ts at $90^\\circ, 270^\\circ$ and minimum at $180^\\circ$.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-07T02:10:10.549Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"58eec626-cc78-43c4-ab0b-20c3f9b61c40","revisionId":"7c45699d-640d-4f3e-94e2-2972608d0080","questionNumber":"4","questionType":"LA","specification":"","difficulty":1,"difficultyBand":null,"totalMarks":2,"sourceTitle":"Trigonometric and Exponential Graphs","calculatorUse":null,"isCalculation":false,"parts":[{"id":"0bec9f6a-1f9d-45e4-87a9-4383eac3fe19","label":"0","content":"On the grid, sketch the curve with equation $y = 2^x$\nGive the coordinates of any points of intersection with the axes.\n\n![A coordinate system for sketching the curve y = 2^x.](a854aa286b14d26d27047ee5893ffaa7_img.jpg)","markscheme":"- Correct shape of an exponential growth curve in the first and second quadrants M1\n- Intersection with y-axis at $(0, 1)$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Draws an increasing exponential curve that never touches the x-axis.","code":"M"},{"id":"p2","marks":1,"description":"Correct y-intercept labeled as $(0, 1)$.","code":"A"}],"finalAnswers":["(0, 1)"],"generatedAt":"2026-05-07T02:10:10.549Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"26ebf2d6-0736-4e21-b5de-ddd25f4a73d7","revisionId":"e0bc9dda-025b-4b6f-a7ca-155f87b14013","questionNumber":"9","questionType":"LA","specification":"The depth of water, $d$ metres, at the entrance to a harbour is given by the formula:\n$d = 5 - 4\\sin(30t)$, where $t$ is the time in hours after midnight on one day.","difficulty":3,"difficultyBand":null,"totalMarks":6,"sourceTitle":"Trigonometric and Exponential Graphs","calculatorUse":null,"isCalculation":true,"parts":[{"id":"74611682-2959-4e90-abe6-8f99fa87a0ae","label":"a","content":"On the axes below, draw the graph of $d$ against $t$ for $0 \\leq t \\leq 12$\n\n![A coordinate grid for plotting the graph of d against t.](d864789b0d8384da1d22fd6a5d76bbdf_img.jpg)","markscheme":"- Calculates at least 3 points correctly (e.g., $t=0, d=5$; $t=3, d=1$; $t=9, d=9$) M1\n- Plots points correctly on the grid M1\n- Draws a smooth sine curve through the points A1\n- Correct amplitude and vertical shift shown A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Calculates key coordinates for the graph.","code":"M"},{"id":"p2","marks":1,"description":"Plots points accurately on the provided grid.","code":"M"},{"id":"p3","marks":1,"description":"Draws a smooth sinusoidal curve.","code":"A"},{"id":"p4","marks":1,"description":"Graph correctly ranges from $d=1$ to $d=9$ with correct period.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-07T02:10:10.549Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]},{"id":"0081853a-612e-4add-ac81-0d81856171bb","label":"b","content":"Find the two values of $t$, where $0 \\leq t \\leq 24$, when the depth is least.","markscheme":"- Identifies depth is least when $\\sin(30t) = 1$ M1\n- $30t = 90 \\implies t = 3$ and $30t = 450 \\implies t = 15$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Identifies $t=3$ from the graph or calculation.","code":"M"},{"id":"p2","marks":1,"description":"Identifies the second value $t=15$ using periodicity.","code":"A"}],"finalAnswers":["3, 15"],"generatedAt":"2026-05-07T02:10:10.549Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths__cie","summarie

Trigonometric and Exponential Graphs

Question 4

Trigonometric and Exponential Graphs Questions

Trigonometric and Exponential Graphs

Question 4

Trigonometric and Exponential Graphs Questions