ntercept","description":"On the x-axis, y = 0. Solve for t, ensuring the constraint t > -2 is satisfied, and substitute it into the x equation."},{"type":"text","order":3,"title":"State the final coordinates","content":"The points where the curve crosses the coordinate axes are:\n- **y-axis crossing:** $(0, -9)$\n- **x-axis crossing:** $(1.641, 0)$ (using 3 decimal places, or $(\\ln(\\sqrt{10}+2), 0)$ in exact form)."}]},{"id":"61e13537-fcab-45ee-91a2-5496d9315cb4:cartesian","label":"Conversion to Cartesian Form","steps":[{"type":"text","order":0,"title":"Express t in terms of x","content":"To find the Cartesian equation of the curve, we first eliminate the parameter t by rearranging the equation for x:\n\n$$x = \\ln(t + 2)$$\n\nApplying the exponential function to both sides to solve for t:\n\n$$e^x = t + 2$$\n\n$$t = e^x - 2$$"},{"type":"text","order":1,"title":"Substitute into y","content":"Now substitute this expression for t into the parametric equation for y to get the Cartesian equation of the curve:\n\n$$y = t^2 - 10$$\n\n$$y = (e^x - 2)^2 - 10$$"},{"type":"text","order":2,"title":"Find the y-axis intercept","content":"The curve crosses the y-axis when x = 0.\n\nWe substitute x = 0 into our Cartesian equation:\n\n$$y = (e^0 - 2)^2 - 10$$\n\n$$y = (1 - 2)^2 - 10$$\n\n$$y = (-1)^2 - 10$$\n\n$$y = 1 - 10 = -9$$\n\nThus, the y-axis intercept point is $(0, -9)$. This earns the second mark A1."},{"type":"text","order":3,"title":"Find the x-axis intercept","content":"The curve crosses the x-axis when y = 0.\n\nWe set our Cartesian equation equal to 0:\n\n$$(e^x - 2)^2 - 10 = 0$$\n\n$$(e^x - 2)^2 = 10$$\n\nTaking the square root of both sides:\n\n$$e^x - 2 = \\pm\\sqrt{10}$$\n\n$$e^x = 2 \\pm \\sqrt{10}$$"},{"type":"text","order":4,"title":"Solve for x","content":"Since $e^x$ must be strictly positive, and $2 - \\sqrt{10} \\approx -1.162 < 0$, we reject the negative root.\n\nTherefore, we have:\n\n$$e^x = 2 + \\sqrt{10}$$\n\nTaking the natural logarithm of both sides:\n\n$$x = \\ln(2 + \\sqrt{10}) \\approx 1.641 \\text{ (3 d.p.)}$$\n\nThus, the x-axis intercept point is $(1.641, 0)$. This earns the first mark M1.","calculator":[{"gdc":{"expression":"ln(2+√(10))","instructions":"In the Run-Matrix menu, enter ln(2+√(10)) and press [EXE]."},"ti84":{"expression":"ln(2+√(10))","instructions":"Press [ln], then enter 2 + [2nd] [x²] (to get √), type 10, close all parentheses, and press [ENTER]."},"desmos":{"expression":"\\ln(2+\\sqrt{10})","instructions":"Type ln(2 + sqrt(10)) into a blank line to find the decimal approximation."},"description":"Evaluate the value of $\\ln(2+\\sqrt{10})$ using a calculator."}]}]}],"generatedAt":"2026-06-09T02:02:56.207Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"f435712a-edcc-4f01-96ee-cdf476af8ac7","revisionId":"d07b18a9-7be1-42eb-8fdf-a82029d7ebe2","questionNumber":null,"questionType":"LA","specification":"A circle is defined by the parametric equations $x = 1 + 3 \\cos \\theta$, $y = -6 + 3 \\sin \\theta$","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"cd8bac85-cd80-44b6-a2f4-7e29636f504d","label":"a","content":"Find the centre and the radius of the circle.","markscheme":"- Rearrange for trig and use $\\cos^2 \\theta + \\sin^2 \\theta = 1$: $\\frac{(x-1)^2}{9} + \\frac{(y+6)^2}{9} = 1$ M1\n- Cartesian form: $(x-1)^2 + (y+6)^2 = 9$ A1\n- Centre $(1, -6)$, Radius $3$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Rearrange for $\\sin \\theta$ and $\\cos \\theta$ and attempt to use the identity $\\sin^2 \\theta + \\cos^2 \\theta = 1$.","code":"M"},{"id":"p2","marks":1,"description":"Correct Cartesian equation $(x-1)^2 + (y+6)^2 = 9$ or equivalent.","code":"A"},{"id":"p3","marks":1,"description":"Correct centre $(1, -6)$ and radius $3$.","code":"A"}],"finalAnswers":["Centre (1, -6), Radius 3"],"generatedAt":"2026-05-06T14:04:51.612Z","modelVer

1.3.7 Parametric equations of curves (A-level only)

Question 3

1.3.7 Parametric equations of curves (A-level only) Questions

1.3.7 Parametric equations of curves (A-level only)

Question 3

1.3.7 Parametric equations of curves (A-level only) Questions