O T R - Bored of Studiescommunity.boredofstudies.org/attachments/12/mathematics/...4 UNIT MATHEMATICS – HARDER 3 UNIT TOPICS 1 – HSC ¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA:

4 UNIT MATHEMATICS – HARDER 3 UNIT TOPICS 1 – HSC

¤©BOARD OF STUDIES NSW 1984 - 1997

©EDUDATA: DATAVER1.0 1995

Harder 3 Unit Topics 1: 1997 - 1991 4U97-3c)!

In a game, two players take turns at drawing, and immediately replacing, a marble from a bag

containing two green and three red marbles. The game is won by player A drawing a green marble, or

player B drawing a red marble. Player A draws first. Find the probability that:

i. A wins on her first draw;

ii. B wins on her first draw;

iii. A wins in less than four of her turns;

iv. A wins eventually.¤

« i) 2

5 ii)

9

25 iii)

1622

3125 iv)

10

19 »

4U97-4b)!

i. Find an expression for cot2A in terms of tanA.

ii. Show that tanA and -cotA satisfy the equation x2 + 2x cot2A - 1 = 0.

iii. Hence, or otherwise, find the exact value of tan8

.

iv. Hence find the exact value of tan16

cot16

.¤

« i) cot2A =1- tan A

tanA

2

2 ii) Proof iii) 2 1 iv) 2 2 1( ) »

4U97-6a)!

The series 1 x x x2 4 4n... has 2n + 1 terms.

i. Explain why 11

1

x x x

x

x

2 4 4n4n + 2

2... .

ii. Hence show that 1

11

1

1

xx x x

xx

2

2 4 4n

2

4n + 2... .

iii. Hence show that, if 0 y 1, then tan y y -y y y

4n + 1tan y +

1

4n + 3

-13 5 4n +

-1 3 5

1

... .

iv. Deduce that 0 11

3

1

5

1

1001 410 3

...

.¤

« i) Since Sa(1- r )

1- rn

n

and a = 1, r = -x2, N = 2n + 1 ii) iii) iv) Proof »

4U97-7a)!

S

R

UTO

The points R and S lie on a circle with centre O and radius 1. The tangents to the circle at R and S

meet at T. The lines OT and RS meet at U, and are perpendicular. Show that OU OT = 1.¤

« Proof »

4U97-7b)!




S

R

U

x

y T

P

QO

The circle (x - r)

2 + y

2 = r

2, with centre Q(r, 0) and radius r, lies inside the circle x

2 + y

2 = 1, with

centre O and radius 1. The point P(r + rcos, rsin) lies on the inner circle, and P and O do not

coincide. The tangent to the inner circle at P meets the outer circle at R and S, and the tangents to the

outer circle at R and S meet at T. The lines OT and RS meet at U, and are perpendicular.

i. Show that OT is parallel to QP.

ii. Show that the equation of RS is xcos + ysin = r(1 + cos).

iii. Find the length of OU.

iv. By using the result of part (a), show that T lies on the curve r2y

2 + 2rx = 1.¤

« i) ii) Proof iii) r(1 + cos) iv) Proof »

4U97-7c)!

y

x

K

JN M

L

O

The parabola x

2 = 4ay touches the circle x

2 + y

2 + 2gx + 2fy + c = 0 at J, and cuts it at K and L. The

midpoint of KL is M, and the line JM cuts the y axis at N, as shown on the diagram.

i. Find a quartic equation whose roots are the x coordinates of J, K and L.

ii. Show that JN = NM.

iii. Hence show that the area of JKN is one-quarter of the area of JKL.¤

« x4 + (16a

2 + 8af)x

2 + 32a

2gx + 16a

2c = 0 ii) iii) Proof »

4U97-8a)!




C

X

E

A

D BF

Triangle ABC is scalene. External equilateral triangles ABF, BCD and CAE are constructed on the

sides of triangle ABC as shown. Lines AD and BE meet at X.

Copy or trace this diagram into your Writing Booklet.

i. Show that BCE = DCA.

ii. Show that BCE DCA.

iii. Show that BDCX is a cyclic quadrilateral.

iv. Show that BXD = DXC = CXE = EXA =

3.

v. Show that CF passes through X.

vi. Show that AD = BE = CF.¤

« Proof »

4U96-3c)!

v

O 1 2 3 4 5 6 7 8 9 t10

A particle moves along the x axis. At time t = 0, the particle is at x = 0. Its velocity v at time t is

shown on the graph.

