)z) -a CC3,3,9

i llli ilitil lil11 iltit Iilil lllil tililflllil1il ilil]ilffi*X71775O1O5,k

Page 05

1$

G A (6,0,0)

[Turn over

J

L

TtT yA (-1,6)

1L7L xB (3,-2)

6. The diagram betow shows the straight tine joining points A and B.

Find the equation of the line AB.

Give the equation in its simptest form.

*g -Lfi1NrrE

?ttu ,-#f q ?,

CyZftg,1o{ n

tS ni4qen-tut:(slrPr,\t; /oui

3: b="fiA -?)u5a (-i,C: toN': ffi=-L

b*6= *ZlC+'D

)F *1:c-LrGLA=c)

nn : 1)-? I

rZ^xl-rZIJ

l-LL

\

-1z' ,<

-

r l|]t llil] lllil lllil llllllllll ]illlllllllll llill llilllllxx747750106r.

Page 06

= -s.3-rt

L

7. ln triangte DEF:

DE = 8 centimetres

EF = 12 centimetres

2slnt=-3

Catcutate the area of triangte DEF.

a

a

Arl4= )-Z

I€J

ebsrr.:C-

rg x l'LY Srn-,

= -L * gntL

g,: l+8xL

-37 3LcmL

NcriSuQsTt)i0-t L- -/-

L/

-/,/\

Y,J?-J

i lllil ltilr lllll illt ffit fflt ilil ilil lilt ilil tilililt*x747750107t

Page 07

[Turn over

i

r

L

r MARKS

8. Sotve, algebraicatty, the inequality

19+x>15+3(x-Z).

lln {LsftTf S lKNu; fn{ft> - €rB

rS -G*rqIC

i()5

I il]t il]l ilt ]]t lllil til] lilll llll lill lllll llll lilltcx747750108*

Page 08

(x*{J,1,

NTTEc$ttY(ftorcd)o) oF ls6qu'or-:ri{ srq?f,

\ /,*, (>"tu \l,^"J l.ff)--i $4lJLTlW'l

fuIt 5lr-6 g? -l

L

^x

#n+JTHIS

MARGIN

J

3

9:

:

:,':E ,',,E,'E

E

:

:.t'

ln the diagram shown betow:

. ABE is a tangent to the circte centre O

. Angte DBE is 58"

<ru2 1

abD= 10 -56nn€. b1

sw3%

OD6=3Zo3lrxr-ua

I-swO{sStfli rsFJ(l.t:

Qot32: IZZ"

Lc*E = t6o *(tLL8z) =JC" rrurnover

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Poge 09

r

L

XL

rftw* Lws

r+

11.

fha'Ut nSrnrc-*

frlvO

VISSMfl-YW

ffi<r?rn& tvtflft)3

10. change the subject of the formuta F =" *4b to b.

C

sr,u A? sr ofS tL++b = i-<*

LL+Eb=Ub'=

b.3

6:f^ ,2-

q-?rb

n ,7:C-T _L

-32txpress ? --aaiI

, a*0, as a singte fraction in its simptest form.

-)a1\ _/e\->;,4-\)\-/\.

'-3U

- Z*Jqn2-L4

f cft-74G3,

_ 2r\-.LC-1

=3.az3_AZ

-zr{qd- \cl /

3-Zfun2-t\Jl

lilffitlllffil[tll] lill lllll lllll lllll llll illlt<X74775O11O"k

Page 10

MARKS

3

#R+"JTHIS

MARGIN

#R+'JTHIS

MARGIN12. Gym members are asked to fitt out a questionnaire to rate the quatity ofservice provided.

They are asked to give a rating on a scale of 1 to 6.

The ratings given by five members were as fottows:

14636

ln its simptest form, the standard devjation of these ratings can be written;

as c/th.2

Find the vatues of a and b.

