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bg-background","children":["$","$L3a",null,{"backHref":"/a-level/old-maths/edexcel/questions","basePath":"/a-level/old-maths/edexcel/legacy-old-alevel-e45fbe19-e001-47e7-9b5f-b0d4bbdb2d55/questions","formulaUrl":"https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2013/Specification%20and%20sample%20assessments/N38210A-GCE-Mathematical-Formulae-Statistical-Tables.pdf","initialIndex":6,"initialQuestionWindow":[{"id":"dd32b2bc-0ef4-4387-bdf6-a9bbf6a2c245","revisionId":"3bc342f9-16f1-43d2-94f2-28bd21a77ff6","questionNumber":"6","questionType":"LA","specification":"Solve the simultaneous equations\n\n$$y = x - 2,$$\n\n$$y^2 + x^2 = 10.$$","difficulty":1,"difficultyBand":"medium","totalMarks":7,"sourceTitle":"Algebra and Functions 1","calculatorUse":null,"isCalculation":true,"parts":[{"id":"d1c8cd28-e2d8-4612-920e-5ef6ffe00c6f","label":"0","content":"Solve the simultaneous equations.","markscheme":"- Substitutes $y = x - 2$ into $y^2 + x^2 = 10$: $(x - 2)^2 + x^2 = 10$ M1\n- Expands: $x^2 - 4x + 4 + x^2 = 10$ M1\n- Simplifies to $2x^2 - 4x - 6 = 0$ or $x^2 - 2x - 3 = 0$ A1\n- Factorises: $(x - 3)(x + 1) = 0$ M1\n- Finds $x = 3$ and $x = -1$ A1\n- Substitutes back to find $y$: $y = 3 - 2 = 1$ and $y = -1 - 2 = -3$ M1\n- Final answers $(3, 1)$ and $(-1, -3)$ A1","marks":7,"order":0,"markingRubric":{"totalMarks":7,"points":[{"id":"p1","marks":1,"description":"Substitutes the linear equation into the quadratic equation.","code":"M"},{"id":"p2","marks":1,"description":"Correctly expands the squared bracket $(x-2)^2$.","code":"M"},{"id":"p3","marks":1,"description":"Obtains a simplified quadratic equation in $x$, e.g., $x^2 - 2x - 3 = 0$.","code":"A"},{"id":"p4","marks":1,"description":"Attempts to factorise or solve the resulting quadratic equation.","code":"M"},{"id":"p5","marks":1,"description":"Finds two correct values for $x$ ($3$ and $-1$).","code":"A"},{"id":"p6","marks":1,"description":"Attempts to find corresponding $y$ values.","code":"M"},{"id":"p7","marks":1,"description":"Finds two correct pairs of $(x, y)$.","code":"A"}],"finalAnswers":["(3, 1), (-1, -3)"],"generatedAt":"2026-05-06T15:23:40.168Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"b7bff7c2-ab8a-4bc3-a23e-e6bae40d3649","revisionId":"230900a8-440f-43c9-99a0-1a6246815781","questionNumber":"7","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":4,"sourceTitle":"Algebra and Functions 1","calculatorUse":null,"isCalculation":true,"parts":[{"id":"d0aef385-a383-4a02-9d3b-a0d2b9dd0bcf","label":"0","content":"Find the set of values of $x$ for which\n\n$$x^2 - 7x - 18 > 0.$$","markscheme":"- Factorises to $(x - 9)(x + 2) > 0$ M1\n- Identifies critical values $x = 9$ and $x = -2$ A1\n- Uses a sketch or test values to find the outside region M1\n- Final answer $x < -2$ or $x > 9$ A1\n\n![Graph of the quadratic inequality x^2 - 7x - 18 > 0. The parabola opens upwards and intersects the x-axis at x = -2 and x = 9. The region between the roots is shaded, indicating the solution set x < -2 or x > 9.](img:e863446d7d263c68be9402afd63668ac_img.jpg)","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Factorises the quadratic expression into $(x - 9)(x + 2)$.","code":"M"},{"id":"p2","marks":1,"description":"Identifies critical values $x = 9$ and $x = -2$.","code":"A"},{"id":"p3","marks":1,"description":"Attempts to find the region outside the roots (e.g., via sketch).","code":"M"},{"id":"p4","marks":1,"description":"Final answer $x < -2$ or $x > 9$.","code":"A"}],"finalAnswers":["x < -2 or x > 9"],"generatedAt":"2026-05-06T15:23:40.168Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"96b7d211-921a-4cd5-8dbd-ffd7e170e240","revisionId":"c76473bc-53e3-46ff-af05-f1c87af9f794","questionNumber":"9","questionType":"LA","specification":"The equation $kx^2 + 4x + (5 - k) = 0$, where $k$ is a constant, has 2 different real solutions for $x$.","difficulty":1,"difficultyBand":"medium","totalMarks":8,"sourceTitle":"Algebra and Functions 1","calculatorUse":null,"isCalculation":true,"parts":[{"id":"d0695099-3f12-4207-84f7-7c1d87cf3eef","label":"a","content":"Show that $k$ satisfies\n\n$$k^2 - 5k + 4 > 0.$$","markscheme":"- Uses discriminant $b^2 - 4ac > 0$ for 2 different real solutions M1\n- Substitutes $a=k, b=4, c=5-k$: $4^2 - 4(k)(5 - k) > 0$ M1\n- Expands and simplifies: $16 - 20k + 4k^2 > 0 \\Rightarrow 4k^2 - 20k + 16 > 0$ A1\n- Divides by 4 to get $k^2 - 5k + 4 > 0$ A1","marks":4,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"States or uses the condition for two real roots $b^2 - 4ac > 0$.","code":"M"},{"id":"p2","marks":1,"description":"Correctly substitutes $a, b, c$ into the discriminant expression.","code":"M"},{"id":"p3","marks":1,"description":"Correctly simplifies the inequality to reach $k^2 - 5k + 4 > 0$.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-06T15:23:40.168Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"discriminant","label":"Use the discriminant","steps":[{"type":"text","order":0,"title":"Check it is quadratic","content":"First, $k\\ne0$. If $k=0$, the equation becomes $4x+5=0$, which has only one solution, so it cannot meet the condition of two different real solutions.\n\nTherefore the equation is quadratic. For a quadratic $ax^2+bx+c=0$ to have two different real solutions, its discriminant must satisfy $$b^2-4ac>0.