Trace or copy this graph into your Writing Booklet.

i. At what time is the acceleration greatest? Explain your answer.

ii. At what time does the particle first return to x = 0? Explain your answer.

iii. Sketch the displacement graph for the particle from t = 0 to t = 9.¤




« i) At t = 6.5 where the curve of v(t) has the greatest gradient. ii) At t = 4, since the curve is symmetric

around x = 2 in the interval 0 t 4 and v t dt( ) 0 . iii)

x

t94

»

4U96-4a)!

By differentiating both sides of the formula 11

1

2 31

x x x xx

x

nn

... , find an expression for

1 2 2 3 4 4 8 2 1 ... n n .¤

« (n – 1)2n + 1 »

4U96-4c)!

Consider a lotto-style game with a barrel containing twenty similar balls numbered 1 to 20. In each

game, four balls are drawn, without replacement, from the twenty balls in the barrel.

The probability that any particular number is drawn in any game is 0.2.

i. Find the probability that the number 20 is drawn in exactly two of the next five games

played.

ii. Find the probability that the number 20 is drawn in at least two of the next five games

played.

Let j be and integer, with 4 20 j .

iii. Write down the probability that, in any one game, all four selected numbers are less than or

equal to j.

iv. Show that the probability that, in any one game, j is the largest of the four numbers drawn is

j

1

3

20

4

.¤

« i) 0.205 (to 3 d.p.) ii) 0.263 (to 3 d.p.) iii)

j

4

20

4

iv) Proof »

4U96-6a)!

Solve 3 2 2 32x x x .¤

« –1 x 2 »

4U96-8b)!

P

T

Q R

The points P, Q, R lie on a straight line, in that order, and T is any point not on the line. Using the fact

that PR - PQ = QR, show that QT - QP > RT - RP.¤

« Proof »




4U96-8c)!

A

K

B

CMD

N

L

i. ABCD is a quadrilateral, and the sides of ABCD are tangent to a circle at points K, L, M, and

N, as in the diagram. Show that AB + CD = AD + BC.

ii. ABCD is a quadrilateral, with all angles less than 180. Let X be the point of intersection of

the angle bisectors of ABC and of BCD. Prove that X is the centre of a circle to which

AB, BC, and CD are tangent.

iii. ABCD is a quadrilateral, with all angles less than 180. Given that AB + CD = AD + BC,

show that there exists a circle to which all sides of ABCD are tangent. You may use the

result of part (b).¤

« Proof »

4U95-4b)!

i. Solve x² > 2x + 1.

ii. Prove by mathematical induction that 2n > n

2 for all integers n 5.¤

« (i) x 1 2 or x 1 2 (ii) Proof »

4U95-5a)!

i. Show that sin x + sin 3x = 2sin 2x cos x.

ii. Hence, or otherwise, find solutions of sin x + sin 2x + sin 3x = 0 for 0 x < 2.¤

« (i) Proof (ii) x 0,2

, ,3

2,2

3,4

3

»

4U95-6a)!

Pat observed an aeroplane flying at a constant height, h, and with constant velocity. Pat first sighted it

due east, at an angle of elevation of 45. A short time later it was exactly north-east, at an angle of

60.

i. Draw a diagram to represent this information.

ii. Find an expression in terms of h for the initial horizontal distance between Pat and the point

directly below the aeroplane.

iii. In what direction was the aeroplane flying? Give your answer as a bearing to the nearest

degree.¤




« (i)

4560

45

x

h

A

B

P

h

(ii) x = h (iii) N55W or 305 »

4U95-6b)!

UM

T

V

P0

r s

W

In the above diagram, a circle with centre O and radius r meets a circle with centre P and radius s at

the points V and W. The straight lines VW and OP meet at M. The point T is arbitrary, and U is the

point on the line OP such that TU is perpendicular to OP.

i. Prove that OP and VW are perpendicular.

ii. Show that OT² - PT² = OU² - PU² and that OM² - PM² = r² - s².

iii. Hence show that T lies on the line VW exactly when OT² - PT² = r² - s².

iv.

Q

PO

F E

D

C

B

A

FAEB, BCAD, and DECF are circles with centres O, P and Q, and radii r, s, and t,

respectively. Using the result of part (iii), or otherwise, show that the straight lines AB, CD,

and EF are concurrent.¤

« Proof »

4U95-7b)!

A fair coin is tossed 2n times. The probability of observing k heads and (2n - k) tails is given by

P2n

k

1

2

1

2k

k 2n k

.

i. Show that the most likely outcome is k = n. That is, show that Pk is greatest when k = n.




ii. Show that P(2n)!