5D = I 76*-712J n- (

--*? nnerrr.-J -- t -t q--tG -3.t t - +2=

I

+b5

b

-\cx-x-)l-+Lr -(+,(, -(t3"- ,-<

t =q

-, -3nz L1

t *{

-i" 2-

5,_7

U_rYqO

+I

+

f ,€ - lrt-f5-t JaSD=

?;WF.t 1r

Irf{-.n' = 3tZd.;

Q=3 b= L

I ll]t il]t iltil ]Iil ffit lllt illt lill ililt lilt lil ffir<x747750111r,

Page 11

--frrrnAiGi

* -/,>e+)-_n^l

13. The graph below shows two straight lines with the equations:

. 3x-!=2

. x+3y=19

The tines intersect at the point P.

Find, atgebraicalty, the coordinates of P.

x3qX-33=U#brO lcl- -= 2b

i1 (= J'.)

3z*3 = L)( * 39= ici

a(D

6

)L

<t 65fiTa1tr a?z'S ,/-jTL Cf] {J3' u- a

Jd\-3 =/-b?" Z*a S:t^ = -5'S

<)

b'5 5-

+,5

I ilt ilil fiil ]til ililil] tlil ilil ililililt il1ilil|r<X74775O112r.

Page 12

? (2's,5'5)

x+3v=19

14. The graph betow shows a parabota with equation of the form y =(x+ a)' +b.

The equation of the axis of

(a) State the vatue of a.

symmetry of the parabota is x=-5.

Prttn (c ut rviD\rd) -rb

L\-,J-

L6{ 6\ 3(L+;f' rb

The point (-3,8) lies on the parabola.

(b) Catculate the vatue of b.

Sul6STl f\)f,: 1=-3 6,'"rD Y 1t /rJi-D

L1 = (;t {-Sf t b<)

5 -> (_3 f\)L6 = Q)'tb6':' +-r bb=*

k=b

tL)

[Turn over for next question

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Poge 13

DONOTJWRITE IN

THISMARGIN

15, ln the d'iagram below:

. TS is para[tel to QR

. TS=5 centimetres

. QR=7 centimetres

. SR=2.6 centimetres

)r- s

fsrG st+n(E

,na-+ ) b:

The tength of PS is -rr centimetres.

Catcutate the vatue of x.

s 2.6.*

q

P

t5

G/C,;-C{* ?l har-: Sruq

SI,N(

(xq 5x+ t< = 7,-{ 7751,'?x a -13 /

-2x ---13 /?tq

2(

_-_ t3

= 6'3

IEND OF QUESTTON PAPERI

I ll]t illlil ilfi lllil ll|il IIt] ]Iil il] tilt ilil til tilt,x74775O1141t

Page 14

w26)

IJ L l-lrvrEjs

b.)c5

FOR OFFICIAL USE

Hati*r"rai

**a]{ftr*t{*ms3#17

x747 175t02

FRIDAY, 5 MAY

2:20 PM - 3:50 PM

MathematicsPaper 2

I lllil il]t lllil ilil lllil ffi] ]til ilil il] ffi,uX74775O2r<

Fitt in these boxes and read what is printed below.

Fu[[ name of centre

Date of birthDay Month Scottish candidate number

Tota[ marks - 50

Attempt ALL questions.

You may use a calcutator.

Fut[ credit wit[ be given onty to solutions which contain appropriate working.

State the units for your answer where appropriate.

Write your answers ctearty in the spaces provided in this bookiet. Additionat space for answers isprovided at the end of this booktet. lf you use this space you must ctearly identify the questionnumber you are attempting

Use blue or black ink.

