Answer:
Height = 5
Minimum number of vertices in a binary tree
whose height is h.
So, there must be At least one node at each of first h levels.
Minimum number of vertices = h
So, Minimum number of vertices of height 5 is 5
The maximum number of nodes in a binary tree of height h = 
Substitute h = 5
The maximum number of nodes in a binary tree of height 5 = 
The maximum number of nodes in a binary tree of height 5 = 63
So, the maximum number of vertices in a binary tree with height 5 is 63
The formula of the sum of an arithmetic sequence:
We have:
substitute:
Answer: b. 21
1. Is false
It is false because -7+9 equal 2 and -9+7 equal -2
The first number is positive, so the second has to be positive too so that their product can be a positive number which is 48.
Suppose the first number is x and the second number is y.
We know that:
y = 4 + 2x (4 more meaning adding 4 to sthing and that sthing is twice x hence the 2x)
Moving on, we have x.y = 48 (we replace y by its other value that has x in it so that we have only one unknown variable that we need to find)
<=> x.(4 + 2x) = 48
<=> 4x + 2x^2 = 48
<=> 2x^2 + 4x - 48 = 0 (we add (-48) on both sides)
Now we have to solve the second degree equation, I don't know the exact names of things since I didn't study math in English, but we have to calculate the value of, say, b^2 - 4ac (ax^2 + bx + c)
in this case b = 4 , a = 2, c = -48
so b^2 - 4ac = 16 + 384 = 400, so the square root of that equals 20
now, we have two possible values for x, suppose x1 and x2
x1 = (-4 + 20)/2.a = (-4 + 20)/4 = 16/4 = 4
x2 = (-4 - 20)/2.a = (-4 - 20)/4 = (-24)/4 = (-6) (which is impossible giving that the two numbers must be positive)
So x has to be equal to x1 => x = 4
We find x, and since y = 4 + 2x <=> y = 4 + 2.4 = 4 + 8 = 12
As a conclusion, x = 4 , y = 12