2 (n!)n 2n 2 .

iii. Using the result of part (a) (iii), show that

1

nP

1

n12

n

.¤

« Proof »

4U95-8a)!

Suppose that p and q are real numbers. Show that pqp q

2

2 2

.¤

« Proof »

4U94-5b)!

A jar contains w white and r red jellybeans. Three jellybeans are taken at random from the jar and

eaten.

i. Write down an expression, in terms of w and r, for the probability that these 3 jellybeans

were white.

Gary observed that if the jar had initially contained (w + 1) white and red jellybeans, then the

probability that the 3 eaten jellybeans were white would have been double that in part (i).

ii. Show that rw w

w

2 2

5.

iii. Using part (a) (v), or otherwise, determine all possible numbers of white and red jellybeans.¤

« (i) w

w r

w 1

w r 1

w 2

w r 2

(ii) Proof (iii) 2 red and 3 white, or 10 red and 4 white »

4U94-6b)!

Y

P

M

X

N

0C

Z

B A

Circles PABC and PMNO intersect at P, and APM, BPN, and CPO are straight lines. BA and MN

produced meet at X, CA and MO produced meet at Y, and CB and NO produced meet at Z, as in the

diagram. Let YAX = .

i. Prove that BPC = .

ii. Prove that OMN = .

iii. Prove that XYAM is a cyclic quadrilateral.




iv. Prove that XYM = BCP.

v. Prove that X, Y and Z are collinear.¤

« Proof »

4U94-7b)!

For all integers nn n n nn

11

1

1

2

1

2 1

1

2, ... let t . That is:

t

t

t

1

2

3

1

2

1

3

1

4

1

4

1

5

1

6

...

i. Show that tn n n nn

1

2

1 1

1

1

2 1... .

ii.

x

y

0

1n

1n+1

1

2n-1

1x

y =

n + 1n n + 2 2n

2n-1

The diagram above shows the graph of the function yx

1

for n x n 2 .

Use the diagram to show that tnn 1

22ln .

[Note that it can be similarly shown that t ln2n .]

iii. For all integers nn nn

1 11

2

1

3

1

4

1

2 1

1

2, ... let s . That is:

s

s

s3

1

2

112

112

13

14

112

13

14

15

16

...

Prove by mathematical induction that s tn n .




iv. Hence find, to three decimal places, the value of 11

2

1

3

1

4...

1

9 999

1

10 000 .¤

« (i) (ii) (iii) Proof (iv) 0.693»

4U94-8a)!

Suppose a > 0, b > 0, c > 0.

i. Prove that a b ab2 2 2 .

ii. Hence prove that a b c ab bc ca2 2 2 .

iii. Given a b c abc a b c a b c ab bc ca3 3 3 2 2 23 ( )( ) , prove that

a b c abc3 3 3 3 .

In parts (iv) and (v), assume x > 0, y > 0, z > 0.

iv. By making a suitable substitution into (iii), show that x y z xyz 31

3( ) .

v. Suppose (1 + x)(1 + y)(1 + z) = 8. Prove that xyz 1 .¤

«Proof»

4U93-6b)! D

S

P

Q

R

A

B C

In the diagram, ABCD is a cyclic quadrilateral and P, Q, R and S are the incentres of triangles ABC,

BCD, CDA and DAB respectively. The incentre of a triangle is the point of intersection of the

bisectors of its three angles. Thus, for example, BP bisects ABX and CP bisects ACB;

similarly BQ bisects DBC and CQ bisects DCB.

i. Copy the diagram.

ii. Prove that PBQ ( = PBC - QBC) = 12 ABD.

iii. Prove that PCQ = PBQ and hence explain why BCQP must be a cyclic quadrilateral.

iv. Prove that SPQ = - BAS + BCQ.

v. Deduce that SPQR is a rectangle.¤

« Proof »

4U93-7a)!

In the next 7 days, called day 1, day 2, ... , day 7, Esther and George must each take 3 days in a row

off work. They choose their consecutive 3 days randomly and independently of each other.

i. Show that the probability that they both have day 1 off together is 1

25.

ii. What is the probability that day 2 is the first day that they both have off together?




iii. Find the probability that Esther and George have at least one day off together.¤

« (i) Proof (ii) 3

25 (iii)

19

25 »

4U93-7b)!

For n = 1, 2, 3, ..., let S 11

r!n

r 1

n

.

i. Prove by mathematical induction that e S ex

n!e dxn

n

x

0

1

.

ii. From (i), deduce that 0 e S3

(n 1)!n

for n = 1, 2, 3, ... . [Remember that e < 3 and

e-x

1 for x 0.]¤

« Proof »

4U93-8b)!

