1999滑铁卢竞赛试题

hours Calculators are permitted , provided they are non-programmable and without graphic displays.Do not open this booklet until instructed to do so. The paper consists of 10 questions, each worth 10 marks.Parts of each question can be of two types. 1-2) or 3 marks each (questions 3-7). Instructions for SHORT ANSWER parts

SHORT ANSWER parts are indicated like this:

Enter the answer in the appropriate box in the answer booklet.be given for a correct answer which is placed in the box. Part marks will be awarded parts are indicated like this:

Finished solutions must be written in the appropriate location in the answer booklet.IBM

Canada Ltd.

Sybase Inc.(Waterloo)

Canadian Institute of Actuaries

for the

Chartered Accountants

NOTE:

1.Please read the instructions on the front cover of this booklet.

2.Place all answers in the answer booklet provided.

3.

For questions marked “

”, full marks will be given for a correct answer which is to be

placed in the appropriate box in the answer booklet. Part marks will be given for work shown . Students are strongly encouraged to show their work.

4.

It is expected that all calculations and answers will be expressed as exact numbers such as 4π, 27+, etc., except where otherwise indicated.

1.(a)If x –––11134=+, what is the value of x ?

(b)If the point P –,32() is on the line 375x ky +=, what is the value of k ?(c)

If x x 220––=, determine all possible values of 1162

–

–x x .2.(a)

The circle defined by the equation x y +()+()=43922

– is moved horizontally until its centre is on the line x =6. How far does the centre of the circle move?

(b)

The parabola defined by the equation P and Q . If a b ,() is the mid-point of the line segment (c)

shown in the diagram.

3.(a)

triangle of side 10 cm?

(b)

of Alphaville decreased by 2.9% during 1996, then increased by 8.9% during 1997, and then increased by 6.9% during 1998. The population of Betaville increased by r % in each of the three years. If the populations of the towns are equal at the end of 1998, determine the value of r correct to one decimal place.

4.(a)

In the diagram, the tangents to the two circles intersect at 90° as shown. If the radius of the smaller circle is 2, and the radius of the larger circle is 5, what is the distance between the centres of the two circles?

(b)

a seat is at its lowest point which is 2 m above the ground. Determine how high the seat is above the ground at t =40 seconds.

5.(a)

A rectangle PQRS has side PQ on the touches the graph of y k x =cos R as shown. If the length of PQ is π3

and the area

of the rectangle is

53

π

, what is the value of (b)

In determining the height, MN , of a tower on an island, two points A and B , 100 m apart, are chosen on the same horizontal plane as N . If ∠=°NAB 108,∠=°ABN 47 and ∠=°MBN 32,

determine the height of the tower to the nearest metre.

6.(a)

The points A , P and a third point Q (not shown) are the vertices of a triangle which is similar to triangle ABC . What are the coordinates of all possible positions for Q

?

(b)

Determine the coordinates of the points of intersection of the graphs of y x =+()1110–log .

7.(a)On the grid provided in the answer booklet, draw the graphs of the functions y x =+–21and y x =–2. For what value(s) of k will the graphs of the functions y x =+–21 and

y x k =+–2 intersect? (Assume x and k are real numbers.)(b)

Part of the graph for y f x =() is shown, 02≤

()212

for all real

values of x , draw the graph for the intervals, –20≤

8.(a)

The equation y x ax a =++22 represents a parabola for all real values of a . Prove that each of these parabolas pass through a common point and determine the coordinates of this point.(b)

The vertices of the parabolas in part (a) lie on a curve. Prove that this curve is itself a parabola whose vertex is the common point found in part (a).

9.

A ‘millennium ’ series is any series of consecutive integers with a sum of 2000. Let m represent the first term of a ‘millennium ’ series.(a)Determine the minimum value of m .(b)Determine the smallest possible positive value of m .

10.

ABCD

is a cyclic quadrilateral, as shown, with side AD d =, where d is the diameter of the circle. AB a =,BC a = and CD b =. If a , b and d are integers a b ≠,(a)prove that d cannot be a prime number.(b)determine the minimum value of d .