gineer's mistake.\n\n(ii) Write down a correct version of the proof that \$\\alpha^2 + \\beta^2\$ is rational.","markscheme":"- Identifies that Step 1 is incorrect because the expansion of $(\\alpha + \\beta)^2$ is $\\alpha^2 + 2\\alpha\\beta + \\beta^2$, which is not equal to $\\alpha^2 + \\beta^2$ unless $\\alpha\\beta = 0$ B1\n\n- Uses the identity $\\alpha^2 + \\beta^2 = (\\alpha + \\beta)^2 - 2\\alpha\\beta$ M1\n- Explains that since $\\alpha + \\beta$ is rational, $(\\alpha + \\beta)^2$ is rational; since $\\alpha\\beta$ is rational, $2\\alpha\\beta$ is rational; and since the difference of two rational numbers is rational, $\\alpha^2 + \\beta^2$ must be rational A1","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"algebraic_identity","label":"Algebraic Identity Approach","steps":[{"type":"text","order":0,"title":"Analyze the mistake","content":"To find the mistake in Step 1, let's look at the algebraic expansion of a squared binomial:\n\n$$(\\alpha + \\beta)^2 = \\alpha^2 + 2\\alpha\\beta + \\beta^2$$\n\nIn Step 1, the engineer states that $\\alpha^2 + \\beta^2 = (\\alpha + \\beta)^2$. Comparing this to our expansion, we can see that this is only true if the middle term $2\\alpha\\beta = 0$, which is not generally the case since we are only told that $\\alpha$ and $\\beta$ are irrational numbers with a rational sum and product.\n\nTherefore, the mistake is that the engineer omitted the middle term $2\\alpha\\beta$ when expanding.","highlights":[{"text":"Step 1: \$\\alpha^2 + \\beta^2 = (\\alpha + \\beta)^2\$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(\\alpha + \\beta)^2 = \\alpha^2 + \\htmlData{anchor=mid-term}{2\\alpha\\beta} + \\beta^2$"},{"latex":"$$\\alpha^2 + \\beta^2 = \\htmlData{anchor=sq-term}{(\\alpha + \\beta)^2} - \\htmlData{anchor=sub-term}{2\\alpha\\beta}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"mid-term"},{"color":"sky","style":"text-color","anchor":"sq-term"},{"color":"amber","style":"text-color","anchor":"sub-term"}],"connections":[{"to":"sub-term","from":"mid-term","color":"amber","label":"rearranged","style":"solid"}]},"order":1,"title":"Establish the identity","description":"To write a correct proof, we must use the correct algebraic identity that links the sum of squares to the sum and product of the variables."},{"type":"text","order":2,"title":"Apply closure properties","content":"Now we use the given information that the sum $\\alpha + \\beta$ and the product $\\alpha\\beta$ are both rational numbers.\n\nBy the closure properties of rational numbers (the product, difference, and power of rational numbers are also rational):\n1. Since $\\alpha + \\beta$ is rational, its square $(\\alpha + \\beta)^2$ must also be rational.\n2. Since $\\alpha\\beta$ is rational, multiplying it by the integer $2$ gives $2\\alpha\\beta$, which must also be rational.\n3. Since both $(\\alpha + \\beta)^2$ and $2\\alpha\\beta$ are rational, their difference $(\\alpha + \\beta)^2 - 2\\alpha\\beta$ must be rational.\n\nThus, $\\alpha^2 + \\beta^2$ is rational.","highlights":[{"text":"rational sum and a rational product","location":"specification"}]}]}],"generatedAt":"2026-06-09T03:59:23.109Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"8746c6b3-2dcc-49a0-aae7-ab8446ead85c","label":"b","content":"Prove by contradiction that the product of any non-zero rational number and any irrational number is irrational.","markscheme":"- Assume that the product of a non-zero rational number $r$ and an irrational number $k$ is a rational number $q$. Let $rk = q$. M1\n- Since $r$ and $q$ are rational, let $r = \\frac{a}{b}$ and $q = \\frac{c}{d}$ where $a, b, c, d \\in \\mathbb{Z}$ and $a, b, d \\neq 0$. M1\n- Rearrange for the irrational number: $k = \\frac{q}{r} = \\frac{c/d}{a/b} = \\frac{bc}{ad}$ M1\n- Since $bc$ and $ad$ are integers and $ad \\neq 0$ (as $a \\neq 0$ and $d \\neq 0$), $k$ must b

Algebraic Methods

Question 93

Algebraic Methods Questions

Algebraic Methods

Question 93

Algebraic Methods Questions