Let f(x) = 1 + x² and let x1 be a real number. For n = 1, 2, 3, ..., define x xf(x )

f (x )n 1 n

n

n

.

[You may assume f '(xn) 0.]

i. Show that │xn + 1 - xn│ 1 for n = 1, 2, 3, ..., .

ii. Graph the function y = cot for 0 < < .

iii. Using your graph from (ii), show that there exists a real number n such that xn = cot n

where 0 < n < .

iv. Deduce that cot n + 1 = cot 2n for n = 1, 2, 3, ... .

[You may assume tan 22tan

1 tan2

.]

v. Find all points x1 such that, for some n, x1 = xn + 1.¤

« (i) Proof (ii)

y

2

(iii) (iv) Proof (v) All points cotm

2 1n

, where m and n are

integers, n 1 and 0 m n. »

4U92-3b)!




D

A

B

C

E In the diagram, the bisector AD of BAC has been extended to intersect the circle ABC at E.

i. Prove that the triangles ABE and ADC are similar.

ii. Show that AB.AC AD.AE .

iii. Prove that AD AB.AC BD.BC2 ¤

« Proof »

4U92-6b)!

Let n be an integer with n 2. i. For i = 1, 2, ..., n suppose xi is a real number satisfying 0 xi .

Use mathematical induction to show that there exist real numbers a a an1 2, , ,..., such that

a i 1 for i = 1, 2, ..., n and such that

sin( ... ) sin sin ... sinx x x a x a x a xn n n1 2 1 1 2 2 .

ii. Deduce that sinnx nsinx where 0 x .¤

« Proof »

4U92-7a)!

M

A A'

X'X

B

North

The diagram shows the road grid of a city.

Ayrton drives exactly 10 blocks from his home, A, to his workplace, B, which is 6 blocks south (S)

and 4 blocks east (E). The route on the diagram is SESSSEEESS.

i. By how many routes can Ayrton drive to work?

ii. By how many different routes can Ayrton drive to work on those days that he wishes to stop

at a shop marked M?

iii. The street marked AA is made one-way westward. How many routes can Ayrton follow if

he cannot drive along AA ?




iv. Suppose that instead of AA the street marked XX is made one-way westward. How

many different routes can Ayrton follow if he cannot drive along XX ?¤

« (i) 210 (ii) 90 (iii) 126 (iv) 126 »

4U92-8b)!

Let n be a positive integer and let x be any positive approximation to n .

Choose y so that xy = n.

i. Prove that x y

n

2

.

ii. Suppose that x n .

Show that x y

2 is a closer approximation to n than x is.

iii. Suppose x n .

How large must x be in terms of n for x y

2 to be a closer approximation to n than x is?¤

« (i) Proof (ii) Proof (iii) n

3x n »

4U91-5a)!

TS

X

P

Y

Q

R In the diagram, the circles XPYS and XYRQ intersect at the points X and Y, and PXQ, PYR, QSY,

PST, and QTR are straight lines.

i. Explain why STQ = YRQ + YPS.

ii. Show that YRQ + YPS + SXQ = .

iii. Deduce that STQX is a cyclic quadrilateral.

iv. Let QPY = and PQY = . Show that STQ = + .¤

« (i) The exterior angle is the sum of the interior opposite angles. (ii) Proof (iii) Proof (iv) Proof »

4U91-6a)!

A nine-member Fund Raising Committee consists of four students, three teachers and two parents.

The Committee meets around a circular table.

i. How many different arrangements of the nine members around the table are possible if the

students sit together as a group and so do the teachers, but no teacher sits next to a student?

ii. One student and one parent are related. Given that all arrangements in (i) are equally likely,

what is the probability that these two members sit next to each other?¤

« (i) 288 (ii) 1

4 »

4U91-7a)!

i. By assuming that cos(A + B) = cosAcosB - sinAsinB, prove the identity

cos A3

4cos A

1

4cos 3A3 .




ii. Show that x = 2 2 cosA satisfies the cubic equation x3 - 6x = -2 provided cos3A =

1

2 2.

iii. Using (ii), find the three roots of the equation x3 - 6x + 2 = 0. Give your answers to four

decimal places.¤

« (i) Proof (ii) Proof (iii) x = 2.2618, -2.6017 or 0.3399 »




Harder 3 Unit Topics 2: 1990 - 1984 4U90-4b)!

Let L x y1 4 5 1 and L x y2 2 3 5 .

i. Find the point P of intersection of the two straight lines L1 0 and L2 0 .

ii. In the Cartesian plane draw the lines L L L L1 2 1 20 0 6 7 , , , , marking the point P.

