Alternative method for qn 5 :

①
Since only a9 y^9 contains term in x^9, i.e

a9 y^9 = a9 (x + 1)^9 = a9 x^9 + ... + a9 1^9

Then
-x^9 = a9 x^9
Comparing coefficients, a9 = -1

Term in x^8 of a9 y^9 = (-1)(9C1 x^8) = -9x^8


②
Similarly, only a9 y^9 and a8 y^8 contain terms in x^8 .

Then
x^8 = -9x^8 + a8 x^8
x^8 = (-9 + a8)x^8
Comparing coefficients, 1 = -9 + a8
a8 = 10

Term in x^7 of a8 y^8 = 10(8C1 x^7) = 80x^7
Term in x^7 of a9 y^9 = (-1)(9C2 x^7) = -36x^7


③
Likewise, only a9 y^9 , a8 y^8 and a7 y^7 contain terms in x^7

Then
-x^7 = 80x^7 - 36x^7 + a7 x^7
-x^7 = (44 + a7) x^7
Comparing coefficients, -1 = 44 + a7
a7 = -45

Term in x^6 of a7 y^7 = -45(7C1 x^6) = -315 x^6
Term in x^6 of a8 y^8 = 10(8C2 x^6) = 280x^6
Term in x^6 of a9 y^9 = (-1)(9C3 x^6) = -84x^6


④
Likewise, only a9 y^9 , a8 y^8 , a7 y^7 and a6 y^6 contain terms in x^6

Then
x^6 = -315x^6 + 280x^6 - 84x^6 + a6 x^6
x^6 = (-119 + a6) x^6
Comparing coefficients, 1 = -119 + a6
a6 = 120


Term in x^5 of a6 y^6 = 120(6C1 x^5) = 720 x^5
Term in x^5 of a7 y^7 = -45(7C2 x^5) = -945 x^5
Term in x^5 of a8 y^8 = 10(8C3 x^5) = 560x^5
Term in x^5 of a9 y^9 = (-1)(9C4 x^5) = -126x^5


⑤
Lastly, only a9 y^9 , a8 y^8 , a7 y^7 , a6 y^6 and a5 y^5 contain terms in x^5

Then -x^5 = 720x^5 - 945x^5 + 560x^5 - 126x^5 + a5 x^5
-x^5 = (209 + a5) x^5
Comparing coefficients, -1 = 209 + a5
a5 = -210


∴ a5 = -210