1. (a) 2x - 5y + x - 2y = 3x - 7y
(b) 5a - {3a - 2 + a + 4} = 5a - {4a + 2} = 5a - 4a - 2 = a - 2
2. Income per day = ₹(x + y). Income after 3 weeks (21 days) = ₹21(x + y)
3. P = -10, so P² - 2P - 100 = (-10)² - 2(-10) - 100 = 100 + 20 - 100 = 20
4. If a + b = 6, then ½a + ½b = ½(a + b) = ½ × 6 = 3
5. Sum of 3x - y + 11 and -y - 11 is 3x - 2y. Subtracting 3x - y - 11 gives: (3x - 2y) - (3x - y - 11) = 3x - 2y - 3x + y + 11 = -y + 11
6. (i) 1
(ii) -3
(iii) 2
(iv) -5
7. Simplify: 2(a + ab) + 3 - ab = 2a + 2ab + 3 - ab = 2a + ab + 3. When a = 5, b = -3: 2(5) + (5)(-3) + 3 = 10 - 15 + 3 = -2
8. 4x²y + 8x²y - 2x²y = (4 + 8 - 2)x²y = 10x²y
9. Solve: (2/21)x + 8 = x + 6 → 8 - 6 = x - (2/21)x → 2 = (19/21)x → x = 42/19. Verification: LHS = (2/21)(42/19) + 8 = 4/19 + 8 = 156/19; RHS = 42/19 + 6 = 42/19 + 114/19 = 156/19 → Verified.
10. To obtain 4ab + b² from a² + ab + b², subtract a² - 3ab.
11. Let breadth = b m, length = (3b - 6) m. Perimeter = 2(l + b) = 2(3b - 6 + b) = 2(4b - 6) = 8b - 12 = 148 → 8b = 160 → b = 20. Length = 3×20 - 6 = 54 m. Dimensions: 54 m × 20 m.
12. 12m² - 9m + 5m - 4m² - 7m + 10 = (12m² - 4m²) + (-9m + 5m - 7m) + 10 = 8m² - 11m + 10
13. Subtract (a² - 4a² + 5a - 6) - (a² - 2a + 1) = ( -3a² + 5a - 6 ) - (a² - 2a + 1) = -3a² + 5a - 6 - a² + 2a - 1 = -4a² + 7a - 7
14. Let base angles = x°, vertex angle = 2x°. Sum of angles: x + x + 2x = 180° → 4x = 180° → x = 45°. Angles: 45°, 45°, 90°.
15. Let number of 50 paise coins = x, 25 paise coins = 25 - x. Total value: 0.50x + 0.25(25 - x) = 30 → 0.50x + 6.25 - 0.25x = 30 → 0.25x = 23.75 → x = 95. But total coins are only 25, so problem is inconsistent. Recheck: if 50 paise coins are x, then 25 paise coins are 25 - x. Value equation: 50x + 25(25 - x) = 3000 paise → 50x + 625 - 25x = 3000 → 25x = 2375 → x = 95 — impossible. Correction: The condition "total number of 25 paise coins is four times that of 50 paise coins" implies 25 paise coins = 4x, 50 paise coins = x, total coins = 5x = 25 → x = 5. So 50 paise coins = 5, 25 paise coins = 20. Value = 5×50 + 20×25 = 250 + 500 = 750 paise = ₹7.50 ≠ ₹30. Problem has conflicting data. Assuming the coin count ratio is correct, the value should be ₹7.50, not ₹30. If value is ₹30, then 50x + 25(25 - x) = 3000 → 25x = 2375 → x = 95 — impossible with 25 coins. Therefore, the problem as stated contains an error. However, if we assume the intended meaning is that there are 25 coins total and the value is ₹30, then it's impossible. Alternatively, if the value is ₹7.50, then 5 coins of 50 paise and 20 coins of 25 paise. Given the inconsistency, no solution exists under both constraints. But since the problem likely intends the ratio, we go with 5 coins of 50 paise and 20 coins of 25 paise, ignoring the ₹30 value as erroneous. Final answer: 5 coins of 50 paise, 20 coins of 25 paise.
Parent Tip: Review the logic above to help your child master the concept of 7th grade algebraic expressions worksheet.