Explain why these four lines define a parallelogram.

iii. If a and b are constants, not both zero, explain why aL bL1 2 0 defines a straight line

through P.

iv. Using part (iii), or otherwise, prove that the diagonal through P of the parallelogram defined

in part (ii) has equation: 7 6 01 2L L . ¤

« (i) P(1, 1) (ii)

L1 = 0

L1 = 6

L2 = 7

L2 = 0

y

x

P

L1 = 0 is parallel to L1 = 6 and L2 = 0 is parallel

to L2 = 7. (iii) L1 = L2 = 0 at P so aL1 + bL2 = 0. (iv) Proof »

4U90-5b)!

i. How many different five figure numbers can be formed from the digits 1,2,3,4,5 without

repetition?

ii. How many of these numbers are greater than 45321?

iii. How many of these numbers are less than 45321? ¤

« (i) 120 (ii) 24 (iii) 95 »

4U90-6b)!

D

P

C

BA

In the diagram, AB is a fixed chord of a circle, P a variable point in the circle and AC and BD are

perpendicular to BP and AP respectively. Copy this diagram into your examination booklet.

i. Show that ABCD is a cyclic quadrilateral on a circle with AB as diameter.

ii. Show that triangles PCD and APB are similar.

iii. Show that P varies, the segment CD has constant length.

iv. Find the locus of the midpoint of CD. ¤

« (i) Proof (ii) Proof (iii) Proof (iv) If M is the midpoint of AB and DBC = , then the midpoint of CD

moves on a circle centre M and radius r cos . »




4U90-7b)!

x

y

1 20 n-1 n

2

1

n

y = x

Consider the graph of the function y x .

i. Show that this curve is increasing for all x0 .

ii. Hence show that 1 22

30

... n xdx n nn

.

iii. Use mathematical induction to show that 1 24 3

6

... n

nn , for all integers

n1.

iv. Use parts (ii) and (iii) to estimate 1 2 ... 10 000 to the nearest hundred. ¤

« (i) Proof (ii) Proof (iii) Proof (iv) 666 700 »

4U90-8a)!

You are given that 2cosAsinB = sin(A + B) - sin(A - B).

Let S 1 2 2 2 2 3cos cos cos .

i. Prove that Ssin sin

2

7

2 .

ii. Hence show that if 2

7, then 1 2 2 2 2 3 0 cos cos cos .

iii. By writing S in terms of cos , prove that cos2

7

is a solution of the polynomial equation

8 4 4 1 03 2x x x . ¤

« Proof »




4U90-8b)!

x

y

0

P(t t12

13, )

Q(t t22

23, )

Consider the curve defined parametrically by x t

2,y t

3.

Let P t t12

13, and P t t1

213, be two distinct points on the curve.

i. Write down the equation of the curve in terms of x and y only.

ii. Show that the equation of the chord PQ is given by

t t y t t t t x t t 01 2 12

1 2 22

12

22 .

iii. Hence, or otherwise, show that the equation of the tangent to the curve at a point

corresponding to t, where t 0, is given by 2y3tx t3 0 .

iv. Let R x y0 0, be a point in the plane such that x0

3 y0

2 0. Prove that there are precisely 3

tangents from R to the curve and sketch this on a diagram. ¤

« (i) y2 = x

3 (ii) (iii) (iv) Proof ,

R

x

y

0

»

4U89-3b)!

A public opinion survey of a certain parliamentary proposition finds 47% of the population in favour,

38% opposed and 15% undecided. Three persons are selected at random. Using the expansion

(pq r)3 p

3q

3 r

33p

2q3q

2r3pq

23qr

23rp

23r

2p6pqr or otherwise, find the

probability that:

i. one person is in favour, one opposed and one is undecided;

ii. exactly two persons are opposed;

iii. at least two persons are of the same opinion, either in favour, or opposed or undecided. ¤

« (i) 0.16704 (ii) 0.268584 (iii) 0.83926 »

4U89-4a)!

i. Write expressions for sin( + ), cos( + ) in terms of sin , sin , cos and cos .

ii. Show that tan() tan tan

1 tan tan.

iii. Hence find tan(

4) in terms of . ¤




« (i) sin( ) sin cos cos sin , cos( ) cos cos sin sin . (ii) Proof (iii) tan

tan

1

1 »

4U89-4b)!

x

y p

A

B

R

O

Q

C

FIGURE NOT TO SCALE

The lines AB and BC in the diagram above have equations 3y = -4x + 20 and 4y = 3x - 15

respectively, and meet at B(5,0). BC makes and angle with the x axis. The line PQ has equation

x = 1 and meets the line AB in Q. BR is the bisector of ABC.

i. Show that AB is perpendicular to BC and then copy the diagram into your examination

booklet.

ii. Use (a) to show that BR has equation y = 7x - 35.

iii. The bisector of PQB has slope 1

3 and meets BR at S. Calculate the co-ordinates of S.

iv. Draw SM and SN perpendicular to AB and BC, meeting AB at M and BC at N, respectively.

