حل السؤال 20، 4MA1/2H يونيو 2020: المخططات الشجرية للأحداث المترابطة

1. 1Let the number of red counters be r. Then blue counters = r + 4. Total counters X = r + (r + 4) = 2r + 4.
2. 2Probability first counter is blue = (r + 4) / (2r + 4)
3. 3Probability second counter is blue = (r + 3) / (2r + 3)
4. 4Probability both counters are blue = [(r + 4)/(2r + 4)] × [(r + 3)/(2r + 3)] = 3/8
5. 5So: (r + 4)(r + 3) / [(2r + 4)(2r + 3)] = 3/8
6. 6Multiply both sides by 8(2r + 4)(2r + 3): 8(r + 4)(r + 3) = 3(2r + 4)(2r + 3)
7. 7Expand left: 8(r² + 7r + 12) = 8r² + 56r + 96
8. 8Expand right: 3(4r² + 6r + 8r + 12) = 3(4r² + 14r + 12) = 12r² + 42r + 36
9. 9Bring all to one side: 8r² + 56r + 96 − (12r² + 42r + 36) = 0
10. 10−4r² + 14r + 60 = 0
11. 11Multiply by −1: 4r² − 14r − 60 = 0
12. 12Divide by 2: 2r² − 7r − 30 = 0
13. 13Factorise: (2r + 5)(r − 6) = 0
14. 14So r = 6 (since r cannot be negative)
15. 15Then blue counters = 6 + 4 = 10
16. 16Total counters X = 6 + 10 = 16

Answer16