The direction field for a differential equation is given on page 1 of the Question 12 Writing Booklet - HSC - SSCE Mathematics Extension 1 - Question 12 - 2021 - Paper 1

Starting with the equation rac{dT}{dt} = k(T - 25) We separate the variables and integrate: rac{1}{T - 25} dT = k dt Integrating both sides yields: $ext{ln}|T - 25| = kt + C$ To find C, we can use the initial condition T(0) = 5, leading to: $C = ext{ln} |5 - 25| = ext{ln}(20)$ When solving for T at t = 8, set T = 10: $10 - 25 = 20e^{8k}$ Resulting in: e^{8k} = -rac{15}{20} From this, k can be solved. Then to find t when T = 20°C, we repeat similar steps to arrive at an approximate t value near 6 minutes.

The graph of T as a function of t will asymptotically approach the room temperature of 25°C, with an initial temperature of 5°C. It will pass through (0, 5), and rise to 20°C, reflecting a moderate upward curve as it approaches the limit.

We first prove the base case for n = 1. ext{LHS} = rac{1}{1 imes 2 imes 3} = rac{1}{6} And for n = 1, ext{RHS} = rac{1}{2(1)(1 + 2)} = rac{1}{6} Assuming true for n = k, we show: $1/(1 × 2 × 3) + 1/(2 × 3 × 4) + ... + 1/(k(k + 1)(k + 2)) ext{ holds true for n = k + 1.}$ This leads to manipulation of the fractions accordingly to demonstrate equality.

To sketch the graph of the function f(x) = 4 - (1 - x^2)², identify x-intercepts by setting f(x) = 0 and solve for x. The y-intercept can be found by evaluating f(0). Plot these intercepts and analyze the behavior by checking the derivative for increasing or decreasing sections.

Begin with the equation y = 4 - (1 - x^2)². Solve for x in terms of y, leading to $x = 1 ext{ or } x = 2 - ext{various roots derived from rearranging.}$ The domain of the inverse should reflect that of the original function, limited to the y-values calculated via f.

The graph of the inverse function should mirror that of the original function. Identify key features based on the intersection of the function with y = x line, swapping the axes appropriately.