Prove SM = SN.

v. Show that S is the centre of a circle tangential to PQ, AB and BC and write down the

equation of the circle. ¤

« (i) Proof (ii) Proof (iii) S(6, 7) (iv) Proof (v) (x - 6)2 + (y - 7)

2 = 25 »




4U89-5a)!

A

B

S

Q

C

Y

R

D P

Let ABCD be a cyclic quadrilateral; AB and DC produced meet at P; DA and CB produced meet at Q

as in the diagram.

Let PR be the internal bisector of APD meeting AD at R and BC at S.

Let QY be the internal bisector of DQC meeting PR at Y as in the diagram.

Copy the diagram into your examination booklet and prove that:

i. QRS = QSR;

ii. QY PR. ¤

« Proof »

4U89-7b)!

i. Find real numbers a and b such that x4x

3x

2x 1 (x

2ax1)(x

2bx 1) .

ii. Given that x cos2

5 i sin

2

5 is a solution of x

4x

3x

2x 1 0, find the exact

value of cos2

5. ¤

« (i) a1 5

2

, b

1 5

2

(ii)

5 1

4

»

4U89-8a)!

Find all values with 0 2 such that sin cos 3 1. ¤

«

2

7

6, »

4U89-8b)!

The difference between a real number r and the greatest integer less than or equal to r is called the

fractional part of r, F(r). Thus F(3.45) = 0.45. Note that for all real numbers r, 0 F(r) < 1.

i. Let a = 2136 log102. Given that F(a) = 7.0738.... 10-5

, observe that F(2a) = 14.1476....

10-5

, F(3a) = 21.2214.... 10-5

.

. Use your calculator to show that log101.989 < F(4223a) < log101.990.

. Hence calculate an integer M such that the ordinary decimal representation of 2M

begins with 1989. Thus 2M

= 1989.... .

ii. Let r be a real number and let m and n be non-zero integers with m n.

. Show that if F(mr) = 0, then r is rational.

. Show that if F(mr) = F(nr), then r is rational.

iii. Suppose that b is an irrational number. Let N be a positive integer and consider the

fractional parts F(b), F(2b), ...., F((N + 1)b).

. Show that these N + 1 numbers F(b), ...., F((N + 1)b) are all distinct.




. Divide the interval 0 x 1 into N subintervals each of length 1

N and show that

there must be integers m and n with m n and 1 m, n N + 1 such that F((m -

n)b) < 1

N.

iv. Given that log102 is irrational, choose any integer N such that 1

Nlog

1990

198910 ; note that in

(i), F(a) < log10

1990

1989. Use (iii) to decide whether there exists another integer M such that

2M

= 1989.... . ¤

« (i) () Proof () 4223 2136 (ii) Proof (iii) Proof (iv) There are an infinite number of integers m such

that 2m = 1989... . »

4U88-3a)!

On a particular island, twenty per cent of all turtles survive for four weeks after hatching. Fifteen

turtles hatch on the same day and are tagged for a study.

i. Find, correct to two significant figures, the probability that:

. all fifteen turtles will survive the four weeks;

. none of the turtles survives the four weeks.

ii. Write down expressions for the probability that:

. no more than three turtles survive the four weeks;

. at least three turtles survive the four weeks; ¤

« (i) () 3.3 10-11

() 3.5 10-2

(ii) ()

( . ) ( . ) ( . ) ( . ) ( . ) ( . ) ( . )08015

1080 020

15

2080 020

15

3080 02015 14 13 2 12 3

()

1 0 8015

10 80 0 20

15

20 80 0 2015 14 13 2

( . ) ( . ) ( . ) ( . ) ( . ) »

4U88-3b)!

The population P of a town increases at a rate proportional to the number by which the town's

population exceeds 1000. This can be expressed by the differential equation dP

dt k(P1000),

where t is the time in years and k is a constant.

i. By differentiation show that P = 1000 + Aekt, where A is a constant, is a solution of this

equation.

ii. The population of the town was 2500 at the start of 1970 and 3000 at the start of 1985. Find

its population at the start of the year 2000.

iii. During which year will the population reach 4000? ¤

« (i) Proof (ii) Approximately 3670 (iii) 2006 »




4U88-7b)!

