The diagram shows the graph of $y = f(x)$ - HSC - SSCE Mathematics Extension 2 - Question 3 - 2007 - Paper 1

For $f(|x|)$, we need to reflect the right side of the graph of $y = f(x)$ over the y-axis and keep the left side unchanged. The graph will be symmetric about the y-axis.

To plot $f(x) - x$, take the graph of $f(x)$ and move it downward by moving each point down by the corresponding y-coordinate of $y = x$. This will create a new graph that intersects the line $y = x$ at the roots of the equation.

To find a cubic polynomial with integer coefficients whose zeros are $2α$, $2β$, and $2γ$, we can use Vieta's formulas. The polynomial can be expressed as: $P(x) = (x - 2α)(x - 2β)(x - 2γ) = x^3 - (2α + 2β + 2γ)x^2 + (4(αβ + αγ + βγ))x - 8αβγ$ This gives us a polynomial with the required integer coefficients.

To find the volume of the solid formed by rotating the shaded region around the y-axis, use the formula for the volume of cylindrical shells: $V = 2π imes ext{(radius)} imes ext{(height)}$ In this case, the radius is $x$ and the height is rac{ ext{log}_e x}{x}, where the bounds are from $x = 1$ to $x = e$: V = ext{Evaluate } 2π imes ext{integral from } 1 ext{ to } e ext{ of } x rac{ ext{log}_e x}{x} ext{ dx}

To find $N$, we resolve forces:

• Vertically: $N - mg ext{cos} θ - mrω^2 ext{sin} θ = 0$ Thus, $N = mg ext{cos} θ - mrω^2 ext{sin} θ$

From the derived equation, for $N > 0$, we require: $mg ext{cos} θ - mrω^2 ext{sin} θ > 0$ This implies: $mg ext{cos} θ > mrω^2 ext{sin} θ$ Therefore, rac{g ext{cos} θ}{r ext{sin} θ} > ω^2 This gives the conditions for $ext{ω}$.