,"description":"We perform the synthetic division as follows:\n\n1. Bring down the first coefficient: $2$\n2. Multiply by the root $2$: $2 \\times 2 = 4$. Add to the next term to get $4 - k$.\n3. Multiply by $2$: $2 \\times (4 - k) = 8 - 2k$. Add to the next term to get $5 + (8 - 2k) = 13 - 2k$.\n4. Multiply by $2$: $2 \\times (13 - 2k) = 26 - 4k$. Add to the last term to find the remainder: $11 + (26 - 4k) = 37 - 4k$."},{"type":"text","order":2,"title":"Equate remainder and solve","content":"The question states that the remainder when divided by $(t - 2)$ is $17$.\n\nWe set our remainder expression equal to $17$ and solve for $k$:\n\n$$\\begin{aligned} 37 - 4k &= 17 \\\\ -4k &= 17 - 37 \\\\ -4k &= -20 \\\\ k &= 5 \\end{aligned}$$\n\nThus, the value of $k$ is $5$.","highlights":[{"text":"remainder when h(t) is divided by (t - 2) is 17","location":"specification"}]}]}],"generatedAt":"2026-06-09T03:01:40.310Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"072b940e-4992-409f-86ca-539b215200a5","revisionId":"16d60f99-4b4f-4308-9460-46b24b987db2","questionNumber":null,"questionType":"LA","specification":"The boundary of a circular wildlife sanctuary is modeled by the equation $x^2 + y^2 - 14x + 8y = 175$, where distances are measured in kilometers.\n\nA straight firebreak path connects two points $A$ and $B$ on the boundary. The angle subtended by the arc $AB$ at the center of the sanctuary is $1.4$ radians.","difficulty":4,"difficultyBand":"medium","totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"26ddb6cb-3526-4d50-947c-f80ca878e43e","label":null,"content":"Calculate the area of the region bounded by the arc $AB$ and the straight path $AB$ (the minor segment).\n\nGive your answer to three significant figures.","markscheme":"- Rearranges the circle equation by completing the square to find $r^2$: $(x - 7)^2 - 49 + (y + 4)^2 - 16 = 175$ M1\n- Identifies the radius squared as $r^2 = 175 + 49 + 16 = 240$ A1\n- Uses the formula for the area of a sector: $\\text{Area} = \\frac{1}{2}r^2\\theta = \\frac{1}{2} \\times 240 \\times 1.4 = 168$ M1\n- Uses the formula for the area of the triangle: $\\text{Area} = \\frac{1}{2}r^2\\sin\\theta = \\frac{1}{2} \\times 240 \\times \\sin(1.4) \\approx 118.25...$ M1\n- Subtracts the triangle area from the sector area to find the segment area: $168 - 118.25... = 49.746...$ M1\n- States the final answer as $49.7$ km$^2$ A1","marks":6,"order":0,"markingRubric":null,"explanation":{"methods":[{"id":"difference_of_areas","label":"Sector minus Triangle Area","steps":[{"type":"text","order":0,"title":"Analyze the problem","content":"We need to find the area of the minor segment bounded by the arc $AB$ and the chord $AB$.\n\nThe **Sector minus Triangle Area** method tells us that the area of a circular segment is found by subtracting the area of the triangle formed by the two radii and the chord from the area of the entire circular sector:\n\n$$\\text{Segment Area} = \\text{Sector Area} - \\text{Triangle Area}$$\n\nOur first step is to find the radius of the circle, $r$, or more specifically, the value of $r^2$. We can find this by rearranging the equation of the circle."},{"type":"text","order":1,"title":"Find radius squared","content":"The given equation of the circle is:\n\n$$x^2 + y^2 - 14x + 8y = 175$$\n\nTo find the radius, we complete the square for both the $x$ terms and the $y$ terms:\n\n$$\\begin{aligned} (x^2 - 14x) + (y^2 + 8y) &= 175 \\\\ \\left[(x - 7)^2 - 49\\right] + \\left[(y + 4)^2 - 16\\right] &= 175 \\\\ (x - 7)^2 + (y + 4)^2 - 65 &= 175 \\\\ (x - 7)^2 + (y + 4)^2 &= 175 + 65 \\\\ (x - 7)^2 + (y + 4)^2 &= 240 \\end{aligned}$$\n\nComparing this to the standard equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$, we identify the radius squared as:\n\n$$r^2 = 240$$","highlights":[{"text":"x^2 + y^2 - 14x + 8y = 175","location":"specification"}]},{"type":"text","order":2,"title":"Calc

1 Pure Mathematics

Question 824

1 Pure Mathematics Questions