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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).