1977-10-04 — Page 19

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育教僑華

數學

1978

「中學會考試題預習專欄

Mathematics

明德社主權

文長波

This course provides the candicates of HKCEE 1978 a general revision in Mathematics (Alternate Syllabus B). A knowledge

of mathematica upto KCEE is assumed, and the subject is developed by a concentric treatment in which Beach exercise is used to illustrate ideas already treated.

Exercise I

Attempt All questions in Section A and any Six questions in Section B

Section A

1. Find the value of x if

logy

*

((1+41og2-10g5),

the base of logarithm being 10.

Form a quadratic equa t'i on whose roots are the square of the roots.

-

60

Express your equation in the form ax

where a, b and care

integers.

F

where

Solve the equation, 2sino tano

0

tana

and

180° < A < 360°, find

the value of

58inA

*:

Осоз

7c08A-

Join

5. In Tigure

ᎪᎠ BE CF DB EC FA

Find the ratio of the

area of the triangles DEF and ABC. (Geometry theorems need: not be quoted when used)

C.

In figure, the circle touches the sides of the triangle ANC. If AB = BC a and AC b, express BX in terms of a, b and (Geometry theorems need. not be quoted when used.)

X.

A and B enter into

partnership with capitals

$4000 and $2400 --

respectively. At

At the ends

of 4 months.. B withdraws $600. If the profits in

that year is $1590, what

share of the profit should I get?

8. An alloy of zinc and tin

contains 35% by weight of

zinc. Find the weight of zinc which must be added. to 400gm of this alloy

if the final percentage of zine is to be 66

Section B.

9. In Figure, QR = ST,

PS and 2 PQR = Z EST. CRX is

a straight line. Prove that

(5). PRKT is a cyclic (b), TX Lisects 4 X".

10. A rectangular plot of Land is surrounded by 700m fence, Three- quarters of the area of the plot is a rectangular plaving field and the remaining aren is a path 10m wide surrounding the playing field. Find the dimension of the plot. 11. If BCD is a rectangular

door hinged along RC, and if AB = 1.5m and BC = 2.5m., calculate the angle between the initial and final positions of the diagonal AC when the door is opened through an angle" of 60°

12. A man borrow 88000 to be

repaid with interest at 97.4. in 2 equal annual instalments, the first. payment being due at the end of the first year, Find the annual payment correct to the nearest $10.

13. If x(x+

1

X

X

is satisfied for all values of x find the values of A. and, B. Hence, find

1

+

2 x

1

3 x

in terms of n.

+

15. In the figure, AC 1 BD,

YX BC. Prove that

•) ¿YAP ZYPAL*

(b) AY = YD = YP.

15. If

show that.

+ Logb). 16. A sells an article to B

at a gain of 10. B sells it to C at a gain of 20%, If C paid $320 more than it cost A, find

(a) how much A, B and respectively paid

for the article.

(b) the respective profits.

for A and B.

新數學 (一) 魯榮家

MODERN MATHEMATICS (1);

This course provides the candidates of H.K.C.E.E.1978 a general revision in Modern Mathematics, Conventional" test papers are set up for rehearsal purpose, with ang- wers given in the following weeks.

TEST ONE

Attempt all questions in

section A and

any six in section B. All workings must be clearly shown, SECTION A (40%)

1. In figure 1, ABCD is a parallelogram with area 72cm,

I, K are the mid-points of

AB and CD, find the area of

the shaded region,

figure 1

Find the value of x if·

2b 3c

a 30

3. Find the value of k such

that

2

x+y+8kx 20y+ 125 × 0 represents a point circle.

4.Find the values of A, B and

C if

2

5x1

- A (x2

+x+2)+(Bx+C ) (x−1)

5. Solve the inequality

(-2) 7-*

6. A function :/B-R is

defined by

f(x) + x + 5

=

find the value of

7. Iftanė

value of

r22 (5) find

the

figure

3ain0 + 4cosů. 5sinë - 6¢os0

8. Find the value of m if the

equation

x(x-1)=(+1)---

has equal roots.

SECTION B (60%)

Prove by mathematical

induction

2n+1

by 11, where

is a natural number.

10. Find the radius and cen-

tre of the cirèle passing through (1,1), (1,3) and centred at 2x 3y+6 = 0,

11. An equilateral triangle

has a perimeter of 30cm. Another triangle is formed by joining the mid-points of the

sides of the given triangle,

another is formed by joining

the mid-points of the sides

of the 2nd triangle and so on

Find

(i) the sum of perimeters

(ii) the sum of areas

of the triangles so foru-

12. The cost of

painted ball

as the cube.

partly

radius

the radiating a

and partly as the square of

If the cost of a

painted of radius 12cm is

$37.5 and that of radius 8cm

$153, find the cost of a

ball whose radius is 4cm.

13. In figure 2, VARCD repre sents a right pyramid with

rectangular base, If the

height VG is 10cm, AB - 8cm

and BC-

angle.

between

6cm, Find th

(i) line VA and plane ABCD (ii) planes VAB and ABCD (iii) planes VAD and ABCO

>

Bem 14. 200 students took examin-

ation in English, Chinese and

Mathematics, 25 passed in

Chinese and English, 15 passed

in Chinese and Mathematics,10

passed in Mathematics and English, 30 passed in English only, 40 passed in Chinese only, 20 passed in Mathematica only, There are altogether 40

passed in Mathematics

(a) If a student is picked at

find the probability

he passed in (i) All three subject (ii) no gubject

(iii) one subject only (b) If a student is picked from those who passed in *ug-+

lish find the probability" that he also passed in Mathe-

matics.

15. Plot the graph of

2

y= x -2x + 1 (-5<x<5) and solve graphically

*

-2x

(11) x2 -x - 6

2

(lii) 2x2 - 5x – 12 – 0

16. In figure 3, ACB is diameter of the circle, centre C, P is a point on the circum- ferenceof the circle. M-2A}

ť CP - 8 and CB

7. (i) Express All, BP and B in

terms of 8 and t.

W

(ii) Show that as I moves on

the circumference of the

circle, the magnitude of BR remains constant. What is the locus of 2 (11) Hence, find the equation

of the locus of R if

(2,3) and B

(6,0).