New Senior Secondary Mathematics
Advanced Exercise Ch. 01: Quadratic Equations I
ADVANCED EXERCISE CH. 01: QUADRATIC EQUATIONS I
[Finish the following questions if you aim at DSE Math Level 4]
Q1 [Misc]
2
Solve the equation ( x − 4 x ) − 5 ( x − 4 x ) + 4 = 0 .
2
2
[Finish the following questions if you aim at DSE Math Level 5]
Q2 [Misc]
2
5
4
3
2
Given that r is a root of the quadratic equation x − 3 x − 5 = 0 , find the value of r − 3r − 5r + 2r − 6r .
Q3 [Misc]
Tom and Mary are due south and due east of a bus stop respectively, and the distance between them is 60 m. They walk towards the bus
stop at the same time with their own constant speeds. After 6 seconds, they are both 27 m away from the bus stop. If Mary walks slower
than Tom by 2 m s-1, find their speeds.
Q4 [Misc]
The figure shows the graph of y = x − 6 x + 10 , where x ≥ 0 . R ( p, q) is a point on the graph. P ( p, 0) and Q (0, q) are points on
2
the x-axis and the y-axis respectively. A rectangle OPRQ is drawn.
(a)
(b)
Express q in terms of p.
If the perimeter of rectangle OPRQ is 12 units, find the possible values of p.
[Finish the following questions if you aim at DSE Math Level 5*]
Q5 [Misc]
By completing the square, solve x − 2ax + ( a − b − 2bc − c
2
Page 1
2
2
2
) = 0 , where a, b and c are real numbers.
New Senior Secondary Mathematics
Advanced Exercise Ch. 01: Quadratic Equations I
Q6 [Misc]
2
The figure shows the graph of y = −2 x + 16 x − 24 which cuts the x-axis at two points M and N, and passes through two points P and
Q which are both above the x-axis. It is given that area of ∆PMN = area of ∆QMN = 12 sq. units.
(a)
(b)
(c)
Find the coordinates of M and N.
Find the coordinates of P and Q.
If R is a point on the graph other than P and Q, is it possible that area of ∆RMN = 12 sq. units? Explain your answer.
Q7 [Misc]
2
2
2
2
Prove that if a, b and c are rational numbers, then the roots of the equation x − 6ax + 9a − 4b + 12bc − 9c = 0 are rational.
[Finish the following questions if you aim at DSE Math Level 5**]
[No question in this chapter can be set at 5** level.]
[Finish the following questions if you want extra knowledge / aim at difficult questions]
Q8 [Misc.]
Prove that for real numbers a and b, if ab = 0 then a = 0 or b = 0 .
Q9 [CE AMath 89I 8] (Modified)
Define the absolute value of a real number x as
 x
(for x ≥ 0)
x =
.

−x
(for x < 0)
(a)
Solve the equation x − 2 = 3 .
(b)
Solve ( x − 2) − 5 x − 2 + 6 = 0 .
2
Answer
Q1 x = 2 ± 5 or 2 ± 2 2
Q5 x = a ± (b + c)
Page 2
Q2 10
2
Q3 Tom: 3.5 m s-1 Mary: 1.5 m s-1 Q4 (a) q = p − 6 p + 10 (b) 1 or 4
Q6 (a) M ( 2, 0) N (6, 0) (b) P (3, 6) Q (5, 6) (c) Yes
Q9 (a) 5 or -1 (b) -1, 0, 4 or 5