uation within the interval $0^\\circ \\le \\phi \\le 180^\\circ$ are:\n\n$$\\phi = 20^\\circ, \\; 100^\\circ, \\; 140^\\circ$$"}]}],"generatedAt":"2026-06-09T09:09:07.922Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"cef9c774-cc2f-4693-a79a-6434cadcb3f2","revisionId":"e44e2d5d-f581-4b66-9018-6636e2384c8d","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"660d452f-49d6-4939-9567-b4cf039d955a","label":"a","content":"Solve, for $-180 \\leq x \\leq 180$, the equation,\n\n$$2 \\cos 2x + 7 \\cos x = 0$$","markscheme":"- Substitutes $\\cos 2x = 2 \\cos^2 x - 1$ M1\n- Forms quadratic: $4 \\cos^2 x + 7 \\cos x - 2 = 0$ M1\n- Factorises to $(4 \\cos x - 1)(\\cos x + 2) = 0$ M1\n- Solves $\\cos x = \\frac{1}{4}$ (rejects $\\cos x = -2$) M1\n- Final answers: $x \\approx 75.5^\\circ, -75.5^\\circ$ (to 1 d.p.) A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Attempt to substitute identity $\\cos 2x = 2 \\cos^2 x - 1$.","code":"M"},{"id":"p2","marks":1,"description":"Formation of a correct 3-term quadratic equation in $\\cos x$.","code":"M"},{"id":"p3","marks":1,"description":"Correct factorization or solution method for their quadratic equation.","code":"M"},{"id":"p4","marks":1,"description":"Identification of $\\cos x = 1/4$ and explicit or implied rejection of the other root.","code":"M"},{"id":"p5","marks":1,"description":"Final answers $x = 75.5^\\circ$ and $x = -75.5^\\circ$ (or similar rounding) provided correctly.","code":"A"}],"finalAnswers":["75.5^\\circ, -75.5^\\circ"],"generatedAt":"2026-05-06T14:04:22.546Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"algebraic_quadratic","label":"Algebraic Approach with Quadratic Formula","steps":[{"type":"text","order":0,"title":"Use double-angle identity","content":"To solve the equation, we first need to express it entirely in terms of $\\cos x$. We can do this by using the double-angle identity for cosine from our formula sheet:\n\n$$\\cos 2x = 2 \\cos^2 x - 1$$\n\nSubstituting this into the given equation:\n\n$$2(2 \\cos^2 x - 1) + 7 \\cos x = 0$$","highlights":[{"text":"2 \\cos 2x + 7 \\cos x = 0","location":"specification"}]},{"type":"text","order":1,"title":"Form a quadratic equation","content":"Now we expand and simplify the expression to form a standard quadratic equation in terms of $\\cos x$:\n\n$$4 \\cos^2 x - 2 + 7 \\cos x = 0$$\n\nRearranging the terms, we get:\n\n$$4 \\cos^2 x + 7 \\cos x - 2 = 0$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$4\\htmlData{anchor=quad-var}{\\cos^2 x} + 7\\cos x - 2 = 0$"},{"text":"This quadratic can be factorised in the same way as $4y^2 + 7y - 2 = 0$:"},{"latex":"$$(\\htmlData{anchor=fact-1}{4 \\cos x - 1})(\\htmlData{anchor=fact-2}{\\cos x + 2}) = 0$"}],"highlights":[{"color":"sky","style":"underline","anchor":"quad-var"},{"color":"sky","style":"text-color","anchor":"fact-1"},{"color":"amber","style":"text-color","anchor":"fact-2"}],"connections":[{"to":"fact-1","from":"quad-var","color":"sky","label":"factorise","style":"solid"}]},"order":2,"title":"Factorise the quadratic","description":"Factorise the quadratic expression into two linear factors."},{"type":"text","order":3,"title":"Solve for cos x","content":"Set each factor to zero to find the possible values for $\\cos x$:\n\n1. From $4 \\cos x - 1 = 0$, we have:\n $$\\cos x = \\frac{1}{4}$$\n\n2. From $\\cos x + 2 = 0$, we have:\n $$\\cos x = -2$$\n\nSince the range of the cosine function is $-1 \\leq \\cos x \\leq 1$, the equation $\\cos x = -2$ has no real solutions. We reject this case."},{"type":"text","order":4,"title":"Find the angles","content":"Now we find the values of $x$ for $\\cos x = 0.25$ within the interval $-180^\\circ \\leq x \\leq 180^\\circ$.\n\nFirst, find the principal value using your calculator in degree mode:\n\n$$x = \

1.7.19 Double angle identities (A-level only)

Question 13

1.7.19 Double angle identities (A-level only) Questions

1.7.19 Double angle identities (A-level only)

Question 13

1.7.19 Double angle identities (A-level only) Questions