SCOTTISHCERTIFICATE OFEDUCATI ON1993

MONDAY, 1O MAY9.30 AM - 11.30 AM

MATHEMATICSHIGHER GRADEPaper I

All questions should be attemPted

1. A is the point (-3,2,4) and B is (-1,3,2)' Find-+

(a) the comPonents of vector AB;

(D) the length of AB.

Relative to the axes shown and with an appropriate scale, Alex stands at the

point (-2,3) where Hartington Road meets Newport Road'

Find the equation of Newport Road which is perpendicular to

Hartington Road.

Breoda is waiting for a bus at the point (-5'1)' Show that Brenda is

standing on NewPort Road'

3. Find the vzlues of ft for which the equatio rt 2* + 4x + k = 0 has real roots'

4. Find the r-oordinate of each of the points on the curve.

g =2x3 -31 -t2x+20 at which ttre tangent is parallel to the x-axis'

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This diagram shows a computer-generateddisplay of a game of noughts and crosses.

Relative to the coordinate axes which have

been added to the display, the "nought" at Ais represented by a circle with equation

(*-z)' +(s, -z)' = +.

(a) Find the centre of the circle at B.(b) Find the equation of the circle at B.

For acute angles P and Q, sin P = Q and

Show that the exact value of sin(n + Q)

One root of the equation 2x3 -3xz + fuc+ 30 = 0Find the value of p and the other roots.

8. The diagram shows the graph of y = f (x).

Sketch the graph of y - 2 - l@) .

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(3)(1)

sinQ = -1.

is 4.6-5

7. is -3.

g. Differentiate 41i *3cos2x.

t7

jH13 make angles of ao and 6o with the positiveshown in the diagram.

(a\ Find the values of a and,b.(D) Hence find the acute angle between the two given lines.

11. The graphs of y = f (x) and y = g(x')intersect at the point A on the y-axis, as

shown on the diagram.

lf e(x)=3x*4Pnd f'(x)=2x-3, find f(x).

12. The vectors a, b and c are defined as follows:

a=2i:-k, b=i+2i + h, s=-j + h

Evaluate a.b * a.c .

t'"tI

r.*3

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(a)

14. A sketc

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Sketch tl:

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(b)

(c)

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(r) From your answer to part (a), make a deduction about the vector b * c.

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13. f(x)=2x-1, g(x)=3-2x and, tdx)=*(t-4.

(a) Find a formula for k(x) where k(x)=f(r(r)).

(b) Find a formula *' 4n(.)).

(c) What is the connection between the functions h and k?

14. A sketch of a cubic function,./, with domain *4 < x ( 4, is shown in thediagram below.

Sketch the graph of the derived function, f ', for the same domain.

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A medical technician obtains this print-out of a wave form generated by an

oscilloscope.

The technician knows that the equation of the first branch of the graph

(for 0 < * S 3) should be of the form y = o"k* .

(a) Find the values of a and, k.

(D) Find the equation of the second branch of the curve (i.e. for 3 S r < 6).

^/1+3,. dx xtd,hence find the exact ^lt**a*

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(4)".h. rf IIt- If J(a)=6sin2 a-cosa, express l@)inthe form 2cos2 a+qcosa+r.

&oe solve, correct to three decimal places, the equation

6sin2a-cosa=5 for 0 1 aSn.

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18. Explain why the equation * + y2 +2x+3y + 5= 0 does not represent a circle.

19. (a) Show that (cosx+.irrr)' = 1+sin2n.

(b) Hence rina J(cosx

+"ir'*)' d* .

The point P (p,k) lies on the curve with equatioo y = log"x.

The point Q @,k) lies on the curve with equation y - ltog"*.Find a relationship between p and q and hence find q whenp = 5.

21. The diagram shows the graph of the function

71x)=L-, x*-7.

Prove that the function/is decreasing for all

values of x except tc = *7.

IEND OF QUESTTON PAPER]

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t'SCOTTI SHCERTIFICATE OFEDUCATTON199 3

MO\IDAY. TO ITAY1.30 Ptil - +-OO Pnt MATITEMATICS

HIGHER GRADEPaper II

Atlqocrtions should be attemPtedMarks

l. The frmfunJ, rhre incorrpletegrqlh is $oun in 6e diagram, is

Find 6c oordinates of thest*frnrypoints and justify theirEmrre-

2- The concrete on the 20 feet by 28 feet rectangular facing of the entrance,to an

underground cavern is to be repainted.

Coordinate axes are chosen as shown in the diagram with a scale of 1 unit equal to.,

1 foot. The roof is in the form of a parabola with equatior y = 18 - tx- .

(a) Find the coordinates of the points A and B.

(6) Calculate the total cost of repainting the facing at d3 per square foot.

3. In an experiment with a ripple tank, a

series of concentric circles with centre

C(4,-l) is formed as shown in the

diagram.The line I with equation ! =2x+6represents a barrier placed in the tank.The largest complete circle touches thebarrier at the point T.

(a) Find the equation of the radius CT.

(D) Find the equation of the largestcomplete circle.

