xt","order":1,"title":"Draw the horizontal line","content":"On your graph, locate $0.75$ on the vertical $y$-axis (which is three-quarters of the way up from $0$ to $1$).\n\nDraw a horizontal line across the grid at $y = 0.75$. This line will cross the cosine curve at two distinct points within the range $0 \\le x \\le 360$."},{"type":"text","order":2,"title":"Read the intersection points","content":"Now, project down vertically from each of the two intersection points to the horizontal $x$-axis to estimate the corresponding angles:\n\n1. **First intersection (between $0^\\circ$ and $90^\\circ$):** The line intersects the curve at approximately $x = 41^\\circ$.\n2. **Second intersection (between $270^\\circ$ and $360^\\circ$):** Due to the symmetry of the cosine graph, the second solution is located at $360^\\circ - 41^\\circ = 319^\\circ$.\n\nThus, our estimated solutions are:\n$$x \\approx 41^\\circ \\quad \\text{and} \\quad x \\approx 319^\\circ$$\n*(Note: Any estimates between $39^\\circ$ to $43^\\circ$ and $317^\\circ$ to $321^\\circ$ are accepted in the exam).*"}]},{"id":"e53406b2-a1de-4a3a-bdf8-21349c7a317b:symmetry_rule","label":"(ii) Graph Symmetry and Reference Angle","steps":[{"type":"text","order":0,"title":"Rearrange the equation","content":"To find the solutions using our graph, we first need to isolate $\\cos(x)$ in the equation:\n\n$$4 \\cos(x) = 3$$\n\nDivide both sides by $4$:\n\n$$\\cos(x) = \\frac{3}{4}$$\n\n$$\\cos(x) = 0.75$$\n\nThis rearranges our problem to finding where the curve $y = \\cos(x)$ crosses the horizontal line $y = 0.75$. This step earns you the first method mark (**M1**).","highlights":[{"text":"4 \\cos(x) = 3","location":"specification"}]},{"type":"text","order":1,"title":"Estimate the first solution","content":"Next, we locate $0.75$ on the $y$-axis of the graph and draw a horizontal line across to meet the cosine curve in the first quadrant (between $0^\\circ$ and $90^\\circ$).\n\nLooking at the graph, the line $y = 0.75$ intersects the curve at approximately:\n\n$$x \\approx 41^\\circ$$\n\nThis first angle is our reference angle in the first quadrant."},{"type":"text","order":2,"title":"Find the second solution","content":"The cosine graph is symmetrical about the line $x = 180^\\circ$. This symmetry means that if we have a solution at $x$, another solution exists at:\n\n$$360^\\circ - x$$\n\nUsing our first estimated solution of $41^\\circ$:\n\n$$\\text{Second solution} = 360^\\circ - 41^\\circ = 319^\\circ$$"},{"type":"text","order":3,"title":"State final answers","content":"The two estimated solutions in the interval $0 \\leq x \\leq 360$ are:\n\n$$x = 41^\\circ \\quad \\text{and} \\quad x = 319^\\circ$$\n\n*(Note: The exam markscheme accepts any values in the range of $\\pm 2^\\circ$ from these answers, so $39^\\circ$ to $43^\\circ$ and $317^\\circ$ to $321^\\circ$ are both acceptable for the final accuracy mark **A1**).* "}]}],"generatedAt":"2026-06-12T13:35:00.772Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"06fb4588-8932-42be-a84b-4df3269c3e19","revisionId":"33e9b87b-b02a-4393-bfff-d92354a37eac","questionNumber":"8","questionType":"LA","specification":"This sketch shows part of the graph with equation $y = pq^x$ where $p$ and $q$ are constants.\n\n![A sketch of an exponential growth curve.](https://assets.mathsgenie.co.uk/questions/4cefe1fd-e3ba-4abe-852d-5b6f46c0b06d.png)\n\nThe points with coordinates $(0, 8)$, $(1, 18)$ and $(1.5, k)$ lie on the graph.\n\nCalculate the values of $p$, $q$ and $k$.","difficulty":3,"difficultyBand":"medium","totalMarks":6,"sourceTitle":"Trigonometric and Exponential Graphs","calculatorUse":"non_calc","parts":[{"id":"68b53bbc-7da7-41a7-9ed2-b1e83583067d","label":"0","content":"","markscheme":"- Substitutes $(0, 8)$ into $y = pq^x$: $8 = pq^0 \\implies p = 8$ M1 A1\n- Substitutes $(1, 18)$ and $p=8$: $18 = 8q^1 \\implies q = 18/8 = 2.25$ M1 A1\n- Substitutes $(1.5, k)$ and values of $p, q$: $k = 8(2.25)^{1.5}$ M1\

Trigonometric and Exponential Graphs

Question 12

Trigonometric and Exponential Graphs Questions

Trigonometric and Exponential Graphs

Question 12

Trigonometric and Exponential Graphs Questions