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1. (a) It is given that ( = {1, 2, 3, 4, 5, ..., k} where k is a positive integer.
P is a subset where P = {x: x is a prime number}.
Q is another subset and Q = {x: the unit digit is 9}
(i) List all the members of Q given that k = 100
(ii) If n(P ( Q) = 4, list the elements of the set P ( Q and hence state the largest possible value of k.
(b) It is given that n(() = 23, n(A ( B) = x, n(A) = y, n(B) = 2y and n(A` ( B`) = 7.
(i) Use a Venn diagram and show that 3y = 16 + x
(ii) Hence find the least possible value of y.
2. Find the equation of the perpendicular bisector of AB where A is (3, 10) and B is (7, 2)
3. The roots of the quadratic equation x2 -
x +
= 0 are ( and (
Without using a calculator, show that
+
=

4. The tangents TA and TD are drawn from a point T to the circle, centre O. The diameter DB and the tangent TA when produced meet at E. Given that ( ADO = 26o, find
(a) ( ADT
(b) ( ATD
(c) ( BAE
(d) ( ACD ( C is a point on the cirlce, centre O)

Tôn Nữ Bích Vân giới thiệu













Solution:
1. (a) (i) Q = {9, 19, 29, 39, 49, 59, 69, 79, 89, 99}
(ii) P ( Q = {19, 29, 59, 79} ; k = 88
(b)





least value of y = 6 [when x = 2]
2. The mid-point of AB is (5, 6). The gradient of the line AB is
= -2. The perpendicular is the
line with gradient
and passing through the
point (5, 6). Equation of the perpendicular of AB is:
y - 6 =
(x - 5)
y =
x +

Alternative method
Let P(x, y) lie on the perpendicular of AB.
Therefore PA = PB
(
=

( x2 - 6x + 9 + y2 - 20y + 100 = x2 - 14x + 49 + y2 - 4y + 4
Simplifying, 8x - 16y + 56 = 0
( y =
x +

3. The quadratic equation whose roots are ( and ( is (x - () (x - () = 0
x2 - (( + ()x + (( = 0
By inspection: ( + ( =

(( =

Now
[Hence shown]
4. (a) (ODT = 90o (tan ( rad. )
(ADT = 90o - 26o
= 64o
(b) TA = TD (tangents from an external point)
(TDA = (TAD (base (s of isos. ()
(ATD = 180o - 64o - 64o (( sum of ()
= 52o
(c) (BAE = (ADB (( in alt. segment) = 26o
(d) (ACD = (ADT (( in alt. segment) = 64o