$$ This earns the first method mark.","highlights":[{"text":"has 2 different real solutions for $x$","location":"specification"}]},{"type":"text","order":1,"title":"Identify the coefficients","content":"Compare $kx^2+4x+(5-k)=0$ with $ax^2+bx+c=0$:\n\n$$a=k,\\\\ b=4,\\\\ c=5-k.$$\n\nThe bracket is important: the whole expression $5-k$ is the constant term.","highlights":[{"text":"$$kx^2 + 4x + (5 - k) = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=general}{b^2-4ac>0}"},{"latex":"\\htmlData{anchor=substituted}{4^2-4(k)(5-k)>0}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"substituted"}],"connections":[{"to":"substituted","from":"general","color":"sky","label":"substitute"}]},"order":2,"title":"Substitute the coefficients","description":"Put the three coefficients into the positive-discriminant condition. This substitution earns the second method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=before}{16-4k(5-k)>0}"},{"latex":"\\htmlData{anchor=expanded}{16-20k+4k^2>0}"},{"latex":"\\htmlData{anchor=collected}{4k^2-20k+16>0}"}],"connections":[{"to":"expanded","from":"before","color":"orange","label":"expand brackets","style":"solid"},{"to":"collected","from":"expanded","color":"teal","label":"reorder terms","style":"solid"}]},"order":3,"title":"Expand the inequality","description":"Expand the product carefully, then write the terms in descending powers of the variable. This correct expansion and simplification earns the accuracy mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=before}{4k^2-20k+16>0}"},{"latex":"\\htmlData{anchor=after}{k^2-5k+4>0}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"after"}],"connections":[{"to":"after","from":"before","color":"green","label":"divide by 4","style":"solid"}]},"order":4,"title":"Divide by four","description":"Every term is divisible by the positive number 4, so dividing does not change the inequality direction. This gives the required result and earns the final accuracy mark."}]}],"generatedAt":"2026-07-21T05:18:07.056Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"bef87615-726a-4a38-93a0-bc8b06d215ea","label":"b","content":"Hence find the set of possible values of $k$.","markscheme":"- Factorises to $(k - 4)(k - 1) > 0$ M1\n- Identifies critical values $k = 4$ and $k = 1$ A1\n- Uses a sketch or test values to find the outside region M1\n- Final answer $k < 1$ or $k > 4$ A1\n\n![Graph of the quadratic inequality (k - 4)(k - 1) > 0. The parabola opens upwards and intersects the x-axis at k = 1 and k = 4. The region between the roots is shaded, indicating the solution set k < 1 or k > 4.](img:efe05e64cc211c110cbc96718fb54d44_img.jpg)","marks":4,"order":1,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Factorises the quadratic expression into $(k - 4)(k - 1)$.","code":"M"},{"id":"p2","marks":1,"description":"Identifies critical values $k = 4$ and $k = 1$.","code":"A"},{"id":"p3","marks":1,"description":"Attempts to find the region outside the roots (e.g., via sketch).","code":"M"},{"id":"p4","marks":1,"description":"Final answer $k < 1$ or $k > 4$.","code":"A"}],"finalAnswers":["k < 1 or k > 4"],"generatedAt":"2026-05-06T15:23:40.168Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"old_maths_alevel","summaries":[{"id":"4fbc8ecb-f0a2-4d66-bd4b-a850b5adac45","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"01b400ec-31ec-460f-945a-502630c1ef58","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d1cff280-ca27-421a-bc31-7eeebb0f6857","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"da1e702a-aace-489d-b10e-c5dbbc571e0c","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"b887ed59-8db5-4955-bfd6-3d9c8155f993","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"dd32b2bc-0ef4-4387-bdf6-a9bbf6a2c245","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"b7bff7c2-ab8a-4bc3-a23e-e6bae40d3649","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e8574c80-4081-43ff-abee-a82de83fcbe3","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"96b7d211-921a-4cd5-8dbd-ffd7e170e240","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"954f2646-ec5f-478b-97ea-1ba6e7ef1c71","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"7a6464dd-1160-4af5-8474-b592dd6d219d","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"976ede9c-8a7c-41c0-8ab4-41d4fe5ad9f3","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"4cf470fc-6a89-45c3-8a96-398751984545","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"60b7b6cb-7ee1-4104-996e-31c37c2a0ac7","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"1da7e784-1cbb-41d7-8f2f-2cf6b37fbb4d","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"4c2df585-9ef5-4818-a099-94ee3af81635","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"e41b3cd3-bb99-4c2c-8dec-f73fc47a9538","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"44004c7d-9aec-462a-bff8-aac56f23f52b","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true}],"title":"Algebra and Functions 1","topicId":"3b4d13b6-7f36-4140-b223-853db52c3763","topicSidebar":"$L3b","topicSlug":"legacy-old-alevel-e45fbe19-e001-47e7-9b5f-b0d4bbdb2d55","topicTitle":"Algebra and Functions 1"}]}],"$L3c","$L3d"]}]

Algebra and Functions 1

Question 6

Algebra and Functions 1 Questions

Algebra and Functions 1

Question 6

Algebra and Functions 1 Questions