1) A parabola with equation y=x^2+bx+c passes through the points (-1,-11) and (3,17). What is c?

2)A point P is randomly placed in the interior of the right triangle below. What is the probability that the area of triangle PBC is less than half of the area of triangle ABC? Express your answer as a common fraction,

3) The first six rows of Pascal's triangle are shown below, beginning with row zero. Except for the 1 at each end, row 4 consists of only even numbers, as does row 2. How many of the first 20 rows have this property? (Don't include row 0 or row 1).

THX FOR THE HELP IN ADVANCE

Guest Aug 15, 2019

edited by
Guest
Aug 15, 2019

#1**+1 **

1) A parabola with equation y=x^2+bx+c passes through the points (-1,-11) and (3,17). What is c?

We have that

(-1)^2 + b(-1) + c = -11 ⇒ 1 -b + c = -11 ⇒ -b + c = -12 (1)

and

(3)^2 + b(3) + c = 17 ⇒ 9 + 3b + c = 17 ⇒ 3b + c = 8 ⇒ -3b - c = -8 (2)

Add (1) and (2) and we get that

-4b = -20

b = 5

So....using (1) -b + c =-12

-5 + c = -12 subtract 10 from both sides

c = -7

CPhill Aug 15, 2019

#2**+1 **

The Area of triangle ABC = (1/2)(B)(AB)

Draw DE parallel to CB such that DE is at a height of (1/2)AB

We have a trapezoid DEBC....the area of this trapezoid is

(1/2) (1/2) (AB)(( DE + BC)

But using similar triangles....DE = 1/2BC

So...the area of trapezoid DEBC = (1/2) (1/2)(AB) ( [1/2 BC] + BC) = (1/4)(AB)( 3/2 )(BC) = (3/8)(AB)(BC)

And if P is located inside this trapezoid.....the area of triangle PBC will be less than (1/2) that of ABC

So.....the probability is [ DEBC] / [ABC ] = (3/8)(AB)(BC) / [ (1/2) (AB) (BC) ] = (3/8)/(1/2) = 3/4

CPhill Aug 15, 2019

#4**+1 **

3) The first six rows of Pascal's triangle are shown below, beginning with row zero. Except for the 1 at each end, row 4 consists of only even numbers, as does row 2. How many of the first 20 rows have this property? (Don't include row 0 or row 1).

The even rows occur at the 2^n rows where n is an integer ≥ 1

So rows 2, 4, 8, 16 have the property that all the entries except fo the beginning and ending 1s are even

CPhill Aug 15, 2019