A

B

E

F

G

D

C

ABCD is a cyclic quadrilateral. BA and CD are both produced and intersect at E. BC and AD

produced intersect at F. The circles EAD, FCD intersect at G as well as at D. Prove that the points E,

G, and F are collinear. ¤

« Proof »

4U88-8a)!

A B

L

E F

GH

D

L

h C

A building is in the shape of a square prism with base edge L metres and height h metres. It stands on

level ground. A base diagonal AC is produced to a point K. From K it is found that the angles of

elevation of F and G are 30 and 45 respectively. Prove that: h

t

2 10

4. ¤

« Proof »




4U88-8b)!

Newton's method may be used to determine numerical approximations to the real roots of the equation

x3 = 2. Let x1 = 2, x2, x3,...xn,... be a series of estimations obtained by iterative applications of

Newton's method.

i. Show that xn1 2

3xn

1

xn2

.

ii. Show algebraically that xx x

xn

n n

n

1

3

3 2 3

222 2 2

3

( ) ( ).

iii. Given that xn 23 , show that x xn n 13 3 22 2( ) .

iv. Show that x12

and 23 agree to at least 267 decimal places. ¤

« Proof »

4U87-5ii)

Five letters are chosen from the letters of the word CRICKET. These five letters are then placed

alongside one another to form a five letter arrangement. Find the number of distinct five letter

arrangements which are possible, considering all possible choices. ¤

« 1320 »

4U87-7i)!

ABC is an isosceles triangle with AB = AC. Let Q be a point on the base BC between B and C. AQ

produced meets the circle through the points A, B, C at P.

a. Prove that triangle BQP is similar to triangle AQC.

b. Show that BP.CQ = PQ.AC .

c. Prove that 1

BP

1

CP

1

PQ.BC

AC. ¤

« Proof »

4U87-7ii)

a. Prove using mathematical induction that for n 1, 11

22

1

32...

1

n2 2

1

n.

b. Prove that 1.45 < 11

22

1

32...

1

9921.99. ¤

« Proof »

4U87-8i)

Write down the general solution of the equation sin2 + cos5 = 0. ¤

«

=(4n +1)

6 or

(4n-1)14

»

4U86-2)

The functions S(x), C(x) are defined by the formulae S(x) = 1

2(e

x e

x) and C(x) =

1

2(e

x e

x) .

i. a. Verify that S´(x) = C(x).

b. Show that S(x) is an increasing function for all real x.

c. Prove that {C(x)}2 = 1 + {S(x)}

2.

ii. a. S(x) has an inverse function, S-1

(x), for all values of x. Briefly justify this

statement.

b. Let y = S-1

(x). Prove that dy

dx x

1

1 2( ).

c. Hence, or otherwise, show that S-1

(x) = log ( ( ))e x x 1 2 .




d. Show that dx

x xe

( )log

20

1

2 2

2 5

1 2

. ¤

« (i) Proof (ii) Proof »

4U86-7i)

Find all x such that cos2x = sin3x, and 0 x 2

. ¤

« x 2

10

, »

4U86-7ii)

Appropriate diagrams should accompany each of your solutions to this section.

Z

S

Y

X

a. In the figure, ZS is the tangent to the circle at Z, and X, Y are any two points on the circle.

By drawing the diameter through Z, or otherwise, prove that YZS = ZXY.

b. In the given figure, XY produced meets ZS at P. The lengths PX, PY and PZ are x, y, z,

respectively. Prove that z2 = xy.

c. Two unequal circles intersect at L, M. The common tangent AB touches the circles at A, B.

Prove that LM produced bisects AB. ¤

« Proof »

4U86-8ii)

A committee of 4 women and 3 men are to be seated at random around a circular table with 7 seats.

