SCOTTISHCERTIFICATE OFEDUCATION

MATHEMAHTGHER GRAD

SCOTTCERTIIEDUCA1991

Paper II 2 hours 30 minutes

INSTRUCTIONS TO CAN DIDATES

hlffilyl. p1lcpfrx,fl be given only where the solution contains appropriate working.

Z fi55e15ohaineO Uy readings from scale drawings wil] not receive any credit.

Paper I 2 hours

FORMULAELIST

The equati on x2 + yz + 2 gx + Zfy + c = 0 represents a circle centre (- g,- f) and

radius .lG, + f2 - c).

The equation (x - a)2 + (y - b)z = 12 represents a circle centre (a, b) and radius r.

ScalarProduct: o.b=lal lbl cos0, where0 istheanglebetweenaandb

or ('') ,r=[;]la.b = a1b1+ azbz* a3D3 wher. a = | orl un,l.;;] t.,1

Trigonometric formulae: sin (A + B) = sin A cos B t cos A sin B

cos (A tB) = cos A cos BT sin A sin B

cos 2A = cos2 A - sin2 A= 2cos2A - 1

= 1- 2sin2A

sin 2A = 2sin A cos ATable of standard derivatives: l@) f '(x)

sinarc acosatc

cosatc -asinaxTable of standard integrals : f (x) [ f fda*

sinar j"orr*+Ca

1ri.ra, + Ca

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prior permission of the publisher Robert Gibson & Sonq Ltd., 17 Fitzroy Place, Glasgow, G3 7SF'

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SCOTTISHCERTIFICATE OFEDI'ICATION1992

MONDAY, 1 1 MAY9.30 AM * 11.30 AM MATHEMATICS

HIGHER GRADEPaper I

1.

2.

All questions should be attempted Marks

Find the equation of the tangent to the curve with equation y = 5x3 - 6x2 atthe 1rcint where x = l. (4)

In the diagram, A is the point (7,0),

B is (-3, -2) and, C (-1, 8).

The median CE and the altitude BD intersect at J.

(a) Find the equations of CE and BD.

(D) Find the coordinates of J.

Fhdfr ilr- 2 is a factor of 13 + kx2 - 4x - 12.

Acurve for which * = 3x2 + lpasses through the point (-1,2).

Express 3l in terms of r.

Find, correct to one decimal place, the value of r between 180 and 270rLich satisfies the equation 3cos(2r * 40)" - 1 = 0.

On e suiteble set of real numbers, functions/ and g are defined by

nlo=h and e(x)=!-2.

FinilJk(-D in its simplest form.

'7- (4 f-ryrcs sio.ro - 3cos*" in the form ft sin(r - a)o where ft > 0 and

O< c< 360. Find the values of k and a.

(O fiDd-ftc rrr-imum value of 5 + sinr" - 3cosro and state a value of *fu ilich this maximum occurs.

9.

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t

AnthrlefwiStr

Gifro

(8)

(3)3.

L

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(4)

10. Ttfol

TTaxi-2

Or

(a)

(D)

11. Di

(3)12. At

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(6)

r*1

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1992

Marks

An ancient Stone Circle has a processional pathway from the Heelstone tothe centre of the Stone Circle. In the picture above, the Heelstone is on theleft and the dotted line represents the processional pathway.

With suitable axes and using the Heelstone as the origin, the equation of theStone Circle is 12 + y2 - 8x - 6y + 27 = 0.

Given that 1 unit represents 15 metres, calculate the distance in metresfrom the Heelstone to the nearest point on the edge of the Circle. (S)

The sketch shows the praph of y = f(x)for -2-( #:( 4.

The function g(x) has the line x = 4 as anaxis of symmetry and g(x) = /(x) for-2-( x -( 4.

On separate sketches, indicate

(a) y = g(x) for -2 -( x -( 10

(6) y = -Zg(x) for 0 -( x-( 8 . (4)

(4)3

11. Differentiate 2xZ + sin2x with respect to r.

12. An incomplete sketch (notdrawn to scale) of thegraph of y = logro(x + a) isshown. Find the value of a. (21

f1992

13. The diagram shows a kite OABC. A isthe point (4,0) and B is the point (4,3).

Calculate the gradient of OC correct totwo decimal places.

Marks

14. (a) Evaluate ["llrr* r*.0

(b) Draw a sketch and explain your answer.

An aircraft flying at a constant speed on a straight flight path takes 2minutes to fly from A to B and 1 minute to fly from B to C. Relative to asuitable set of axes, A is the point (-1,3,4) and B is the point (3, 1,-2). Findthe coordinates of the point C. (3)

AB is a tangent at B to the circle withcentre C and equation

(x - 212 + b, - 2)2 -- ZS.

The point A has coordinates (10,8).

Find the area of triangle ABC.

(3)

(s)

17. Calcu}offtsodoes nt

The I

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Iflc

19. Theequi

Ske'

18.

15.

(s)

17.

1992

Calculate the least positive integer value

does not cut or touch the r-axis.

The diagram shows representatives of twovectors, a and. b, inclined at an angle of 60'.

lf lal = 2 ar'.d I al = g, evaluate a.(a + b).

Marks

(4)

(4)

18.

(3)

19. The line with equation y = r isequation y = f(x). The parabola

Sketch the graph of y = l'@).

a tangent at the origin to the parabola withhas a maximum turning point at (a, b\-

IEND OF QUESTTON PAPER]

MONDAY, 1 1 MAY1.30 PM - 4.00 PM

All questions should be attempted

part of the graph of the curve with equation

MATHEMATICSHIGHER GRADEPaper II

Marks

6

, (11)

scale ofaircraft

1 unit to 2and satellite

3. Bioka stxenda

AfaclailThe

I-2

I3

the coordinatespoints where the

the r and y

; mordinates of the'points and justify

sritable set of coordinate axes with athe positions of a transmitter mast, ship,

2

(a)

whpro(307(6)

in the diagram below.

