ntiate exponent"}]},"order":1,"title":"Differentiate the curve","description":"Use the chain rule: differentiate the exponent, then multiply by the unchanged exponential function. Obtaining this derivative earns the first M1 mark."},{"type":"text","order":2,"title":"Find the tangent gradient","content":"Substitute $x=3$ into the derivative:\n\n$$m=\\left.\\frac{dy}{dx}\\right|_{x=3}=\\frac13e^{\\frac13\\times3}=\\frac{e}{3}.$$\n\nThis explicit gradient calculation earns the second M1 mark.","highlights":[{"text":"at the point where $x = 3$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"y-e=\\htmlData{anchor=bracket}{\\frac{e}{3}(x-3)}"},{"latex":"y-e=\\htmlData{anchor=expanded}{\\frac{e}{3}x-e}"},{"latex":"y=\\htmlData{anchor=result}{\\frac{e}{3}x}"}],"highlights":[{"color":"green","style":"text-color","anchor":"result"}],"connections":[{"to":"expanded","from":"bracket","color":"orange","label":"expand bracket","style":"solid"},{"to":"result","from":"expanded","color":"green","label":"add e to both sides","style":"dashed"}]},"order":3,"title":"Form the tangent equation","description":"Substitute the point (3, e) and gradient e/3 into point-gradient form. Expand the bracket, then add e to both sides. Obtaining the final equation earns the A1 mark. Its constant term is zero, so putting x equal to zero gives y equal to zero; hence the line passes through the origin."}]}],"generatedAt":"2026-07-21T04:11:46.141Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"7c42e3fd-0056-45cc-b061-d475d97fa7a1","revisionId":"31a0fbd7-cd88-4ee7-9276-ab7f7e6aced5","questionNumber":"28","questionType":"LA","specification":"","difficulty":4,"difficultyBand":"medium","totalMarks":6,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":null,"parts":[{"id":"98575f0c-9dd8-45ee-b2a6-3cdcc8c02678","label":"0","content":"Find the coordinates of the point of intersection of the curves $y = e^x$ and $y = 3 - 2e^{\\frac{1}{2}x}$","markscheme":"- Set equal: $e^x = 3 - 2e^{\\frac{1}{2}x}$ M1\n- Form quadratic in $e^{\\frac{1}{2}x}$: $(e^{\\frac{1}{2}x})^2 + 2e^{\\frac{1}{2}x} - 3 = 0$ M1\n- Factorise: $(e^{\\frac{1}{2}x} + 3)(e^{\\frac{1}{2}x} - 1) = 0$ M1\n- $e^{\\frac{1}{2}x} = 1$ (since $e^{\\frac{1}{2}x} \\neq -3$) A1\n- $\\frac{1}{2}x = 0 \\Rightarrow x = 0$ M1\n- $y = e^0 = 1$. Point is $(0, 1)$ A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Equates the two curve equations.","code":"M"},{"id":"p2","marks":1,"description":"Recognises the quadratic form in $e^{0.5x}$.","code":"M"},{"id":"p3","marks":1,"description":"Factorises the quadratic.","code":"M"},{"id":"p4","marks":1,"description":"Identifies the valid solution $e^{0.5x} = 1$.","code":"A"},{"id":"p5","marks":1,"description":"Solves for $x=0$.","code":"M"},{"id":"p6","marks":1,"description":"Finds $y=1$ and states coordinates $(0,1)$.","code":"A"}],"finalAnswers":["(0, 1)"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"substitution_and_factorising","label":"Substitution and factorising","steps":[{"type":"text","order":0,"title":"Set the curves equal","content":"At an intersection, both curves have the same $x$-coordinate and the same $y$-coordinate. Therefore, equate their expressions for $y$:\n\n$$e^x=3-2e^{\\frac12x}$$\n\nThis earns the first **M1** mark.","highlights":[{"text":"point of intersection","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=exp_x}{e^x}=\\left(\\htmlData{anchor=half_exp}{e^{\\frac12x}}\\right)^2$"},{"latex":"$$\\left(\\htmlData{anchor=quad_term}{e^{\\frac12x}}\\right)^2+2e^{\\frac12x}-3=0$"},{"text":"This earns the second M1 mark."}],"highlights":[{"color":"sky","style":"text-color","anchor":"exp_x"},{"color":"amber","style":"text-color-with-background","ancho

Exponentials and Logarithms

Question 17

Exponentials and Logarithms Questions

Exponentials and Logarithms

Question 17

Exponentials and Logarithms Questions