TS, the question will not appear in "O" level Emaths and Add
Maths as it is a
combination of quadratic function, inequality and area, and
probability. It can be
tested for H1 Maths.
Nevertheless, the question is fun.
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Question
Given that a and b are real numbers such that 0 < a < 1
and 0< b < 1 find the probability that x^2 + sqrt(a)x + b = 0
have real roots.
Steps
(1) 1x^2
+ sqrt(a)x +
b = 0
ax^2 +
bx + c = 0
(Quadratic Equation)
So, a = 1 , b =
sqrt(a) , c = b
(2) For real roots, b^2 - 4ac > = 0
[sqrt(a)]^2
- 4(1)(b) > = 0
a
- 4b > = 0
b < = a/4
(3) Since a and b are real numbers such that 0 < a <
1 and 0< b < 1,
draw a 1 x 1 box with "b" as the
vertical axis and "a" as the horizontal axis.
The area of the box is 1 x 1 = 1
units^2
(4) Next, draw the inequality b
< = a/4 in the box.
First, draw the line b = a/4.
When a = 1, b = 1/4
So, the inequality will be the
area below the line b = a/4.
The required area will
be the area of the triangle enclosed by the line
b = a/4 , a = 1 and b =
0
(5) Area of the triangle = 1/2 x base x height
= 1/2 x 1/4 x 1
= 1/8 units^2
(6) Hence, given that a and b are real numbers such
that 0 < a < 1 and 0< b < 1
the probability that x^2 +
sqrt(a)x + b = 0 have real roots
area of the
triangle
= -----------------------------
area of the
square box
1/8 units^2
=
-------------------------
1 units^2
=
1/8