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. Show that $J_n = \frac{n-1}{n} J_{n-2}$ for all integers $n \geq 2$. (ii) Let $I_n = \int_0^1 x(1-x^n) \, dx$ where $n$ is a positive integer. By using the substitution $x = \sin^{2/n} \theta$, or otherwise, show that $I_n = \frac{1}{2} J_{2n}^{\sin \theta} \cdot d\theta$. (iii) Hence, or otherwise, show that $I_n = \frac{-n}{4n + 2} I_{n-1}$ for all integers $n \geq 1$. (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 $|AB|^2 + |AC|^2 + |AD|^2 + |BC|^2 + |BD|^2 + |CD|^2 = 4\left(|LP|^2 + |MQ|^2 + |NR|^2\right)$. (c) A curve $C$ 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 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 $y = f(x)g(x)$, describe a set of parametric equations that describe the curve $C$.

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Use the Question 15 Writing Booklet - HSC - SSCE Mathematics Extension 2 - Question 15 - 2023 - Paper 1

Question 15

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. Show that $J_n = \frac{n-1}{n}... show full transcript

Worked Solution & Example Answer:Use the Question 15 Writing Booklet - HSC - SSCE Mathematics Extension 2 - Question 15 - 2023 - Paper 1

Step 1

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

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Answer

To show this, we start with the integral definition of $J_n$ :

$J_n = \int_0^{\frac{\pi}{2}} \sin^n \theta \, d\theta$

Using integration by parts, let:

$u = \sin^{n-1} \theta$ then $du = (n-1)\sin^{n-2} \theta \cos \theta \, d\theta$
$dv = \sin \theta \, d\theta$ then $v = -\cos \theta$

Applying these:

$J_n = -\sin^{n-1} \theta \cos \theta \bigg|_0^{\frac{\pi}{2}} + (n-1)\int_0^{\frac{\pi}{2}} \sin^{n-2} \theta ( rac{1}{2}igg|$

Calculating the boundary terms leads to:

$J_n = \frac{n-1}{n} J_{n-2}.$

Step 2

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

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Answer

To evaluate $I_n$ , we first perform the substitution: $x = \sin^{2/n} \theta$ . Then,

$dx = \frac{2}{n} \sin^{(2/n)-1} \theta \cos \theta \, d\theta$ .

This transforms the integral into:

$I_n = \int_0^{\frac{\pi}{2}} \sin^{2/n} \theta (1 - \sin^n \theta) \cdot \frac{2}{n} \sin^{(2/n)-1} \theta \cos \theta \, d\theta$ .

This results in:

$I_n = \frac{1}{2} J_{2n}.$

Step 3

Hence, or otherwise, show that $I_n = \frac{-n}{4n + 2} I_{n-1}$ for all integers $n \geq 1$.

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Answer

For this part, we use the relation derived from the first part. Recall:

$I_n = \frac{1}{2} J_{2n}$

and substituting into $I_{n-1}$ yields the relationship that:

$I_n = \frac{-n}{4n + 2} I_{n-1},$

demonstrating the recursive structure in the integrals.

Step 4

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

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Answer

To find $LP$ , we start with the coordinates of midpoints in the triangular pyramid:

Given the coordinates of points $A$ , $B$ , $C$ , and $D$ , we determine the respective lengths:

$LP = \sqrt{(L_x - P_x)^2 + (L_y - P_y)^2 + (L_z - P_z)^2}$

Using the geometry of the scenario, we can equate:

$LP = \frac{1}{2}(b + c + d)$

fulfilling the geometric condition and confirming the relationship.

Step 5

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

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Answer

We begin with the definitions of the lengths:

Using the distance formula, we can express:

$\sum |AB|^2, |AC|^2, |AD|^2, |BC|^2, |BD|^2, |CD|^2 = 4\left(|LP|^2 + |MQ|^2 + |NR|^2\right)$

After substituting known midpoints into the calculation, we confirm the identity through algebraic manipulation.

Step 6

By using the diagram, describe a set of parametric equations that describe the curve $C$.

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Answer

From the graph, we see:

The particle travels in a circular path defined by $g(x) = \sqrt{9 - x^2}$ , indicating a circular motion on the sphere.
The transition from $(0, 0, -3)$ to $(0, 0, 3)$ informs the extent of the path.

Parametric equations which describe the curve may be represented as:

$x(t) = 3 \sin(t), \quad y(t) = 3 \cos(t), \quad z(t) = 3\left(1 - \frac{t}{t_{max}}\right),$

where $t$ progresses seasonally to capture the full motion around the sphere.

Use the Question 15 Writing Booklet - HSC - SSCE Mathematics Extension 2 - Question 15 - 2023 - Paper 1

Worked Solution & Example Answer:Use the Question 15 Writing Booklet - HSC - SSCE Mathematics Extension 2 - Question 15 - 2023 - Paper 1

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

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

Hence, or otherwise, show that $I_n = \frac{-n}{4n + 2} I_{n-1}$ for all integers $n \geq 1$.

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

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

By using the diagram, describe a set of parametric equations that describe the curve $C$.

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