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

Number of players above limit = 41.

Since there are 13 teams, the maximum number of players allowed over the limit for each team is 3. Therefore, the maximum number of players above the limit without penalty is: $3 \times 13 = 39.$ Since there are 41 players above the age limit, at least one team will be penalised as they exceed the allowed number of 39.

First, we need to find the derivative of the curve: $y' = \frac{d}{dx}(x \cdot \arctan(x))$ Using the product rule, we have: $y' = \arctan(x) + x \cdot \frac{1}{1 + x^2}.$

Next, we evaluate the derivative at x = 1: $y'(1) = \arctan(1) + 1 \cdot \frac{1}{1 + 1^2} = \frac{\pi}{4} + \frac{1}{2} = \frac{\pi}{4} + 0.5.$

The slope of the tangent at the point (1, \frac{\pi}{4}) is therefore: $m = \frac{\pi}{4} + 0.5.$

Using the point-slope form, we plug in our point and slope: $y - \frac{\pi}{4} = m(x - 1).$

Simplifying this gives our final tangent line equation in the form y = mx + c.