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$k$ is a constant, has two distinct real solutions for $x$.","difficulty":null,"difficultyBand":"hard","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"1b8d0ee9-f17c-430d-b0e4-50c6013a646d","label":null,"content":"Find the set of possible values for $k$.","markscheme":"- Uses the condition for two distinct real solutions, $b^2 - 4ac > 0$ M1\n- $16 - 4(k + 5)(k + 2) > 0$ A1\n- Expands to $16 - 4\\left(k^2 + 7k + 10\\right) > 0$ and rearranges to $k^2 + 7k + 6 < 0$ M1\n- $(k + 6)(k + 1) < 0$, with critical values $k = -6$ and $k = -1$ A1\n- $-6 < k < -1$ A1\n- States that $k \\neq -5$, since the equation must be quadratic for the discriminant condition to apply B1\n\nThe reasoning mark is for excluding $k = -5$: at $k = -5$ the equation reduces to the linear equation $4x - 3 = 0$, which has a single root. Final answer $\\{k : -6 < k < -5\\} \\cup \\{k : -5 < k < -1\\}$.","marks":6,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"56e69f37-b3d0-4f24-b08e-d75c3660a5c1","revisionId":"13970ff2-394c-47d8-8fc4-c0c969bcc316","questionNumber":null,"questionType":"LA","specification":"","difficulty":null,"difficultyBand":"hard","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"207a3129-1552-46aa-9e99-d50d0e672485","label":null,"content":"Find the set of values of $a$ for which the equation\n$ax^2 + 7x + 4 = 0$\nhas no real roots.","markscheme":"- Uses $b^2 - 4ac < 0$ to give $49 - 16a < 0$ M1\n- Rearranges to $49 < 16a$, or divides by $-16$ and reverses the inequality M1\n- $a > \\dfrac{49}{16}$ A1\n\nAccept $\\left\\{a : a > \\frac{49}{16}\\right\\}$ or $a > 3.0625$. Note $a \\neq 0$ is automatic here, since $a = 0$ does not satisfy $a > \\frac{49}{16}$.","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"79b33904-ece0-4df1-a119-0a973d55a8b5","revisionId":"8a22b779-5ac2-4d19-b43c-563f266fc9e2","questionNumber":null,"questionType":"LA","specification":"A circle has equation\n$x^2 + y^2 = 10y - 6x - 9$","difficulty":null,"difficultyBand":"hard","tier":null,"totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"5e2969da-5f47-49cd-9538-1e77c304858f","label":"a","content":"By completing the square in $x$ and in $y$, express the equation of the circle in the form\n$(x - a)^2 + (y - b)^2 = k$\nwhere $a$, $b$ and $k$ are constants to be found.","markscheme":"- Collects all terms on one side: $x^2 + 6x + y^2 - 10y + 9 = 0$ M1\n- Completes the square in both variables: $(x + 3)^2 - 9 + (y - 5)^2 - 25 + 9 = 0$ M1\n- $(x + 3)^2 + (y - 5)^2 = 25$, so $a = -3$, $b = 5$, $k = 25$ A1","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"ae8fa48f-9941-431a-91de-93561743678b","label":"b","content":"Write down the coordinates of the centre of the circle and its radius.","markscheme":"- Centre $(-3, 5)$ B1\n- Radius $5$ B1\n\nFollow through from their answer to (a). Award B0 for a radius of $25$.","marks":2,"order":1,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"a_level_maths__aqa","summaries":[{"id":"8e3022c9-9ad3-4593-9340-e8ab87f62ba0","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"fb42769c-80c0-434b-964d-32d8959b45fe","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"52a37ba6-3fd4-4a41-be5a-64b294d7b11f","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"0d44383a-7b1d-4707-8627-054115b7ef2c","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"188b59cb-3af6-4ef5-9777-f269ab4b88c8","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"2b3c048d-236c-4670-a3bf-5133722b4b6d","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"2c701fb7-c7ad-456e-aada-7b83a601a8ea","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"ae1ee63f-7a01-4e77-8cbb-7fa84b5e929d","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"c079b5ea-11f0-4b79-9755-12ec5ee2892a","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a083b46e-aa16-4e02-81f8-ac05814cc28d","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"a1965528-b29a-4064-b9d8-29556a9ccfd4","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"9a9de4f4-eeb3-49c5-938d-c02a74b8964a","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"556fcb8f-7ea0-47c8-8ab7-7700b4c15287","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"8c5e576e-aebd-4d00-a341-9092c3972611","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"04027f4f-6fe0-423e-8a28-e6ae748907bf","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"26e32529-8249-42e6-a058-408a0ff4d360","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"dbbdc94b-8000-449f-b991-c2ccaa461290","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"24a47b91-a657-4122-a39a-ac11729480b4","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"89fd2d57-fe2e-4420-aae5-adeb88c92302","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"70b49dc8-1f59-480b-884f-19c1abcea4e5","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"8d0aec80-3775-4fb6-a162-824cccfc97c0","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"0a41db84-0db5-4fa7-a452-8fe9583ae604","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"MC","isCalculation":true},{"id":"13b06d6a-6601-43b9-af2e-d864313b3d73","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"186e70d7-1010-4f8b-81a8-7f67bbfdecbd","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"1ede2c17-616a-4091-b5f5-ce32b5940dca","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"1f2bd0a6-9a46-4396-ba23-ef56bd211127","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"24094c1e-7cfb-493e-a8f7-edf4023600af","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"245a450a-4173-4c61-8813-7c78e4324840","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"2e516584-9e76-4930-a0a4-5c0fd774084d","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"34348a61-0ba6-489d-a231-d5d3af8c050c","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"34bcbdeb-3b07-45a8-a3fd-17142fb5616f","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"34d84a29-1b91-4323-9026-e17ddf355047","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"3fc6dac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Quadratic functions","topicId":"5cb60c26-e0fd-410b-ba56-ccca81129656","topicSidebar":"$L3a","topicSlug":"quadratic-functions","topicTitle":"1.5.3 Quadratic functions"}]}],"$L3b","$L3c"]}]

1.5.3 Quadratic functions

Question 7

1.5.3 Quadratic functions Questions

1.5.3 Quadratic functions

Question 7

1.5.3 Quadratic functions Questions