12. 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 $y = x \arctan(x)$ at the point with coordinates $\left( 1, \frac{\pi}{4} \right)$. Give your answer in the form $y = mx + c$.

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12. Use the Question 12 Writing Booklet - HSC - SSCE Mathematics Extension 1 - Question 12 - 2022 - Paper 1

Question 12

12. 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... show full transcript

Worked Solution & Example Answer:12. 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 differential equation $\frac{dy}{dx} = \frac{-x - 2y}{x^{2} + y^{2}}$ , we need to calculate the slopes at the given points P, Q, and R. The directional slopes can be computed as follows:

At Point P: Calculate the coordinates (x, y) for P and substitute into the equation for $\frac{dy}{dx}$ .
At Point Q: Repeat the same process as above for the coordinates of Q.
At Point R: Again, substitute the coordinates for point R into the directional field equation.
Plotting the Direction Field: Use the calculated slopes at P, Q, and R to draw small line segments that represent the direction of the function in the specified regions.

Step 2

Will any team be penalised? Justify your answer.

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In total, there are 41 players above the age limit across 13 junior teams. To determine whether any team will be penalised, we calculate the maximum number of players that can be above the age limit for each team:

Total players = 41
Teams = 13
If each team has a maximum of 3 players above the age limit, the total possible number above the limit is: $3 \times 13 = 39$

Since 41 players exceed this total, at least one team must have more than 3 players above the age limit. Therefore, at least one team will be penalised.

Step 3

Find the equation of the tangent to the curve $y = x \arctan(x)$ at the point with coordinates $\left( 1, \frac{\pi}{4} \right)$$

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Answer

To find the equation of the tangent line at point (\left( 1, \frac{\pi}{4} \right)), we need the derivative of the function:

Differentiate the Function: Using the product rule, $y' = \arctan(x) + \frac{x}{1 + x^{2}}$
Evaluate the Derivative at x = 1: $y'(1) = \arctan(1) + \frac{1}{1 + 1^{2}} = \frac{\pi}{4} + \frac{1}{2}$
Calculate the Slope (m): The slope at this point is, m = substituting the value of arctan(1) yields: $m = \frac{\pi}{4} + \frac{1}{2}$
Using Point-Slope Form to Find Equation: Using point-slope form, $y - y_1 = m(x - x_1)$ $y - \frac{\pi}{4} = m(x - 1)$ This is the equation of the tangent in the desired form.

12. Use the Question 12 Writing Booklet - HSC - SSCE Mathematics Extension 1 - Question 12 - 2022 - Paper 1

Worked Solution & Example Answer:12. 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 \(\left( 1, \frac{\pi}{4} \right)\)$

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