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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 dy/dx = -x - 2y / x^2 + y^2 On the diagram on page 1 o... 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 draw the direction field for the differential equation ( \frac{dy}{dx} = \frac{-x - 2y}{x^2 + y^2} ), calculate the slopes at the points P, Q, and R given in the booklet.
- Point P: Calculate the slope at P, for example, if P = (x1, y1), substitute these values into the equation to find ( \frac{dy}{dx} ).
- Point Q: Similarly, calculate the slope at Q using its coordinates.
- Point R: Repeat for R.
After calculating the values, draw arrows indicating the direction of the field at each point.
Step 2
Will any team be penalised? Justify your answer.
Answer
To determine if any team will be penalized, first we need to ascertain the number of players per team:
- Total players = 41
- Total teams = 13
- Average players per team = ( \frac{41}{13} \approx 3.15 )
Since any team with more than 3 players above the age limit is penalised, it’s possible for some teams to exceed the limit. For example, if one team had 4 players above the age limit, that team would be penalized. Therefore, yes, at least one team will be penalized under the condition given.
Step 3
Find the equation of the tangent to the curve y = x arctan(x) at the point with coordinates (1, π/4)
Answer
To find the equation of the tangent, we first need to find the derivative of the function ( y = x \arctan(x) ):
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Differentiate using the product rule:
- Let ( u = x ) and ( v = \arctan(x) )
- Then ( \frac{du}{dx} = 1 ) and ( \frac{dv}{dx} = \frac{1}{1+x^2} )
- By the product rule: [ \frac{dy}{dx} = \frac{du}{dx} v + u \frac{dv}{dx} = \arctan(x) + x \cdot \frac{1}{1+x^2} ]
- Substitute ( x = 1 ): [ \frac{dy}{dx} \Big|_{x=1} = \arctan(1) + 1 \cdot \frac{1}{2} = \frac{\pi}{4} + \frac{1}{2} ]
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Find the equation of the tangent line:
- The slope at the point is given as ( m = \frac{\pi}{4} + \frac{1}{2} )
- Using point-slope form ( y - y_1 = m(x - x_1) ): [ y - \frac{\pi}{4} = \left(\frac{\pi}{4} + \frac{1}{2}\right)(x - 1) ]
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Convert to slope-intercept form (y = mx + c):
- Rearranging gives the equation of the tangent.