4$ back into the equation for $x$ to find the corresponding $x$-coordinate. This gives us the first point: $(\\ln 9, 0)$."},{"type":"text","order":3,"title":"Crosses the $y$-axis","content":"Set $x = 0$ to find the $y$-intercept:\n\n$$\\ln(2t + 1) = 0$$\n\nTaking the exponential of both sides:\n\n$$2t + 1 = e^0$$\n$$2t + 1 = 1$$\n$$2t = 0 \\implies t = 0$$\n\nSince $t = 0$ satisfies the condition $t > -0.5$, we use this value."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\t\thexcolor{purple}{t} = \t\thexcolor{purple}{0}$"},{"latex":"$$y = \\htmlData{anchor=t-sub2}{\t\thexcolor{purple}{t}}^2 - 16$"},{"latex":"$$y = (\\htmlData{anchor=t-val2}{\t\thexcolor{purple}{0}})^2 - 16 = -16$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"t-val2"},{"color":"purple","style":"text-color","anchor":"t-sub2"}],"connections":[{"to":"t-sub2","from":"t-val2","color":"purple","label":"substitute","style":"solid"}]},"order":4,"title":"Calculate the $y$-intercept","description":"Substitute $t = 0$ into the equation for $y$ to obtain the $y$-coordinate. This yields the second point: $(0, -16)$."},{"type":"text","order":5,"title":"State final answers","content":"The points where the curve crosses the coordinate axes are:\n\n- On the $x$-axis: **$(\\ln 9, 0)$**\n- On the $y$-axis: **$(0, -16)$**"}]},{"id":"cartesian_conversion","label":"Cartesian Form Conversion","steps":[{"type":"text","order":0,"title":"Express $t$ in terms of $x$","content":"To convert the parametric equations into a single Cartesian equation, we first rearrange the equation for $x$ to express $t$ in terms of $x$.\n\nStarting with:\n$$x = \\ln(2t + 1)$$\n\nWe apply the exponential function $e$ to both sides to eliminate the natural logarithm:\n$$e^x = 2t + 1$$\n\nNow, solve for $t$:\n$$\\begin{aligned} 2t &= e^x - 1 \\\\ t &= \\frac{e^x - 1}{2} \\end{aligned}$$"},{"type":"text","order":1,"title":"Substitute $t$ into $y$","content":"Now we substitute this expression for $t$ into our equation for $y$ to get the Cartesian equation of the curve:\n\n$$y = t^2 - 16$$\n\nSubstituting $t = \\frac{e^x - 1}{2}$ gives:\n$$y = \\left(\\frac{e^x - 1}{2}\\right)^2 - 16$$\n\nSimplifying this expression:\n$$y = \\frac{(e^x - 1)^2}{4} - 16$$"},{"type":"text","order":2,"title":"Find the $y$-intercept","content":"To find where the curve crosses the $y$-axis, we set $x = 0$ in our Cartesian equation:\n\n$$y = \\frac{(e^0 - 1)^2}{4} - 16$$\n\nSince $e^0 = 1$:\n$$\\begin{aligned} y &= \\frac{(1 - 1)^2}{4} - 16 \\\\ y &= 0 - 16 \\\\ y &= -16 \\end{aligned}$$\n\nSo, the $y$-intercept is at the point **$(0, -16)$**. This earns A1."},{"type":"text","order":3,"title":"Find the $x$-intercept","content":"To find where the curve crosses the $x$-axis, we set $y = 0$ in our Cartesian equation:\n\n$$0 = \\frac{(e^x - 1)^2}{4} - 16$$\n\nNow, solve for $x$:\n$$\\begin{aligned} \\frac{(e^x - 1)^2}{4} &= 16 \\\\ (e^x - 1)^2 &= 64 \\\\ e^x - 1 &= \\pm 8 \\end{aligned}$$\n\nThis gives us two possibilities:\n1. $e^x - 1 = 8 \\implies e^x = 9 \\implies x = \\ln 9$\n2. $e^x - 1 = -8 \\implies e^x = -7$ (which has no real solutions since $e^x$ must be positive)\n\nTherefore, we have $x = \\ln 9$, giving the $x$-intercept **$(\\ln 9, 0)$**. This earns A1."}]}],"generatedAt":"2026-06-11T03:31:24.307Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"def190c5-d557-4cd0-abdb-af12b45047d6","label":"b","content":"Find an equation for the tangent to the curve when $t = 1$.","markscheme":"- At $t = 1$, $x = \\ln 3$ and $y = 1^2 - 16 = -15$ B1\n- $\\frac{dx}{dt} = \\frac{2}{2t+1}$, $\\frac{dy}{dt} = 2t$ M1\n- $\\frac{dy}{dx} = \\frac{2t}{(2/(2t+1))} = t(2t + 1)$. At $t=1$, $m = 1(3) = 3$ M1\n- $y - (-15) = 3(x - \\ln 3)$ M1\n- $y = 3x - 3\\ln 3 - 15$ A1","marks":5,"order":1,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Finding coordinates $(\\ln 3, -15)$.","code":"B"},{"id":"p2","marks":1,"description":"Correct differentiation of $x$

8.4 Points of Intersection

Question 2

8.4 Points of Intersection Questions

8.4 Points of Intersection

Question 2

8.4 Points of Intersection Questions