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$c$.","markscheme":"- Substitutes $x = 3$, $y = 5$ into $x^2 + y^2 = a$: $3^2 + 5^2 = a$ M1\n- Finds $a$: $a = 9 + 25 = 34$ M1\n- Substitutes both solutions into $y = bx + c$: $5 = 3b + c$ and $-5.4 = -2.2b + c$ M1\n- Eliminates $c$: $10.4 = 5.2b$, so $b = 2$ M1\n- Finds $c$: $5 = 3(2) + c$, so $c = -1$. Final answers: $a = 34$, $b = 2$, $c = -1$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Substitutes the integer solution into the first equation to obtain $3^2 + 5^2 = a$.","code":"M","acceptable":["$$3^2 + 5^2 = a$","$$9 + 25 = a$"]},{"id":"p2","marks":1,"description":"Evaluates the sum to find $a = 34$.","code":"M","acceptable":["$$a = 34$"]},{"id":"p3","marks":1,"description":"Substitutes both given solutions into the linear equation.","code":"M","acceptable":["$$5 = 3b + c$ and $-5.4 = -2.2b + c$","Equivalent simultaneous equations in $b$ and $c$"]},{"id":"p4","marks":1,"description":"Eliminates $c$ and solves for $b$.","code":"M","acceptable":["$$10.4 = 5.2b$","$$b = 2$"]},{"id":"p5","marks":1,"description":"Gives all three correct values.","code":"A","acceptable":["$$a = 34$, $b = 2$, $c = -1$"]}],"finalAnswers":["$$a = 34$, $b = 2$, $c = -1$"],"generatedAt":"2026-09-22T15:28:00.270Z","modelVersion":"gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/review/10107d63-ada1-4a56-95ac-c3fc2c5c8d97.mp4","durationSeconds":165,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/review/10107d63-ada1-4a56-95ac-c3fc2c5c8d97.jpg"}},{"id":"f0facfac-eb63-41a0-a2ae-91f6f0f4d387","revisionId":"9fe15457-433b-45b0-90bc-a6bbeb9fd309","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"medium","tier":null,"totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"3cdb8129-a4b2-4b17-84d0-c9d598bc4f08","label":null,"content":"Solve the simultaneous equations\n\n$$x^2 + y^2 = 24$$\n\n$$x = y - 4$$\n\nGive your answers in exact form.","markscheme":"- Substitutes $x = y - 4$ into the first equation: $(y - 4)^2 + y^2 = 24$ M1\n- Expands and simplifies to a quadratic equation: $2y^2 - 8y - 8 = 0$ or $y^2 - 4y - 4 = 0$ M1\n- Uses the quadratic formula to find $y$: $y = 2 + 2\\sqrt{2}$ or $y = 2 - 2\\sqrt{2}$ M1\n- Substitutes $y$ values back to find $x$: $x = -2 + 2\\sqrt{2}$ and $x = -2 - 2\\sqrt{2}$ M1\n- Final answers: $x = -2 + 2\\sqrt{2}, y = 2 + 2\\sqrt{2}$ and $x = -2 - 2\\sqrt{2}, y = 2 - 2\\sqrt{2}$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Substitutes $x = y - 4$ into the first equation to obtain $(y - 4)^2 + y^2 = 24$.","code":"M","acceptable":["$$(y - 4)^2 + y^2 = 24$"]},{"id":"p2","marks":1,"description":"Expands and simplifies to a quadratic equation.","code":"M","acceptable":["$$2y^2 - 8y - 8 = 0$","$$y^2 - 4y - 4 = 0$"]},{"id":"p3","marks":1,"description":"Uses the quadratic formula to find both values of $y$.","code":"M","acceptable":["$$y = 2 + 2\\sqrt{2}$ and $y = 2 - 2\\sqrt{2}$"]},{"id":"p4","marks":1,"description":"Substitutes the $y$ values back to find both values of $x$.","code":"M","acceptable":["$$x = -2 + 2\\sqrt{2}$ and $x = -2 - 2\\sqrt{2}$"]},{"id":"p5","marks":1,"description":"Gives both simultaneous solutions in exact form.","code":"A","acceptable":["$$x = -2 + 2\\sqrt{2}, y = 2 + 2\\sqrt{2}$ and $x = -2 - 2\\sqrt{2}, y = 2 - 2\\sqrt{2}$"]}],"finalAnswers":[],"generatedAt":"2026-09-22T14:59:10.625Z","modelVersion":"gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"280c9511-9a5a-4086-9148-b19674d63a15","revisionId":"ea7e0c50-a413-460a-9329-37ef4e2f5f8b","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"medium","tier":null,"totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"474cae55-fb51-44b5-bea5-555414b6784a","label":null,"content":"Verify that the solutions of the simultaneous equations\n\n$$x^2 + y^2 = 17$$\n\n$$y = x - 3$$\n\nare $(4,1)$ and $(-1,-4)$.","markscheme":"- Substitutes $x = 4$ and $y = 1$ into the first equation: $4^2 + 1^2 = 16 + 1 = 17$ M1\n- Verifies the second equation for $(4,1)$: $1 = 4 - 3$ M1\n- Substitutes $x = -1$ and $y = -4$ into the first equation: $(-1)^2 + (-4)^2 = 1 + 16 = 17$ M1\n- Verifies the second equation for $(-1,-4)$: $-4 = -1 - 3$ M1\n- Concludes that both $(4,1)$ and $(-1,-4)$ satisfy both simultaneous equations A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Substitutes $x = 4$ and $y = 1$ into the first equation and obtains $16 + 1 = 17$.","code":"M","acceptable":["$$4^2 + 1^2 = 17$","$$16 + 1 = 17$"]},{"id":"p2","marks":1,"description":"Verifies the second equation for $(4,1)$.","code":"M","acceptable":["$$1 = 4 - 3$","$$1 = 1$"]},{"id":"p3","marks":1,"description":"Substitutes $x = -1$ and $y = -4$ into the first equation and obtains $1 + 16 = 17$.","code":"M","acceptable":["$$(-1)^2 + (-4)^2 = 17$","$$1 + 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