R (7,2,31

c (12,-4,11

n,o,-s,-tf6@/

T of the transmitter mast is the origin, the bridge B on the ship is; (5,-5,-l), the centre C of the dish on the top of a mou-ntain is the

*.1t2,-1,D'and the reflector R on the aircraft is the point (7,2,3).

Find the distance in kilometres from the bridge of the ship to themflector on the aircraft.

Three minutes earlier, the aircraft was at the point M(-2,4,8'5). Findfu speed of the aircraft in kilometres per hour.

Prove that the direction of the beam TC is perpendicular to thedirection of the beam BR.

Calculate the size of angle TCR.

4- (*t'!ud

(t

(q

GI

2

3

5

(13)(4

10

3.

1992

MarhsBiologists calculate that, when the concentration of a particular chemical ina sea loch reaches 5 milligrams per litre (mg/l), the level of pollutionendangers the life of the fish.A factory wishes to release waste containing this chemical into the loch. It isclaimed that the discharge will not endanger the fish.

The Local Authority is supplied with the following information:

l. The loch contains none of this chemical at present.

2, The factory manager has applied to discharge waste once perweek which will result in an increase in concentration of 2'5 mg/lof the chemical in the loch.

3. The natural tidal action will remove 40Yo of the chemical from theloch every week.

(") Show that this level of discharge would result in fish being endangered.

When this result is announced, the company agrees to install a cleaningprocess that reduces the concentration of chemical released into the loch by

lor*rnr* the calculations you would use to check this revised application.Should the Local Authority grant permission?

(8)

4. (a) For a particular radioactive substance, the mass m (in grarns) at time r(in years) is given by

in = moe-o'ozt

where mo is the original mass.

If the original mass is 500 grams, find the mass after 10 years.

(6) The half-life of any material is the time taken for half of the mass todecay.

Find the half-life of this substance.

(r) Illustrate all of the above information on a graph.3

3

(8)

tt

|,&4rt"?sft re92

r*fir:.:' -

fl'-Gl[hfird e zoo intend to build a

*iE rhc shaPe of a cuboidfl,rq-: floor- Ttre volume of therlf,*tTfohc 500 m3-

Marks

6

(10)

(2,r,1')

(3,-1,2)

f,t f a rGtres is the length-d- cdgc of the floor.lo: that the a;rea A

, rqtrc mctres of nettingqrilrcd is given bY

..^ll.= *+ 2ooox

0! F-d tte dimensions oftrc aviary to ensure thatt[c cost of netting isanirrirnised.

6. The

Example

oector prod.uct, a x b, or a =[l,,.l"". , = [I) is derined bv

(arbr-arbr) [";J [o'J

oxb=l "ru,-rru, l.[a1b, -arb1 ,,|

(a) If c and b are as shown in the diagramand c = ax b, evaluate c.

(D) By considering a.c an,d b.c, what canbe concluded about c? 4

(71

a,

.)l

"=[iJ,andb=[]

( ,"2 - 3xo ) [-)thenaxb= lsxl-t1 - 1x, l= l-tl

[ 1*o-zx(-l)/ \2)

When

(4,1,0)

i-

12

7. (a) 1

(D) '!

(c) I

(d) l

i

I

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l

(.il

I

OI

1992

(a\ Solve the equation 3sin 2x" = 2sin x" for 0-< r-< 360.

(b) The diagram below shows parts of the graphs of sine functions/and g.

State expressions for f(x) and g(x).(c) IJse your answers to part (a) to find the coordinates of A and B.

@ Hence state the values of x in the interval 0 -< r -<360 for which3sin 2xo ( 2sin ro.

Marks4

1

2

3

(10)

8. A ship is sailing due north at a constant speed.

When at position A, a lighthouse L is observed on abearing of a". One hour later, when the ship is atposition B, the lighthouse is on a bearing of 6o.

The shortest distance between the ship and thelighthouse during this hour was d miles.

(a) Prove that AB = d ^ -tar.ao 2tanbo

(b) Hence prove that AB = bP*-$stna"srnbo

(c) Calculate the shortest distance from the ship to the lighthouse when thebearing,*-a" ?n! b" are 060o and 135" respectively and ihe constant speedof the ship is 14 miles per hour.

(8)

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k'A:\A

3

2

-2

-J

l3

7

1992

Marks

BALLOON

GONDOLA

Arpherical hot-air balloon has radius3ll feet. Cables join the balloon to thegmdola which is cylindrical withdimeter 6 feet and height 4 feet. Theop of the gondola is 16 feet below thebottom of the balloon.

C.oordinate axes are chosen as shownin the diagram. One of the cables isrelrresented by PB and PBA is astraight line.

(a) Find the equation of the cable PB.

(D) State the equation of the circle representing the balloon.(c) Prove that this cable is a tangent to the balloon and find the coordinates

of the point P.

1+

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9.

3

1

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10.

7992

MarksAn artist has been asked to design a window made from pieces of colouredglass with different shapes. To preserve a balance of colour, each shapemust have the same area. Three of the shapes used are drawn below.

Relative to tr,y-axes, the shapes are positioned as shown below. The artistdrew the curves accurately by using the equation(s) shown in each diagram.

(a) Find the area shadedunderY=2x -rcz.

(b) Use the area found inpart (a) to find thevalue of p.

Prove that q satisfiesthe equationcosq * sinq = 0'081 andhence find the value ofq to 2 significantfigures.

4

(c)

IEND OF QUESTTON PAPER]