= 11 - 6 = 5$$\n\nThus, the value of $k$ is $5$."}]}],"generatedAt":"2026-06-13T03:29:55.807Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"db90ba4a-08bb-4b06-a524-e1f0b9559966","revisionId":"6fbdc859-3681-408a-bc1e-7ff41b88dd93","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"hard","totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"91c39e88-c7ed-49c0-ae67-3e6d48ec1364","label":"a","content":"Calculate the gradient of the line that passes through the points $(-5, 3)$ and $(1, -9)$.","markscheme":"- Uses $m = \\frac{-9 - 3}{1 - (-5)}$ M1\n- $\\frac{-12}{6} = -2$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Shows use of $\\frac{\\Delta y}{\\Delta x}$ with the given coordinates.","code":"M"},{"id":"p2","marks":1,"description":"Correct final gradient value of -2.","code":"A"}],"finalAnswers":["-2"],"generatedAt":"2026-05-06T14:06:10.796Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"gradient_formula","label":"Use the gradient formula","steps":[{"type":"text","order":0,"title":"Recall the formula","content":"Gradient means the change in $y$ divided by the change in $x$:\n\n$$m=\\frac{y_2-y_1}{x_2-x_1}$$\n\nKeep the coordinates in the same order in the numerator and denominator.","highlights":[{"text":"Calculate the gradient","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$m=\\dfrac{\\color{@sky}{\\htmlData{anchor=y_values}{y_2-y_1}}}{\\color{@orange}{\\htmlData{anchor=x_values}{x_2-x_1}}}$"},{"latex":"$$m=\\dfrac{\\color{@sky}{\\htmlData{anchor=y_substitution}{-9-3}}}{\\color{@orange}{\\htmlData{anchor=x_substitution}{1-(-5)}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"y_substitution"},{"color":"orange","style":"text-color","anchor":"x_substitution"}],"connections":[{"to":"y_substitution","from":"y_values","color":"sky","style":"solid"},{"to":"x_substitution","from":"x_values","color":"orange","style":"solid"}]},"order":1,"title":"Substitute the coordinates","description":"Use (-5, 3) as the first point and (1, -9) as the second. Writing this correct substitution earns the M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$m=\\dfrac{\\color{@sky}{-9-3}}{\\color{@orange}{1-(-5)}}$"},{"latex":"$$-9-3=\\color{@sky}{-12}$"},{"latex":"$$1-(-5)=\\color{@orange}{6}$"},{"latex":"$$m=\\dfrac{\\color{@purple}{-12}}{\\color{@purple}{6}}$"},{"latex":"$$m=\\color{@green}{-2}$"}]},"order":2,"title":"Calculate the gradient","description":"Work out the vertical and horizontal differences, then divide. The negative answer is sensible because the line goes down as x increases; the final value earns A1."}]}],"generatedAt":"2026-07-19T16:47:17.893Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"e4226afc-b597-4b82-99dc-0d7f478f0cd9","revisionId":"5276e340-8194-4d8d-a35f-e644f6184cd3","questionNumber":null,"questionType":"LA","specification":"The line $CD$ passes through the points $C(1, 8)$ and $D(-2, k)$.\n\nThe gradient of $CD$ is $2$.","difficulty":3,"difficultyBand":"hard","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"a308765a-b7ce-4e98-ab2c-6c963c375de6","label":"a","content":"Work out the value of $k$.","markscheme":"- Sets up equation $2 = \\frac{k - 8}{-2 - 1}$ M1\n- Simplifies to $2 = \\frac{k - 8}{-3}$ and multiplies by -3 to get $-6 = k - 8$ M1\n- $k = 2$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Sets up the gradient equation correctly.","code":"M","acceptable":["2 = (k - 8) / (-3)","2 = (8 - k) / (1 - (-2))"]},{"id":"p2","marks":1,"description":"Simplifies denominator and multiplies out.","code":"M","acceptable":["-6 = k - 8","6 = 8 - k"]},{"id":"p3","marks":1,"description":"Final value for k.","code":"A"

Gradient of a Line

Question 29

Gradient of a Line Questions

Gradient of a Line

Question 29

Gradient of a Line Questions