SSCE HSC Mathematics Extension 2 Polynomial equations: Suppose that a and b are positiv

Question

Suppose that a and b are positive real numbers, and let
$f(x) = \frac{a + b + x}{3(ab)^{\frac{1}{3}}}$
for $x > 0$.

(i) Show that the minimum value of $f(x)$ occurs when $x = \frac{a + b}{2}$.

(ii) Suppose that $c$ is a positive real number.
Show that $rac{(a+b+c)}{3\sqrt{abc}} \geq \frac{2}{\sqrt[3]{ab}}$ and deduce that $\frac{a+b+c}{3} \geq \sqrt{abc}.$

(iii) Suppose that the cubic equation $x^3 - px^2 + qx - r = 0$ has three positive real roots. Use part (ii) to prove that $p \geq 2\sqrt{r}.$

(iv) Deduce that the cubic equation $x^3 - 2x^2 + x - 1 = 0$ has exactly one real root.

---

The point $P(\sec \theta, b\tan \theta)$ is a point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. The line through $P$ perpendicular to the x-axis meets the hyperbola, $\frac{x}{a} - \frac{y}{b} = 0$, at $A(\sec \theta, b\sec \theta)$ and $B(a\sec \theta, -b\sec \theta)$ respectively. A second line through $P$, with gradient $\tan \alpha$, meets the hyperbola at $Q$ and meets the asymptotes at $C$ and $D$ as shown. The asymptote $\frac{x}{a} - \frac{y}{b} = 0$ makes an angle $\beta$ with the x-axis at the origin, as shown.

(i) Show that $AP \times PB = b^2$.

(ii) Show that $CP = AP \cos \beta$ and that $PD = \frac{PB \cos \beta}{\sin(\alpha - \beta)}$.

(iii) Hence deduce that $CP \times PD$ depends only on the value of $\alpha$ and not on the position of $P$.

(iv) Let $CP = p, CP = q$. 
Show that $p = q$.

(v) A tangent to the hyperbola is drawn at $T$ parallel to $CD$. This tangent meets the asymptotes at $U$ and $V$ as shown. Show that $T$ is the midpoint of $UV$.

Suppose that a and b are positive real numbers, and let $f(x) = \frac{a + b + x}{3(ab)^{\frac{1}{3}}}$ for $x > 0$ - HSC - SSCE Mathematics Extension 2 - Question 8 - 2005 - Paper 1

Worked Solution & Example Answer:Suppose that a and b are positive real numbers, and let $f(x) = \frac{a + b + x}{3(ab)^{\frac{1}{3}}}$ for $x > 0$ - HSC - SSCE Mathematics Extension 2 - Question 8 - 2005 - Paper 1

Show that the minimum value of $f(x)$ occurs when $x = \frac{a + b}{2}$

Show that \(\frac{(a+b+c)}{3\sqrt{abc}} \geq \frac{2}{\sqrt[3]{ab}}\) and deduce that $\frac{a+b+c}{3} \geq \sqrt{abc}.$

Use part (ii) to prove that $p \geq 2\sqrt{r}.$

Deduce that the cubic equation $x^3 - 2x^2 + x - 1 = 0$ has exactly one real root.

Show that $AP \times PB = b^2$.

Show that $CP = AP \cos \beta$ and that $PD = \frac{PB \cos \beta}{ \sin(\alpha - \beta)}$.

Hence deduce that $CP \times PD$ depends only on the value of $\alpha$ and not on the position of $P$.

Show that $p = q$.

Show that T is the midpoint of UV.

Join the SSCE students using SimpleStudy...