[S3] Math question

1. Simplify the following expressions, where x is a positive integer, and
express your answers with positive indices.
a. 3^(-2x) * 27^(x+1) / 9^(-2x)
= 3^(-2x) * 3^(3x+3) / 3^(-4x)
= 3^(-2x+3x+3+4x)
= 3^(5x+3)

b. [ 2a^(3-2x) / 5a^(4x-1)]^-2
= [2/5 * a^(3-2x-4x+1)]^-2
= (2/5)^(-2) a^-2(4-6x)
= 25/4 * a^(12x-8)
= 25a^(12x-8) /4

2.Factorize the following expressions.
a. (b-1)^2 - 10(b-1) + 25
= [(b-1) - 5]^2
= (b-6)^2

b. 6b^2 - 17/2 b + 3
= (12b^2 - 17b + 6)/2
= (3b-2)(4b-3)/2

c. 20x^2 + 44xy - 15y^2
= (10x-3y)(2x+5y)

3.In each of the following, determine whether AB and CD are parallel,
perpendicular or neither of them
a. A(0,8), B(-4,3), C(5,0) and D(1,-5)
b. A(-6,2), B(0,5), C(-1,4) and D(3,-4)

4. In each of the following. P is a point on the line segment AB and it
divides AB in the given ratio. Find the coordinates of P.
a. A(-1,4), B(5,-2), AP:PB=1:2
P
= ((-1*2 +5*1)/(1+2) , (4*2+-2*1)/(1+2)
= (1,2)

b. A(-3,-1), B[6,7/2], AP:PB = 2:7
P
= ((-3*7+6*2)/(2+7) , (-1*7+7/2*2)/(2+7))
= (-1,0)

5. It is given that A(-2,3), B(4,-5) and C(12,1) are the vertices of a triangle.
a. Find the lengths of AB, BC and AC
(Leave your answers in surd form if necessary.)
AB
= √[(-1-12)^2+(3-1)^2]
= √(13^2+2^2)
= √173
BC
=√[(4-12)^2 + (-5-1)^2]
= √(8^2 + 6^2)
= √10^2
= 10
AC
= √[(-2-12)^2+(3-1)^2]
= √(14^2+2^2)
= √200
= 10√2

b. Is triangleABC a right-angled isosceles triangle? Explain your answer.
Since AB =/= BC, AB=/=AC, AC=/=BC
Therefore, it is not isosceles triangle.
Since root of sum of square two smaller sides
= √(173+100)
= √273
=/= √200
Hence, it is not a right angled triangle.