Before leaving the exam'ination room you must give this book to the Invigitator; if you do not,you may [ose at[ the marks for this paper.

rum[trYear

r ffiil t]til |ilil ffiil tilil ffit 1ilil ilfi il]t l]ll il] til

{so4J

Town

Forename(s) Surname Number of seat

r<X74775O2O1:k

FORMULAE LIST

The roots of

Sine rute:

Votume of a pyramid:

Standard deviation:

ar'+bxic=0are.r= -b+

2 t2 2 ;tct' = b'+ c- -')-bccos r1 or cos,J =

a _b _csinl sinB sinC

t) ) 2D +C -OCosine rute:

Area of a triangte: A= tabsrnC

Volume of a sphere: v -- tnr'

Volume of a cone: V =lnrlh

1' = !,ltt

/'"' (h')'zlt^ - -- '-

or , = !--tl-, where r is the sample slze.

I ilt il]t fiil ]til ll]t tilt ]lil ililt il]t lilt ilt il]t<x747754202)<

Poge 02

r(x-r)'7n*1

2a

in+.JTHIS

MARGIN

Total marks - 50

Attempt ALL questions

1. Find r', the magnitude of vector , = [-]:ll.l

lvl = /(ra-)' r(-rq)L.-*(s)'

aorf t+51 '

II/oo )

=23

2. A necklace is vatued at f,1200.

Its value is expected to increase by 4.5% per year over the next 3 years.

Catcutate the expected value of the necktace after this time.

Give your answer to the nearest pound.

.,.- o.')nFSrrL- -. I'l ,T, r'!l '9: ,"

1

fiflrorrFii- r

iahlnurli-'i- L- I OD J

= lZ<>a r l" oQ53t,

= lt f Lat,3q1S s

-= L l36q (se".t3\[orto)

I llll |ilil ll]r rffi il]t [] ]il lill tilt tilt ffi lll,.X74775O2O3?k

Page 03

[Turn over

1 / 1t'l

DO NOTWRITE IN

THIS

MARGIN

3. A piece of [and is in the shape of a triangle as shown.

P 180m

. PQ= 250 metres

. PR= 180 metres

. angte QPR= 147"

The owner wishes to bu'itd a fence along the side QR.

Catculate the tength of the fence.

CrL --

*"f r*-MACl- = Z5,S t K.."- Q"'ZSc>xt{c:xGsr

t+o3p'3stlJ L"-zEo '3r(( = lftz*,.^

t-UffiV c( lS +tZ??""'' (gO

= t+ t5 o^ (senx'6:r n"

I flil tflil tilt ]il ffil tilt ]til ililt il]ilil| lil llllt<x747750204*

Poge 04

P)

180 m

4. Sotve the equation Lxz +5x-4:0.Give your answers correct to one decimal ptace.

-- Lr: c>

<-L = c)C)

,-- *6-r-7-,L +bx

2

J-=l-

-bf Jb'-QGt- =

)-c;

: s! {rT

= -S+t?t 6Q* -S-v*-+

t

, 14 .__-.-_

*t;t J5'a +0)(+

]'tflv\*(tDf)

5.

Lt

: b"b o( '' 3'IA theatre group sotd 4830 tickets for their show.

This was 15% more than they sotd last year.

How many tickets did they setl last year?

-aI I Sra

,CLl*I bofo

- +63ou-83c -: lt<'-- UL*l+Z r/oc: ' LLoc>

I ffil ililt lllil ]til tilil tilillilt illt il]t ilil ]il til).X74775O2O5)k

Page 05

flcx#, SoLI-l t*trL|*Yfr\''L a t+Z*bc

[Turn over

E

6E'ri-

fitE.

t;

6. A spherica[ sweet is made by coat'ing a caramel sphere eventy with chocotate.

A cross-section of the sweet is shown below.

The diameter of the sweet is 24 mil[imetreschocotate coating is 3 mittimetres.

Catcutate the votume of the chocotate coating.

