Question 12 (16 marks) Use the Question 12 Writing Booklet. (a) A direction field is to be drawn for the differential equation $$ \frac{dy}{dx} = \frac{-x - 2y}{x^2 + y^2} $$ On the diagram on page 1 of the Question 12 Writing Booklet, clearly draw the correct slopes of the direction field at the points P, Q and R. (b) A sports association manages 13 junior teams. It decides to check the age of all players. Any team that has more than 3 players above the age limit will be penalised. A total of 41 players are found to be above the age limit. Will any team be penalised? Justify your answer. (c) Find the equation of the tangent to the curve $$ y = x \cdot \arctan(x) $$ at the point with coordinates $$ \left( 1, \frac{\pi}{4} \right) $$ . Give your answer in the form $y = mx + c$.

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Question 12 (16 marks) Use the Question 12 Writing Booklet - HSC - SSCE Mathematics Extension 1 - Question 12 - 2022 - Paper 1

Question 12

Question 12 (16 marks) Use the Question 12 Writing Booklet. (a) A direction field is to be drawn for the differential equation $$ \frac{dy}{dx} = \frac{-x - 2y}{x... show full transcript

Worked Solution & Example Answer:Question 12 (16 marks) Use the Question 12 Writing Booklet - HSC - SSCE Mathematics Extension 1 - Question 12 - 2022 - Paper 1

Step 1

A direction field is to be drawn for the differential equation

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Answer

To draw the direction field for the equation $\frac{dy}{dx} = \frac{-x - 2y}{x^2 + y^2}$ , we first evaluate the slopes at specific points, such as P, Q, and R.

Select points: Choose coordinates for P, Q, and R from the Writing Booklet.
Calculate slopes: For example, at point P with coordinates (x_p, y_p):
$slope = \frac{-x_p - 2y_p}{x_p^2 + y_p^2}$ Repeat for Q and R.
Draw the direction field: Use the calculated slopes to sketch the lines representing the direction field.

Step 2

Will any team be penalised? Justify your answer.

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Answer

To determine if any team will be penalised, we first calculate the total number of players exceeding the age limit.

Total players above age limit: There are 41 players found above the age limit.
Teams' total players: Each team has an average of 41 / 13 = 3.15 players above the age limit.
Apply the threshold: Since any team with more than 3 penalised players will be penalised, at least one team will be penalised as the average indicates there is at least one team with more than 3.

Step 3

Find the equation of the tangent to the curve y = x arctan(x) at the point with coordinates (1, π/4)

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Answer

To find the tangent line to the curve at the given point, we first need to determine the slope of the curve using differentiation.

Differentiate the function:
$y = x \cdot \arctan(x)$
Using the product rule:
$\frac{dy}{dx} = \arctan(x) + x \cdot \frac{1}{1+x^2}$
Evaluate at $x = 1$ :
$\frac{dy}{dx} \bigg|_{x=1} = \arctan(1) + 1 \cdot \frac{1}{2} = \frac{\pi}{4} + \frac{1}{2}$
Find the slope: Calculate the value to get the slope $m$ .
Use point-slope form: With slope $m$ and point $(1, \frac{\pi}{4})$ , the equation of the tangent is:
$y - \frac{\pi}{4} = m(x - 1)$
Rearranging gives the required equation in the form $y = mx + c$ .

Question 12 (16 marks) Use the Question 12 Writing Booklet - HSC - SSCE Mathematics Extension 1 - Question 12 - 2022 - Paper 1

Worked Solution & Example Answer:Question 12 (16 marks) Use the Question 12 Writing Booklet - HSC - SSCE Mathematics Extension 1 - Question 12 - 2022 - Paper 1

A direction field is to be drawn for the differential equation

Will any team be penalised? Justify your answer.

Find the equation of the tangent to the curve y = x arctan(x) at the point with coordinates (1, π/4)

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