22

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(8)

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4. Ana

equl

FxA

(a) Fir

(D) FiI

eqr

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An array of numbers """n ^ (l l) t. "r*o a matrix.

The eigenvalues of the matrix l, =(" 1l ,* defined to be the roots of rhe(c d)- -

equation (" -.)(a - *) - bc = 0.

Exenapm

In order to find the eigenvalues of the matrix B =(l 3 )[+ z)

solve (t- .)(z - *)- 4x 3 = o

solution: Z-3x+ *' -12 =O

*' -3*-10 = o

(r+z)(x-s)=ox=-2 or r=5

so the eigenvalues ofB are _2 arrd S

(a) Findtheeigenvaluesof C=G ,

(D) Find the varue of r for which the eigenvalues of the matrix "

= [; ,r)

*"equal.

(3)

(s)

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Anodclof acrystalwas made from a

crbeof dde 3 units bv slicing off the

cGd P to leave a triangularfuArc.Oourrlinate axes have been

imoduced as shown in the diagram.

Thc point A divides OP in the ratio 1:2.

noine B and C similarly divide RP and

SF respectively in the ratio 1:2.

(a) Find the coordinates of A, B and C.

Calculate the area of triangle ABC.

Calculate the percentage increase or decrease in the surface area of the

crystal compared with the cube.

6. The diagram below shows the graph of y --Zsinzx+l for 0 < x3n.

(a) Find the coordinates of A and B (as shown in the diagram) by solving

an appropriate equation algebraically.

(D) The points (0, 2) and (zr, 0) are joined by a straight line t' In how many

points does I intersect the given graph?

(c) C is the point on the given graph with an *-coordinat e of f . Explain

whether C is above, below or on the line I'

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7. The diagram shows the plans for a proposed

new racing circuit. The designer wishes tointroduce a slip road at B for cars wishing toexit from the circuit to go into the pits. Thedesigner needs to ensure that the two sectionsof road touch at B in order that drivers maydrive straight on when they leave the circuit.

road does

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Relative to appropriate axes, the part of the circuit circled above is shown

below. This part of the circuit is represented by a curve with equation

! =5-2*2 - *3 and the proposed slip road is represented by a straight linewithequationy=-4x-3.

Proposedsliproad

(a) Find algebraically the coordinates of B.

(6) Justify the designer's decision that this direction for the slipallow drivers to go straight on.

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8. Secret Agent 004 has been captured and his captors are giving him a

25 milligram dose of a truth serum every 4 hours. lSo/o of t}l.e truth serumpresent in his body is lost every hour.

(a) Calculate how many milligrams of serum remain in his body after4 hours (that is, immediately before the second dose is given).

(e) It is known that the level of serurn in the body has to be continuouslyabove 20 milligrams before the victim starts to confess. Find how manydoses are needed before the captors should begin their interrogation.

(c) I.etu, be the amount of serurn (in milligrams) in his body just after his

oft do".. Show that un*1 = Q' 522un + 25.

(d) It is also known that 55 milligrams of this serum in the body will provefatal, and the captors wish to keep Agent 004 alive. Is there anymaximum length of time for which they can continue to administer thisserum and still keep him alive?

9. A builder has obtained a large supply of 4 metrerafters. He wishes to use them to build some holidaychalets. The planning department insists that thegable end of each chalet should be in the form of anisosceles triangle surmounting two squares, as shownin the diagram.

(a) lf 0" is the angle shown in the diagram and A is the area (in squaremetres) of the gable end, show that

a=8(z+sin g"-2cos oJ.

(b) Express 8 sin 0 " -1 6 cos 0" in the form f sin(0 - c)" .

(c) Find algebraically the value of 0 for which the area of the gable end is30 square metres.

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lIa*s10. When the switch in this circuit was closed, the computer printed out a graph

of the current flowing (.I microamps) against the time (t seconds).This graph

is shown in figure 1.

switch

l-'---I U toaT F> computerI

- intertaceI L.->

In order to determine the equation ofthe graph shown in figure 1, values of

logrlwere plotted against logrt and the

best fitting straight line was drawn as

shown in figure 2.

(a) Find the equation of the line shown in figure 2 in terms of logrl and

log "t.

(b) Hence or otherwise show that I and / satisfy a relationship of the form

I = ktr stating the values of k and r.

(3)

(4)

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11. An oil production platform, 9{3 km offshore, is to be connected by apipeline to a refinery on shore, 100 km down the coast from the platform as

shovm in the diagram.

Oil Production Platform -:-

E/r t-Refinery

The length of underwater pipeline is r km and the length of pipeline on landis y km. It costs {2 million to lay each kilometre of pipeline underwater and

{1 million to lay each kilometre of pipeline on land.

(a) Show that the total cost of this pipeline is {C(x) million where

/1c(x) = zx + too - (.' - z+t)' .

(b) Show that x = 18 gives a minimum cost for this pipeline.Find this minimum cost and the corresponding total length of thepipeline.

IEND OF gUESTrOr,{ PAPER]

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SCOTTICERTIFEDUCA'199 4

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