Give your answer correct to 3 significant figures.

and the thickness of the

/ -,

6,q S?,tneg = 4rrc,s

6-lA*MuAelLsf I

Or*u, rf4 (3

-+*3 rqx t2-

37Z3V';L,.rni

\Lu,rn* SnrnLL ;tr ftd€-U-;r133 n'3

=1-x3'[-^J3

C-r;fr?lrt't'r

= 3C 5L'o'3 ffirf

?z:grL - 3c:>sl cB

z tLl 6)"+<r L+llc,r,# (S sf ;

Iilil iltil lllil]lillilll lllll llilllllllllll llill llllllll,<x747750206,k

Page 06

#R+'JTHIS

MARCIN

7. Triangtes A and B are shown below.

The triangtes are ptaced together to form the larger triangte shown betow.

I tilil illil ffi ]]t fiil [Il ]ilt ililt ilIil iltil il ffi,<X74775O2A77t

Poge 07

}Tls this [arger triangte right-angted?

Justify your answer.

92* = t+sqgrt tt = t+2{

b( flrc c^6d-) vffL'€ o( QYnftAffi,

Srnlrt' gt + 8L+ 9z T)#iTE tlwffi- T{rNiL-L rs No1- {J(*ff*frrsG-,'ry

[Turn over

-l

:,,

i.,

.t

'lr

(b)

,o*orJWRITE IN

THISMARGIN

Triangtes A and B are shown betow.

The triangtes are ptaced together to form the [arger triangte shown betow.

ls this [arger triangte right-angted?

Justify your answer.

(a

CoS fr--t'?- ^1- ^'Lb +o-- qJbc

? (l++ Lu1 * 3 C"

(2* r* +j-a

J-L-It

A - (':*,;l r lv \ -: t-.L,S'1"r-r _ LL-.r t ltr) _ \.r

5+,62

l+ 6 S:l- "t"aO'= lot'z

Qtx*7-rt4)u- I 5.-+

26L..tr LrrJ

t zurlSp tZtYre {(ttr*"tt; ls Nlf Qarr^-

trrrn over

I tilt ilil fiilr ]]t tilt ffit 1ilil ilfl iltil ililt ffi lil*x747750207*

Page 07

lc r<)

'16 cm

Alternative Answer to Question 7

8. ln the diagram betow, & unO & r"pr"r"nt the vectors c and d respectivety.

->(a) Express PR 'in terms of c and d.

il -- c1 -c-

a

a

TP=PQ

V is the midpoint of PR

-)Expness TV in terms of c and d.

Give your answelin simplest form. *n-r -L (r

2_

-i- *L (d -.-)L

(b)

;)

T?CL

GI

Tv -a

d LCa+ JdZ

. :ct - *a.?I tilil tlllil ffil lllil lilillllll llill lffi lllllllill llil lill).X74775O2O8r.

Poge 08

The line QP is extended to T.

^t

7/-'iPv=

MARKS

1

(^\-r(e(+z) , fyYqrl

t*-Ls a C15)ffi -{-gqilff+2) \^"-

(a) Factorise 4x2 -25. -L

@r)'- (s)

(b) Hence simptify #*

A"))l'-,t -lc

'1 (= Lx t:)kr'Z-

(Ltt+! G:c-5)

{o€ffir*ce oG g, sqJu n,&)

I ll]t til]t tilt lilt ffit flil ]ilt ilil tilt tilfi iltffir.x74775O2O9,k

Poge 09

[Turn over

***JTHIS

MARGIN

J

r

3

L

10. ln the diagram betow D, E and F represent the positions of Dunbridge,Eartsford and Fairtown respectivety.

F

Dunbridge is 15 kitometres west of Earlsford.

From Dunbridge, the bearing of Fairtown is 126".

From Eartsford the bearing of Fairtown is 230".

160 * (:q n+ul? lc+

Catcutate the djstance between Dunbridge and Fairtown.

Do not use a scale drawing.

