1. Solve the system using elimination method: Given: $$5a - 3b + c = -3$$ $$7a + 2b - 3c = -35$$ $$a - 6b + 7c = 51$$ Step 1: Eliminate variables systematically. Multiply the third equation by 3 to align with second equation for $c$: $$3a - 18b + 21c = 153$$ Step 2: Add the second and modified third equation to eliminate $c$: $$7a + 2b - 3c + 3a - 18b + 21c = -35 + 153$$ $$10a - 16b + 18c = 118$$ (Note: Error in elimination, revise) Better approach: Multiply first equation by 3: $$15a - 9b + 3c = -9$$ Add to second equation: $$7a + 2b - 3c = -35$$ Sum: $$22a - 7b = -44$$ Step 3: Use third equation and above result to solve: $$a - 6b + 7c = 51$$ Express $a$ from the equation $22a -7b = -44$: $$a = \frac{-44 + 7b}{22}$$ Step 4: Substitute $a$ in third equation: $$\frac{-44 + 7b}{22} - 6b + 7c = 51$$ Multiply through by 22: $$-44 + 7b - 132b + 154c = 1122$$ $$-44 - 125b + 154c = 1122$$ $$-125b + 154c = 1166$$ Step 5: From first equation (multiply by 3) and second equation sum, solve for $b$ and $c$. From step 2: $$22a - 7b = -44$$ Use original first equation for elimination: $$5a - 3b + c = -3$$ Express $c$: $$c = -3 - 5a + 3b$$ Substitute $c$ into equation from step 4: $$-125b + 154(-3 - 5a + 3b) = 1166$$ $$-125b - 462 - 770a + 462b = 1166$$ $$(-125b + 462b) - 770a - 462 = 1166$$ $$337b - 770a = 1628$$ Step 6: Substitute $a = \frac{-44 + 7b}{22}$: $$337b - 770 \times \frac{-44 + 7b}{22} = 1628$$ Multiply both sides by 22: $$337 imes 22 b - 770(-44 + 7b) = 1628 imes 22$$ $$7414b + 33880 - 5390b = 35816$$ $$(7414b - 5390b) + 33880 = 35816$$ $$2024b = 35816 - 33880$$ $$2024b = 1936$$ $$b = \frac{1936}{2024} = \frac{121}{127}$$ Step 7: Calculate $a$: $$a = \frac{-44 + 7 \times \frac{121}{127}}{22} = \frac{-44 + \frac{847}{127}}{22} = \frac{\frac{-44 \times 127 + 847}{127}}{22} = \frac{\frac{-5588 + 847}{127}}{22} = \frac{\frac{-4741}{127}}{22} = \frac{-4741}{2794}$$ Step 8: Calculate $c$: $$c = -3 - 5a + 3b = -3 - 5 \times \left(-\frac{4741}{2794}\right) + 3 \times \frac{121}{127}$$ $$= -3 + \frac{23705}{2794} + \frac{363}{127} = -3 + 8.49 + 2.86 = 8.35$$ Final solution: $$a = -\frac{4741}{2794} \approx -1.697, \quad b = \frac{121}{127} \approx 0.953, \quad c \approx 8.35$$ --- 2. Tap fills tank in 6 hours. Fraction filled in $a$ hours: $$\text{Fraction} = \frac{a}{6}$$ --- 3. (a) Sum of first 45 terms of AP where $a=70$ and $d = -21$: $$S_n = \frac{n}{2}[2a + (n-1)d]$$ $$S_{45} = \frac{45}{2}[2 \times 70 + 44 \times (-21)] = \frac{45}{2}[140 - 924] = \frac{45}{2} \times (-784) = 45 \times (-392) = -17640$$ 3. (b) 12th term of GP with $a r^{11} = 63$ and $r = -3$: Solve for $a$: $$a = \frac{63}{r^{11}} = \frac{63}{(-3)^{11}} = \frac{63}{-177147} = -\frac{1}{2811}$$ 12th term is $a r^{11} = 63$ --- 4. Calculate sum: $$\sum_{i=1}^4 (x_i - y_i - c)$$ Given data: | i | x_i | y_i | c | |---|-----|-----|---| | 1 | 6 | 8 | -7| | 2 | 9 |11 | -7| | 3 |12 |14 | -7| | 4 |15 |17 | -7| Calculate termwise: $$(6 - 8 - (-7)) + (9 - 11 - (-7)) + (12 - 14 - (-7)) + (15 - 17 - (-7))$$ $$= (6 - 8 + 7) + (9 - 11 + 7) + (12 - 14 + 7) + (15 - 17 + 7)$$ $$= 5 + 5 + 5 + 5 = 20$$ --- 5. For equation $(y-9)/(?) = 10$, insufficient data to solve; please clarify denominator.