1. Visualize 6.376 on the number line up to 4 decimal places using successive magnification. Step 1: Identify the range between integers 6 and 7 on the number line. Step 2: Zoom in to divide the segment into 10 equal parts for the first decimal place; 6.3 is selected. Step 3: Further divide the segment between 6.3 and 6.4 into 10 parts; select 6.37. Step 4: Divide the segment between 6.37 and 6.38; select 6.376. Step 5: Precision up to 4 decimal places: 6.3760. 2. Construct frequency distribution table with class interval 10 for the data. Data points: 11,31,41,24,14,42,29,19,30,34,8,42,27,35,3,41,32,43,5,48,33,22,49,23,25,6,36,6,5,31,21,29,7,34,18,28,44,28,39,20 Class intervals: 0-9,10-19,20-29,30-39,40-49 Count frequencies: 0-9: 3,5,6,6,5,7 = 6 10-19: 11,14,19,18,15,21,20 = 7 20-29: 24,29,27,23,25,28,28,29,22 = 9 30-39: 31,32,34,33,36,31,34,35,39,30 = 10 40-49: 41,42,42,41,43,48,44,49 = 8 | Class Interval | Frequency | | 0-9 | 6 | | 10-19 | 7 | | 20-29 | 9 | | 30-39 | 10 | | 40-49 | 8 | 3. Show that $$\frac{\sqrt5 + \sqrt3}{\sqrt5 - \sqrt3} + \frac{\sqrt5 - \sqrt3}{\sqrt5 + \sqrt3} = 8$$. Step 1: Let $$A = \frac{\sqrt5 + \sqrt3}{\sqrt5 - \sqrt3}$$ and $$B = \frac{\sqrt5 - \sqrt3}{\sqrt5 + \sqrt3}$$. Step 2: Find $$A + B = \frac{(\sqrt5 + \sqrt3)^2 + (\sqrt5 - \sqrt3)^2}{(\sqrt5 - \sqrt3)(\sqrt5 + \sqrt3)}$$. Step 3: Calculate numerator: $$(\sqrt5 + \sqrt3)^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15}$$ $$(\sqrt5 - \sqrt3)^2 = 5 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15}$$ Sum = $$8 + 2\sqrt{15} + 8 - 2\sqrt{15} = 16$$. Step 4: Calculate denominator: $$(\sqrt5)^2 - (\sqrt3)^2 = 5 - 3 = 2$$. Step 5: Therefore, $$A + B = \frac{16}{2} = 8$$. 4. Find value of $$p$$ given mean is 21.7 for data: | x | 10 | 15 | 20 | 25 | 30 | 35 | | y | 6 | 8 | 15 | p | 8 | 4 | Step 1: Mean formula $$\bar{x} = \frac{\sum fx}{\sum f}$$ where $$f = y$$ and $$x$$ is data values. Step 2: Compute $$\sum fx$$ excluding $$p$$: $$10\times6 + 15\times8 + 20\times15 + 30\times8 + 35\times4 = 60 + 120 + 300 + 240 + 140 = 860$$ Step 3: Include $$p$$: $$25 \times p$$. Step 4: Total frequency $$= 6+8+15+p+8+4 = 41 + p$$. Step 5: Mean given: $$21.7 = \frac{860 + 25p}{41 + p}$$. Step 6: Multiply both sides: $$21.7(41 + p) = 860 + 25p$$. Step 7: $$889.7 + 21.7p = 860 + 25p$$. Step 8: Rearrange: $$889.7 - 860 = 25p - 21.7p$$. Step 9: $$29.7 = 3.3p$$. Step 10: $$p = \frac{29.7}{3.3} = 9$$. 5. Calculate number of cylindrical bottles to fill from hemispherical bowl. Given: Diameter of bowl $$= 18 cm$$, radius $$r = 9 cm$$ Diameter of bottle $$= 2 cm$$, radius $$= 1 cm$$, height $$h = 6 cm$$. Step 1: Volume of hemisphere $$V = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi (9)^3 = \frac{2}{3} \pi 729 = 486 \pi$$. Step 2: Volume of cylinder $$V_{cyl} = \pi r^2 h = \pi (1)^2 (6) = 6\pi$$. Step 3: Number of bottles = $$\frac{486\pi}{6\pi} = 81$$. 6. Find 6 different solutions for $$3x + 4y = 8$$. Step 1: Express $$y$$ in terms of $$x$$: $$4y = 8 - 3x \Rightarrow y = \frac{8 - 3x}{4}$$. Step 2: Select 6 values of $$x$$ and compute $$y$$: $$x=0, y=2$$; $$x=4, y=-1$$; $$x=8, y=-4$$; $$x=-4, y=5$$; $$x=2, y=0.5$$; $$x= -8, y=8$$. Final answers: Q13: Visualized 6.376 up to 4 decimals. Q14: Frequency table with frequencies 6,7,9,10,8. Q15: Expression equals 8. Q16: $$p=9$$. Q17: 81 bottles. Q18: Six solutions: (0,2), (4,-1), (8,-4), (-4,5), (2,0.5), (-8,8).