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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
Answer
To create a direction field for the differential equation ( \frac{dy}{dx} = \frac{-x - 2y}{x^2 + y^2} ), evaluate the slopes at the given points P, Q, and R using the equation. For each point, substitute the coordinates into the differential equation to find the slope at that point. For example, at point P with coordinates ( (x_1, y_1) ), calculate ( m = \frac{-x_1 - 2y_1}{x_1^2 + y_1^2} ). Repeat for points Q and R, and then sketch the direction field with the calculated slopes.
Step 2
Will any team be penalised? Justify your answer.
Answer
Number of players above limit = 41. Number of teams = 13. Using the principle of pigeonhole, since each team can have a maximum of 3 players above the age limit, the maximum number of players that can be distributed among teams without exceeding the limit is 3 * 13 = 39 players. Since there are 41 players above the age limit, at least one team will have more than 3 players above the limit, thus, at least one team will be penalised.
Step 3
Find the equation of the tangent to the curve
Answer
To find the derivative of ( y = x \cdot \arctan(x) ), use the product rule: [ \frac{dy}{dx} = \arctan(x) + x \cdot \frac{1}{1+x^2} ] Substituting ( x = 1 ), we get: [ \frac{dy}{dx} = \arctan(1) + 1 \cdot \frac{1}{1+1^2} = \frac{\pi}{4} + \frac{1}{2} ] Next, find the y-coordinate at x = 1, which is ( y = \frac{\pi}{4} ). Using the point-slope form, the equation of the tangent line can then be expressed in the form ( y = mx + c ) after calculating for ( c ).