1. Problem: Calculate the percentage error in measuring a string of length 400 cm when rounded to the nearest metre and nearest 100 m. Step 1: Convert 400 cm to metres: $$400\text{ cm} = 4\text{ m}$$ (a) Nearest metre: - The maximum error when rounding to the nearest metre is half a metre: $$\pm 0.5\text{ m}$$ - Percentage error = $$\frac{0.5}{4} \times 100 = 12.5\%$$ (to 3 s.f.) (b) Nearest 100 m: - The maximum error when rounding to the nearest 100 m is half of 100 m: $$\pm 50\text{ m}$$ - Percentage error = $$\frac{50}{4} \times 100 = 1250\%$$ (to 3 s.f.) 2. Problem: Evaluate $$\frac{98.5^2 + \sqrt{79000}}{700 \times 5^2 \times 5.87}$$ Step 1: Calculate numerator: - $$98.5^2 = 9702.25$$ - $$\sqrt{79000} \approx 281.42$$ - Sum = $$9702.25 + 281.42 = 9983.67$$ Step 2: Calculate denominator: - $$5^2 = 25$$ - $$700 \times 25 \times 5.87 = 700 \times 146.75 = 102725$$ Step 3: Divide numerator by denominator: - $$\frac{9983.67}{102725} \approx 0.0971$$ (to 4 s.f.) 3. Problem: Given the 3rd term $$T_3 = 48$$ and 6th term $$T_6 = 14$$ of a geometric progression (G.P), find: Step 1: Recall formula for $$n$$th term of G.P: $$T_n = ar^{n-1}$$ where $$a$$ is first term and $$r$$ is common ratio. Step 2: Write equations: - $$T_3 = ar^2 = 48$$ - $$T_6 = ar^5 = 14$$ Step 3: Divide $$T_6$$ by $$T_3$$ to find $$r^3$$: - $$\frac{ar^5}{ar^2} = r^3 = \frac{14}{48} = \frac{7}{24}$$ Step 4: Find $$r$$: - $$r = \sqrt[3]{\frac{7}{24}} \approx 0.613$$ Step 5: Find $$a$$ using $$T_3 = ar^2$$: - $$a = \frac{48}{r^2} = \frac{48}{(0.613)^2} \approx \frac{48}{0.376} = 127.66$$ Step 6: Find sum of first 5 terms $$S_5$$: - Formula: $$S_n = a \frac{1-r^n}{1-r}$$ - Calculate $$r^5 = (0.613)^5 \approx 0.087$$ - $$S_5 = 127.66 \times \frac{1-0.087}{1-0.613} = 127.66 \times \frac{0.913}{0.387} \approx 127.66 \times 2.36 = 301.3$$ Step 7: Write first 4 terms: - $$T_1 = a = 127.66$$ - $$T_2 = ar = 127.66 \times 0.613 = 78.25$$ - $$T_3 = 48$$ (given) - $$T_4 = ar^3 = 127.66 \times (0.613)^3 = 127.66 \times 0.230 = 29.36$$ Final answers: (a) First term $$a \approx 127.66$$ (b) Common ratio $$r \approx 0.613$$ (c) Sum of first 5 terms $$S_5 \approx 301.3$$ (d) First 4 terms: $$127.66, 78.25, 48, 29.36$$