What is the probability that all the women will be seated together? ¤

« 1

5 »

4U86-8iii)

The function f(x) is given, for x > 0, by f(x) = 2logex - x

21

x.

a. Show that the only zero of f(x) occurs at x = 1.

b. Let g(x) xloge x

x2 1, for x > 0 and x 1. Show that 0 < g(x) <

1

2. ¤

« Proof »

4U85-2i)

a. Find the turning points of the cubic polynomial p(x) = x3 - x

2 - 5x - 1, and without attempting

to solve the equation, show that the equation p(x) = 0 has three distinct real roots, two of

which are negative.

b. Sketch the graph of p(x).

c. Starting with the approximation x = 0, use one application of Newton’s method to estimate a

root of the equation p(x) = 0.

d. What initial approximation would you use to estimate the positive root of p(x) = 0 by

Newton’s method? State briefly your reasons for this choice. ¤




« (a) 5

3

202

27,

is a relative minimum and (-1, 2) is a relative maximum (b)

53

20227,

x

y

(-1, 2)

-1

3-2

(c) x -0.2 (d) x0 = 3. The tangent to the curve at x0 = 3

cuts the x-axis at a point closer than x0 3 . »

4U85-5i)

A thin wire of length L is cut into two pieces, out of which a circle and a closed square are to be

formed, so that the sum of the areas of the circle and square so formed is a minimum. Show that this

minimum value is L

2

4( 4). ¤

« Proof »

4U85-7i)

Given that sinx + siny = 2sinAcosB, find values for A and B in terms of x and y. Solve the equation

sin + sin2 + sin3 + sin4 = 0, giving all solutions in the interval 0 2. ¤

« A (x y)12 , B (x y)1

2 , 0, 25 ,

2, 4

5 , , 65 , 3

2 ,85

»

4U85-7ii)

a. A, B, C are three points lying on a given circle, and P is another point in the same plane.

Write down two different angle tests to determine whether A, B, C, P are concyclic (i.e. P

also lies on the given circle).

b. In an acute-angled triangle with vertices L, M, N, the foot of the perpendicular from L to MN

is P, and the foot of the perpendicular from N to LM is Q. The lines LP, QN intersect at H.

. Draw a clear diagram showing the given information.

. Prove that PHM = PQM.

. Prove that PHM = LNM.

. Produce MH to meet LN at R. Prove that MR is perpendicular to LN.

c. What general result about triangles is proved in (b)? ¤

« (a) 1: APB = ACB or BPC = BAC, 2: APC + ABC = 180 or BAP + BCP = 180 (b) ()

M P N

L

Q

H

() Proof () Proof () Proof (c) The altitudes of a triangle

are concurrent. »

4U85-8i)




a. In how many ways can 4 persons be grouped into two pairs to play a set of doubles tennis?

b. The eight members of a tennis club meet to play two simultaneous sets of doubles tennis on

two separate but otherwise identical courts. In how many different ways can the members of

the club be selected for these two sets of tennis? ¤

« (a) 3 (b) 315 »

4U85-8ii)

a. Show that for k 0, 2k + 3 > 2 ( )( )k k 1 2 .

b. Hence prove that for n 1, 11

2

1

3

12 1 1 ... ( )

nn .

c. Is the statement that, for all positive integers N, 1

101

10

kk

N

true? Give reasons for your

answer. ¤

« (a) Proof (b) Proof (c) 1

k10

k 1

n10

for N (5 10 1) 19 2 The statement is not true. »

4U84-7i)

In how many ways can the five letters on the word CONIC be arranged in a line so that the two

(indistinguishable) C’s are separated by at least one other letter? ¤

« 36 »

4U84-7ii)

It is given that x, y, z are positive numbers. Prove that:

a. x2 + y

2 2xy;

b. x2 + y

2 + z

2 - xy - yz - zx 0.

Multiply both sides of the inequality (b) by (x + y + z) to obtain

c. x3 + y

3 + z

3 3xyz.

Deduce from (c) or prove otherwise, that

d. (x + y + z)(x-1

+ y-1

+ z-1

) 9.

Suppose that x, y, z satisfy the additional constraint that x + y + z is equal to 1. Is it true that the

minimum value of the expression x-1

+ y-1

+ z-1

is equal to 9? Justify your answer. ¤

« (a) Proof (b) Proof (c) Proof (d) Yes »

4U84-8i)

Write down expressions for sin( + ), cos( + ) in terms of sin, cos, sin, cos.

Deduce that tan( + ) = tan tan

1 tantan, and

tan( ) tan tan tan tantantan

1 tan tan tan tan tantan.

By means of the substitution t = tan, transform the equation sin4 + asin2 + bcos2 + b = 0 into a

cubic equation in t. (a, b are real constants, a 2). Suppose the roots of the transformed equation are

tan, tan, tan . Show that + + is a multiple of . ¤

« sin( ) sin cos cos sin + , cos( ) cos cos sin sin ,

(a 2)t bt (a 2)t b 03 2 »

4U84-8ii)

A woman travelling along a straight flat road passes three points at intervals of 200m. From these

points she observes the angle of elevation of the top of the hill to the left of the road to be respectively

30°, 45°, and again 45°. Find the height of the hill. ¤

« 200m »

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