1.) Let R be the region in the xy-plane that is enclosed by the graphs of y = x³ and y = x² – 6x + 8.

a) Find the volume of the solid generated when R is rotated about the x-axis.
First, we need to find the limits of integration. From the equation of the upper bound, y = x³, we get x = y^1/3.

From the equation of the lower bound, y = x² – 6x + 8, we get x = 2 +√(y + 2) and x = 2 – √(y + 2).

So the volume of the solid generated when R is rotated about the x-axis is given by the definite integral:
∫[2-√(y+2), 2+√(y+2)] π(y^2/3)² dy

b) Find the volume of the solid generated when R is rotated about the y-axis.
First, we need to find the limits of integration. From the equation of the upper bound, y = x³, we get x = y^1/3.

From the equation of the lower bound, y = x² – 6x + 8, we get x = 2 + √(y + 2) and x = 2 – √(y + 2).

So the volume of the solid generated when R is rotated about the y-axis is given by the definite integral:
2π ∫[2-√(y+2), 2+√(y+2)] x(y^1/3) dy

2.) Solve the equation tan²x = 3 for -π≤ x ≤ π

tanx = ±√3
For tanx = √3
x=π/3, x= -2π/3

For tanx = -√3
x=-π/3, x= 2π/3

3.) Consider the quadratic -4x²+120x-800

a) 
i) Find the roots.

x=10, x=20

ii) Hence express the quadratic in the form y= a(x-x1)(x-x2)
y= -4(x-10)(x-20)

b)
i) Find the coordinates of the vertex.
(15,100)

ii) Hence express the quadratic in the form y= a(x-h)²+k
y= -4(x-15)²+100

iii) Write down the equation of the axis of symmetry
x=15

iv) Write down the maximum value of y
Ymax= -100

c) Write down the y- intercept of the quadratic
y= -800

4.) Solve the equation: (ln x)² – lnx² +1 = 0

x²-2x+1=0
x= 1
lnx = 1
Hence, x = e

5.) Solve the equation 2sin²x = sin x for 0≤ x ≤ 2π

2sin²x – sinx = 0
sinx(2sinx – 1) = 0
Sinx = 0
and
2sinx -1 = 0
Sinx = 1/2
For sinx, x = 0, x= π, x = 2π
For sinx = 1/2, x=π/6, x= 5π/6

6.) Let f(x) = 2x² – 12x +10. Find the tangent and normal line at x = 2.

Tangent line = y + 6 = -4(x -2)
Normal line = y + 6 = ¼ (x – 2)