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12 marks) Use the Question 12 Writing Booklet - HSC - SSCE Mathematics Extension 1 - Question 12 - 2022 - Paper 1
Question 12
12 marks) Use the Question 12 Writing Booklet. (a) A direction field is to be drawn for the differential equation dy/dx = -x - 2y / x^2 + y^2 On the diagram on pa... show full transcript
Worked Solution & Example Answer:12 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
Answer
To draw the direction field for the differential equation, we need to analyze the given equation:
dy/dx = \frac{-x - 2y}{x^2 + y^2}
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Determine the slopes at the specified points P, Q, and R based on the equation.
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Substitute the coordinates of points P, Q, and R to find their corresponding slopes:
- For point P, let’s say (x,y) = (a,b). Compute the slope:
( m_P = \frac{-a - 2b}{a^2 + b^2} )
- For point Q, let’s say (x,y) = (c,d). Compute the slope:
( m_Q = \frac{-c - 2d}{c^2 + d^2} )
- For point R, let’s say (x,y) = (e,f). Compute the slope:
( m_R = \frac{-e - 2f}{e^2 + f^2} )
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On the provided diagram, draw small line segments at each point with the corresponding slope.
Step 2
Will any team be penalised? Justify your answer.
Answer
To determine if a team will be penalised, we need to understand how many players each team has:
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The total number of players is 41, and there are 13 teams.
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Calculate the average number of players per team:
( \text{Average players per team} = \frac{41}{13} \approx 3.15 )
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Since the age limit is that no team can have more than 3 players above the age limit, and if each team had to even share the players uniformly, the fact that 3.15 rounds up means at least some teams must have more than 3 players exceeding the age limit.
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Therefore, at least one team will definitely be penalised.
Step 3
Find the equation of the tangent to the curve y = x * arctan(x)
Answer
To find the equation of the tangent line at the point (1, π/4):
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First, compute the derivative of the function:
( y = x \cdot \arctan(x) )
Using the product rule:
( y' = \arctan(x) + x \cdot \frac{1}{1+x^2} )
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Substitute x = 1:
( y'(1) = \arctan(1) + 1 \cdot \frac{1}{1+1^2} = \frac{\pi}{4} + \frac{1}{2} = \frac{\pi}{4} + 0.5 )
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Now use the point-slope form of a line to find the tangent:
( y - \frac{\pi}{4} = m (x - 1) )
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The slope m is obtained from y'(1). After rearrangement, we end with the final equation in the form ( y = mx + c ).