to move the factor of one half into the logarithm as a power. This completes the second method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=\\htmlData{anchor=root_source}{\\log_5\\left(\\color{@green}{9^{1/2}}\\right)}$"},{"latex":"$$x=\\htmlData{anchor=root_target}{\\log_5\\left(\\color{@green}{3}\\right)}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"root_source"},{"color":"green","style":"text-color","anchor":"root_target"}],"connections":[{"to":"root_target","from":"root_source","color":"green","label":"evaluate the square root"}]},"order":3,"title":"Evaluate the square root","description":"The square root of nine is three, so replace the logarithm's argument with the integer three. This exact final form earns the accuracy mark (A1)."}]}],"generatedAt":"2026-07-20T16:58:27.520Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"119996f2-9c31-4cd5-a233-1319e0864977","revisionId":"6fb440ba-a627-4701-a9ab-7b9f732ffbf8","questionNumber":"24","questionType":"LA","specification":"A curve has the equation $y = e^{2x}$","difficulty":4,"difficultyBand":"medium","totalMarks":8,"sourceTitle":"Question Set 14: Exponentials and Logarithms","calculatorUse":null,"parts":[{"id":"9fa55f0d-4ca4-4209-a096-89cf2424c16a","label":"a","content":"Find, in terms of $a$, the equation of the tangent to the curve at the point $(a, e^{2a})$","markscheme":"- $\\frac{dy}{dx} = 2e^{2x}$ M1\n- $m = 2e^{2a}$ M1\n- $y - e^{2a} = 2e^{2a}(x - a) \\Rightarrow y = 2e^{2a}x - 2ae^{2a} + e^{2a}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Differentiates the curve equation.","code":"M"},{"id":"p2","marks":1,"description":"Finds the gradient at $x=a$.","code":"M"},{"id":"p3","marks":1,"description":"Correct equation in terms of $a$.","code":"A"}],"finalAnswers":["y = 2e^{2a}x - 2ae^{2a} + e^{2a}"],"generatedAt":"2026-05-12T02:14:06.626Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"tangent_equation","label":"Find the tangent equation","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=e^{2x}$"},{"latex":"$$\\frac{dy}{dx}=2e^{2x}$"}]},"order":0,"title":"Differentiate the curve","description":"The chain-rule factor is 2 because the derivative of 2x is 2. Differentiating to obtain the gradient earns the first method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\left.\\frac{dy}{dx}\\right|_{x=a}=2e^{2a}$"},{"latex":"$$m=2e^{2a}$"}]},"order":1,"title":"Find the tangent 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"},"explanation":{"methods":[{"id":"substitute_origin","label":"Substitute the origin","s

Exponentials and Logarithms

Question 16

Exponentials and Logarithms Questions

Exponentials and Logarithms

Question 16

Exponentials and Logarithms Questions