SSCE HSC Mathematics Extension 2 Vector equation of a curve: Question 15 (16 marks) Use the Q

Question

Question 15 (16 marks) Use the Question 15 Writing Booklet

(a) Let $J_n = \int_0^{\frac{\pi}{2}} \sin^n 	heta \, d	heta$ where $n \geq 0$ is an integer.

Show that $J_n = \frac{n-1}{n} J_{n-2}$ for all integers $n \geq 2$.

(ii) Let $I_n = \int_0^1 x(1-x^2) \, dx$ where $n$ is a positive integer.

By using the substitution $x = \sin 	heta$, or otherwise, show that $I_n = \frac{1}{2} J_{2n} 	heta$.

(iii) Hence, or otherwise, show that $I_n = \frac{-n}{4+n} I_{n-1}$ for all integers $n \geq 1$.

Question 15 (continued)

(b) On the triangular pyramid $ABCD$, $L$ is the midpoint of $AB$, $M$ is the midpoint of $AC$, $N$ is the midpoint of $AD$, $P$ is the midpoint of $CD$, $Q$ is the midpoint of $BD$ and $R$ is the midpoint of $BC$.

(i) Show that $LP = \frac{1}{2}(b+c+d)$.

(ii) It can be shown that

$MQ = \frac{1}{2}(b+c-d)$ and $NR = \frac{1}{2}(b+c-d)$. (Do NOT prove these.)

Prove that $|AB| + |AC| + |AD|^2 + |BC| + |BD|^2 + |CD|^2 = 4 (|LP|^2 + |MQ|^2 + |NR|^2)$.

Question 15 (continued)

(c) A curve $	ext{C}$ spirals 3 times around the sphere centred at the origin and with radius 3, as shown.

A particle is initially at the point $(0,0,-3)$ and moves along the curve $	ext{C}$ on the surface of the sphere, ending at the point $(0,0,3)$.

By using the diagram below, which shows the graphs of the functions $f(x) = \cos(\pi x)$ and $g(x) = \sqrt{9-x^2}$, and considering the graph $y = f(x)g(x)$, give a suitable set of parametric equations that describe the curve $	ext{C}$.

Question 15 (16 marks) Use the Question 15 Writing Booklet (a) Let $J_n = \int_0^{\frac{\pi}{2}} \sin^n \theta \, d\theta$ where $n \geq 0$ is an integer - HSC - SSCE Mathematics Extension 2 - Question 15 - 2023 - Paper 1

Worked Solution & Example Answer:Question 15 (16 marks) Use the Question 15 Writing Booklet (a) Let $J_n = \int_0^{\frac{\pi}{2}} \sin^n \theta \, d\theta$ where $n \geq 0$ is an integer - HSC - SSCE Mathematics Extension 2 - Question 15 - 2023 - Paper 1

Prove that $|AB| + |AC| + |AD|^2 + |BC| + |BD|^2 + |CD|^2 = 4 (|LP|^2 + |MQ|^2 + |NR|^2)$

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