ent gradient","description":"At the point with x-coordinate a, substitute x = a into the gradient function. This earns M1 for substituting x = a into the gradient function."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y-e^{2a}=2e^{2a}(x-a)$"},{"latex":"$$y-e^{2a}=2e^{2a}x-2ae^{2a}$"},{"latex":"$$y=2e^{2a}x-2ae^{2a}+e^{2a}$"}]},"order":2,"title":"Use the tangent formula","description":"Use point-gradient form with point (a, exponential of 2a) and gradient 2 times exponential of 2a. Substituting both the point and gradient earns the final A1 for the correct tangent equation."}]}],"generatedAt":"2026-07-21T04:04:20.629Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"fdc32f64-9868-4439-b69d-07bc8a81591e","label":"b","content":"Find the value of $a$ for which this tangent passes through the origin.","markscheme":"- Pass through $(0,0) \\Rightarrow 0 = 2e^{2a}(0) - 2ae^{2a} + e^{2a}$ M1\n- $e^{2a}(1 - 2a) = 0 \\Rightarrow a = 0.5$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Substitutes $(0,0)$ into the tangent equation.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $0.5$.","code":"A"}],"finalAnswers":["0.5"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"substitute_origin","label":"Substitute the origin","steps":[{"type":"text","order":0,"title":"Use the origin","content":"A line passes through the origin when the coordinates $x=0$ and $y=0$ satisfy its equation. Substitute these coordinates into the tangent equation found in part (a).","highlights":[{"text":"passes through the origin","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=origin_sub}{0=2e^{2a}(0)-2ae^{2a}+e^{2a}}"},{"latex":"\\htmlData{anchor=factorised}{e^{2a}(1-2a)=0}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"origin_sub"},{"color":"amber","style":"text-color-with-background","anchor":"factorised"}],"connections":[{"to":"factorised","from":"origin_sub","color":"sky","label":"factorise","style":"solid"}]},"order":1,"title":"Substitute and factorise","description":"Substitute the coordinates of the origin, then take out the common factor. This substitution earns the M1 mark."},{"type":"text","order":2,"title":"Solve for a","content":"The exponential factor $e^{2a}$ is never zero. Therefore the other factor must be zero:\n\n$$1-2a=0$$\n\nso\n\n$$a=0.5.$$\n\nThis earns the A1 mark."}]}],"generatedAt":"2026-07-21T04:06:34.228Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"01b9b577-d336-48bf-8a0a-5024f1e2607b","label":"c","content":"Hence, find the set of values of $m$ for which the equation $e^{2x} = mx$ has no real solutions.","markscheme":"- Tangent gradient at $a=0.5$ is $m = 2e^{2(0.5)} = 2e$ M1\n- No solutions when gradient is less than tangent but non-negative: $0 \\leq m < 2e$ M1 A1\n![](https://assets.mathsgenie.co.uk/booklet/e08c70529b6a04c2c36f30edcff90a3002ee52d665ea13e2267f961026a682aa.jpg)","marks":3,"order":2,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Identifies the critical gradient $m=2e$.","code":"M"},{"id":"p2","marks":1,"description":"Considers the lower bound of $m=0$.","code":"M"},{"id":"p3","marks":1,"description":"Final range $0 \\leq m < 2e$.","code":"A"}],"finalAnswers":["0 \\leq m < 2e"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"tangent_comparison","label":"Compare with the tangent","steps":[{"type":"text","order":0,"title":"Identify the boundary","content":"The line $y=mx$ always passes through the origin. The curve is always above the $x$-axis, but a negative-gradient line can meet it when $x<0$.\n\nFrom the earlier part, the boundary **positive** line is the one through the origin that is tangent to the curve at $x=0.5$. We next find its gradie

Exponentials and Logarithms

Question 17

Exponentials and Logarithms Questions

Exponentials and Logarithms

Question 17

Exponentials and Logarithms Questions