F4 maths

1. y = -2x^2 + 12x + c

It intersects the x-axis at two points.

So, -2x^2 + 12x + c = 0 has two distinct real roots

Discriminant > 0

(-12)^2 - 4(-2)c > 0

144 + 8c > 0

16 + c > 0

c > -16


2. y = ax^2 - 6x + 5 does not intersect the x-axis.

That is, ax^2 - 6x + 5 = 0 has no real root

Discriminant < 0

(-6)^2 - 4a(5) < 0

36 - 20a < 0

9 - 5a < 0

5a > 9

a > 9/5


3. 2x^2 - 4x + (c - 1) = 0 has real roots,

Discriminant >= 0

(-4)^2 - 4(2)(c - 1) >= 0

16 - 8(c - 1) >= 0

24 - 8c >= 0

8c <= 24

c <= 3

So, the possible values of c are 1, 2 or 3.