F.2 Maths(open-ended Qs)

Question 1:

I). four different factors of degree 1
y = (x +a)(x + b)(x+c)(x + d) where a, b, c, and d are real numberS.
e.g. y = (x +1) (x - 3) (x -5) (x + 7)

II). two different factors of degree 2
y = (ax^2 + bx + c) (dx^2 + ex + f) where a, b, c, d, e and f are real numberS.
e.g. y = (x^2 + 2x - 5) (x^2 - 3x - 9)

III).one factor of degree 2 and two factors of degree 1
y = (ax^2 + bx + c)(x + d)(x + e) where a, b, c, d, and e are real numberS.
e.g. y = (x^2 + 2x - 5) (x -5) (x + 7)


Question 2:

Two simultaneous linear equations in two unknowns have 0, 1 or infinite number of solutions. How do we know that? State your reasoning.

Two linear equations with different slopes will have one solution.
(These two lines will intersect at a point., (x, y) That gives the solution.)

Two linear equations with the same slopes, but different y – intercept will have no solution.
(These two lines are parallel to each other and do not intersect. There is no solution.)

Two linear equations with the same slopes, and same y – intercepts will have infinite solution.
(These two lines are the same. One superimposes on the other. There is infinite  solution.
e.g. 2x + y – 5 = 0
      4x + 2y – 10 = 0