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12. (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 - HSC - SSCE Mathematics Extension 1 - Question 12 - 2022 - Paper 1
Question 12
![12.-(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-HSC-SSCE Mathematics Extension 1-Question 12-2022-Paper 1.png](https://cdn.simplestudy.cloud/assets/backend/uploads/question/subject_1095_paper_15699_q12_67b626dc.jpg)
12. (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 ... show full transcript
Worked Solution & Example Answer:12. (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 - HSC - SSCE Mathematics Extension 1 - Question 12 - 2022 - Paper 1
Step 1
A direction field is to be drawn for the differential equation
Answer
To draw the direction field for the equation ( \frac{dy}{dx} = \frac{-x - 2y}{x^2 + y^2} ), evaluate the slope at points P, Q, and R.
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Point P: Evaluate the slope at P:
Substitute the coordinates of point P into the equation to find ( \frac{dy}{dx} ). -
Point Q: Repeat the process for point Q.
Substitute the coordinates of point Q into the equation. -
Point R: Finally, evaluate the slope at point R similarly.
Using these slopes, draw short line segments at each point to depict the direction field.
Step 2
Will any team be penalised? Justify your answer.
Answer
To determine if any team will be penalised, consider the total number of players and the regulation.
- Number of players above age limit: There are 41 players above the age limit.
- Players per team: If 41 players are distributed among 13 teams, we calculate the average number of players per team:
[ \text{Average} = \frac{41}{13} \approx 3.15 ] - Since a team can have more than 3 players above the limit without penalty, if any team has 4 or more, it will be penalised.
Thus, at least one team will be penalised.
Step 3
Find the equation of the tangent to the curve at the point with coordinates (1, π/4).
Answer
To find the equation of the tangent at ( (1, \frac{\pi}{4}) ):
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Find the derivative: First, find ( \frac{dy}{dx} ) for ( y = x \cdot \arctan(x) ):
Using the product rule, ( \frac{dy}{dx} = \arctan(x) + x \cdot \frac{1}{1+x^2} ). -
Evaluate the slope at x=1:
Substitute ( x = 1 ):
[ \frac{dy}{dx} \bigg|_{x=1} = \arctan(1) + 1 \cdot \frac{1}{1+1^2} = \frac{\pi}{4} + \frac{1}{2} = \frac{\pi}{4} + 0.5 ]
This gives the slope ( m ). -
Equation of the tangent line: Use the point-slope form ( y - y_0 = m(x - x_0) ) where ( (x_0, y_0) = (1, \frac{\pi}{4}) ):
Rearranging will provide the line's equation in the form ( y = mx + c ).