sky}{0}) = -3(1+\\color{@sky}{0})^{-4} = \\htmlData{anchor=f1}{\\color{@purple}{-3}}$"},{"latex":"$$f''(\\color{@sky}{0}) = 12(1+\\color{@sky}{0})^{-5} = \\htmlData{anchor=f2}{\\color{@green}{12}}$"},{"latex":"$$f'''(\\color{@sky}{0}) = -60(1+\\color{@sky}{0})^{-6} = \\htmlData{anchor=f3}{\\color{@amber}{-60}}$"}],"highlights":[{"color":"pink","style":"text-color","anchor":"f0"},{"color":"purple","style":"text-color","anchor":"f1"},{"color":"green","style":"text-color","anchor":"f2"},{"color":"amber","style":"text-color","anchor":"f3"}],"connections":[]},"order":2,"title":"Evaluate at $x = 0$","description":"Now, we evaluate the function and each of its derivatives at $x = 0$ to find the coefficients of our series."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$f(x) \\approx \\htmlData{anchor=f0-term}{\\color{@pink}{f(0)}} + \\htmlData{anchor=f1-term}{\\color{@purple}{f'(0)}}x + \\frac{\\htmlData{anchor=f2-term}{\\color{@green}{f''(0)}}}{2!}x^2 + \\frac{\\htmlData{anchor=f3-term}{\\color{@amber}{f'''(0)}}}{3!}x^3$"},{"latex":"$$f(x) \\approx \\htmlData{anchor=f0-sub}{\\color{@pink}{1}} + (\\htmlData{anchor=f1-sub}{\\color{@purple}{-3}})x + \\frac{\\htmlData{anchor=f2-sub}{\\color{@green}{12}}}{2}x^2 + \\frac{\\htmlData{anchor=f3-sub}{\\color{@amber}{-60}}}{6}x^3$"}],"highlights":[{"color":"pink","style":"text-color","anchor":"f0-term"},{"color":"purple","style":"text-color","anchor":"f1-term"},{"color":"green","style":"text-color","anchor":"f2-term"},{"color":"amber","style":"text-color","anchor":"f3-term"},{"color":"pink","style":"text-color","anchor":"f0-sub"},{"color":"purple","style":"text-color","anchor":"f1-sub"},{"color":"green","style":"text-color","anchor":"f2-sub"},{"color":"amber","style":"text-color","anchor":"f3-sub"}],"connections":[{"to":"f0-sub","from":"f0-term","color":"pink","label":null,"style":"solid"},{"to":"f1-sub","from":"f1-term","color":"purple","label":null,"style":"solid"},{"to":"f2-sub","from":"f2-term","color":"green","label":null,"style":"solid"},{"to":"f3-sub","from":"f3-term","color":"amber","label":null,"style":"solid"}]},"order":3,"title":"Apply Maclaurin formula","description":"The Maclaurin series expansion formula for the first four terms is given by:"},{"type":"text","order":4,"title":"Simplify final terms","content":"Simplifying each coefficient gives us our final ascending series expansion:\n\n$$f(x) \\approx 1 - 3x + 6x^2 - 10x^3$$"}]}],"generatedAt":"2026-06-09T09:45:39.924Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":65}},{"id":"076792c8-14d4-495d-ac2a-629067b2aecf","revisionId":"749b603f-11e7-48e8-a73e-c457d0c747e5","questionNumber":null,"questionType":"LA","specification":"The sensitivity $S$ of a precision transducer depends on the frequency $\\omega$ of pressure oscillations according to the model $$S(\\omega) = (1 + k\\omega)^{-\\frac{1}{2}}, \\quad |k\\omega| < 1$$ where $k$ is a constant. Given that the binomial expansion of $S(\\omega)$ in ascending powers of $\\omega$ up to the term in $\\omega^3$ is $$1 + 0.6\\omega + P\\omega^2 + Q\\omega^3$$","difficulty":4,"difficultyBand":"medium","totalMarks":9,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"ef37ac96-f53b-4e52-8052-c3941f1a200c","label":"a","content":"(i) find the value of $k$,\n\n(ii) find the value of the constant $P$ and the constant $Q$.","markscheme":"- Identifies the general term as $\\binom{n}{1}(k\\omega)$ and equates to $0.6\\omega$: $n = -\\frac{1}{2} \\implies (-\\frac{1}{2})k = 0.6$ M1\n- Solves to find $k = -1.2$ A1\n- Sets up the calculation for $P$: $P = \\frac{(-\\frac{1}{2})(-\\frac{3}{2})}{2!}k^2 = \\frac{3}{8}(-1.2)^2$ M1\n- Calculates $P = 0.54$ A1\n- Sets up the calculation for $Q$: $Q = \\frac{(-\\frac{1}{2})(-\\frac{3}{2})(-\\frac{5}{2})}{3!}k^3 = -\\frac{5}{16}(-1.2)^3$ M1\n- Calculates $Q = 0.54$ A1","marks":6,"order":0,"markingRubric":null,"explanation":{"methods":[{"id":"binomial_expansion","label":"Standard Binomial Expansion","ste

3.3.7 Sequences and series (A-level only)

Question 1

3.3.7 Sequences and series (A-level only) Questions

3.3.7 Sequences and series (A-level only)

Question 1

3.3.7 Sequences and series (A-level only) Questions