Mathematics Magazine for Grades 1-12  

Grade 10

Theory:

A quadratic function is any function equivalent to one of the form

f(x) = ax² + bx + c

Here are some examples of quadratic functions

Ø       f(x) = x² – 9

Ø       f(x) = x² + 2x +1

A quadratic equation is any equation equivalent to one of the form

ax² + bx + c = 0

Here are some examples of quadratic equations

Ø       x² – 9 = 0

Ø       x² + 2x +1 = 0

Solutions from the Previous Issue:

1.        Prove that

Solution:

=

      =

=

2.        a + b + c= π. Prove that
cos(a) ∙cos(a) +cos(b) ∙ cos(b) + cos(c) ∙cos(c) + 2cos(a) ∙cos(b) ∙ cos(c) = 1

Solution:

a+b = pi - c

=>  cos(a + b) = cos(π - c)

=>  cos(a + b) = -cos(c)

=>  cos(a) ∙cos(b) - sin(a) ∙sin(b) = - cos(c)

=>  cos(a) ∙cos(b) + cos(c) = sin(a) ∙sin(b)

=>  (cos(a) ∙ cos(b) + cos(c))²  = (sin(a) ∙sin(b))²

=>  [cos(a) ∙cos(b)]² +2.cos(a) ∙cos(b) ∙cos(c) + cos²(c)

=>  [1 -  cos²(a)] ∙ [1 - cos²(b)]

=>   cos(a) ∙cos(a) + cos(b) ∙cos(b) + cos(c) ∙cos(c) + 2cos(a) ∙cos(b) ∙cos(c) =1

3.        Prove that: sin(a) + sin(b) - sin(c) =

sin(a + b + c) + 4sin sin sin  

Solution:

        sin(a) + sin(b) =

        sin (a + b + c) – sin (c) = 2

        Then

        sin(a)+ sin(b) + sin(c) - sin(a + b + c)

        =

        =2                

        sin(a)+ sin(b) + sin(c) =

        = sin(a + b + c) +2     

4.        Solve: 2∙tan(x) = sin(4x) - 2.sin(2x) ∙tan(x) ∙cos(2x) ∙tan(x)

Solution:

2∙tan(x) = sin(4x) - sin(4x) ∙tan² (x)

<=>      2∙tan(x) = sin(4x) ∙ (1 - tan²(x))

<=>     2∙sin(x) ∙ cos(x) = sin(4x) ∙ (cos² (x) - sin² (x))  and cos(x) not 0

<=>     sin(2x) - 2∙sin(2x). cos(2x) ∙ cos(2x)  = 0 and cos(x) not 0

<=>     sin(2x) ∙ [1 - 2∙cos² (2x)] = 0 and cos(x) not 0

<=>     2∙sin(x) ∙ cos(x) ∙ [1 - 2∙ cos² (2x)] = 0 and cos(x) not 0

<=>     sin(x) = 0  or cos(2x) = or cos(2x) =-

<=>     x = k∙ π  or 2x =

<=>     x = k∙ π  or x =

5.        Prove that: sin(2∙arctan(x)) =

Solution:

Let arctan(x) = a => tan(a) = x

sin(2 arctan(x)) = sin(2a) = 2sin(a)cos(a) = 2 tan(a).cos(a).cos(a) 

 =

Because:

Proposed Exercises:

Plot the inequalities:

1.        (x - 1)² + (y+2)²  ≥  9

2.        2x + 3y – 5 ≤ 0

3.        1 < x² + y² ≤ 4

4.        y ≤ x³ +2x – 1