230'

D'Si mrLLl $&LJffi<t}-.o ftr(,rct-J/-, i'-'-

e5tr'r3 5t"re

slNio q

jL = l( Srn (1-C

Si", ia Ha q'q3e?8s

Ouo-'SfuMtq,qFkn @o r)

St'vft

j-

S/Fl+4

l:>

I lilt iltil ll]l lllll lllll llll llll llll lllll lllll llll llllr<X74775O21O:t

Poge 10

\ 1V,Nk\.€t[ <,rv6 ?f,tl"'I D?itJ riloflo?tasa€ M^I

ll'tr-to Lalqnnts:h JL

l-11 . A straight line has equation 3x -5y - 1 0 = 0.

Find the gradient of this [ine.

#n+"JTHIS

MARGIN

3f -5"-,o -= oJ

*L"l = *3xJ

5n=3xt/o-toZ

tO

G-,c+5)

CHn?Aaf",/3

:g€L+ g =*rr1 L+

f\a = Crqtugb{ = 3.5

12. Express a,n the form x".ilx

I=)'-E/=L

R)L-

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Page 11

y*r\ = t!J\-

{

[Turn over

|[?]rr>

(usa;ftS.Prcfs

/nt\/=lur'\-

-)c*

DoNor:lWRITE IN

THISMARGIN

13. Two identicatshapes are used to form a [ogo.

Each shape is part of a circte.

. The circtes have centres C, and Cr.

. The rad'ius of each circte is 14 centimetres.

. The logo has half-turn symmetry about the

. AB is 48 centimetres [ong.

Catcutate the height of the [ogo.

height

mid-point of AB.

2- ,,L -l*7*'- l? €

2-f.1)L z >La* 2 J.ut3

Kt1-

1.*

ffi Oiff(ifiT- it>P

V+gt(ffi=11++]x= l+2.4L c-^

(z$s) rtquo?)

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Page 12

14. The diagram below shows part of a circle, centre O.

AB

The radius of the circte is 6-4 centimetres.

Major arc AB has length 31.5 centimetres.

Catcutate the size of the rg[!e>r angte AO9.

rnftJo(fl[C-

- - [nn, kL: 1oth-rtr1Yt

" L ftL{1i6-1,A)

,J* : 3 i'f

{nQJo{*{4eC-

I ilil tlllil ll]t ]lil ilil tilt ]til ililI tilt tilt til ll]r<x747750213r<

Poge 13

]l.-D

3t'5 + 2< 3"r?x iZK

3i.S :Jec

x x O,lila

: 28a"a"iil u

= 5i Sc,n

[Turn over

,o*or]RITE INTHIS

ffiL

MARKS I,iR+i"il

15. A wind turbine has three btades as shown betow.

The height, /z metres, of the tip of btade A above the ground'in each rotationis given by

h= 40 + 23cos-ro, 0 <.r < 360

where x is the angte btade A has turned ctockwise from its vertical position.

(a) Catcutate the height of the tip of blade A after it has turned through an

angte of 60".

h = i{<3-r- 23C'sC o

= $1,5 n^

r rilt il]t ll]t ]]l ilt [] illt illl lll] lllil lll llll,vX74775O214t<

Page 14

15. (continued)

(b) Find the minimum height of the tip of btade A above the ground.

#irNrrnurn fffuffi lS ilrt6.J fri;l46(t*-"tuar')h = +o -l- ZSG-"s (8O

- lTm

(c) Catcutate the vatues of x for which the tip of blade A is 61 metres abovethe ground.

GIG( -L+T

=ZlZI-,Z3

IEND OF QUESTTON PAPERI

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Page 15

h c +o t- ZsG=)l

Su6>Tr:rr;T€ Vr= 6lv?A Jt--'fl() ftGq>'€

tJ-o f&@fr.Iprtj,-ffitfi6{,{ly^

[t:+:q)\T c'/Ge.-{36c:-2t-'t'1-a ?35'q

23Gs1.=}3Ce,s;C =

23Gs dG5 )(

*z (-o,s-/ (*)

= 2+'n?ur 335'1'

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