8-4ee7-9276-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","anchor":"quad_term"}],"connections":[{"to":"half_exp","from":"exp_x","color":"sky","label":"rewrite","style":"solid"},{"to":"quad_term","from":"half_exp","color":"amber","label":"form quadratic","style":"solid"}]},"order":1,"title":"Make a quadratic","description":"At an intersection the two expressions for $y$ are equal. Rewrite $e^x$ as a square, then rearrange into quadratic form."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\left(\\htmlData{anchor=quad}{e^{\\frac12x}}\\right)^2+2e^{\\frac12x}-3=0$"},{"latex":"$$\\left(\\htmlData{anchor=factor_a}{e^{\\frac12x}+3}\\right)\\left(\\htmlData{anchor=factor_b}{e^{\\frac12x}-1}\\right)=0$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"quad"},{"color":"orange","style":"text-color-with-background","anchor":"factor_a"},{"color":"orange","style":"text-color-with-background","anchor":"factor_b"}],"connections":[{"to":"factor_a","from":"quad","color":"orange","label":"factorise","style":"solid"},{"to":"factor_b","from":"quad","color":"orange","label":"factorise","style":"solid"}]},"order":2,"title":"Factorise the quadratic","description":"The quadratic in $e^{\\frac12x}$ has roots $1$ and $-3$, so write it as two linear factors. This factorisation earns the third M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=neg_factor}{e^{\\frac12x}+3}=0$"},{"latex":"$$\\htmlData{anchor=neg_value}{e^{\\frac12x}}=-3$"},{"latex":"$$\\htmlData{anchor=pos_factor}{e^{\\frac12x}-1}=0$"},{"latex":"$$\\htmlData{anchor=pos_value}{e^{\\frac12x}}=1$"}],"highlights":[{"color":"red","style":"text-color-with-background","anchor":"neg_value"},{"color":"green","style":"text-col

Exponentials and Logarithms

Question 21

Exponentials and Logarithms Questions

Exponentials and Logarithms

Question 21

Exponentials and Logarithms Questions