JEE Main | 2014Math miscellaneousHard
Question
If the coefficients of x3 and x4 in the expansion of (1 + ax + bx2) (1 - 2x)18 in powers of x are both zero, then (a, b) is equal to
Options
A.
B.
C.
D.
Solution
(1 + ax + bx2) (1 - 2x)18
(1 + ax + bx2)[18C0 - 18C1(2x) + 18C2(2x)2 - 18C3(2x)3 + 18C4(2x)4- .... ]
Coeff. of x3 = - 18C3. 8 + a × 4. 18C2 - 2b × 18 = 0
- 36 b = 0
= -51 × 16 × 8 + a × 36 × 17 - 36b = 0
= -34 × 16 + 51a - 3b = 0
= 51a - 3b = 34 × 16 = 544
= 51a - 3b = 544 ....(i)
Only option number (2) satisfies the equation number (i).
(1 + ax + bx2)[18C0 - 18C1(2x) + 18C2(2x)2 - 18C3(2x)3 + 18C4(2x)4- .... ]
Coeff. of x3 = - 18C3. 8 + a × 4. 18C2 - 2b × 18 = 0
- 36 b = 0= -51 × 16 × 8 + a × 36 × 17 - 36b = 0
= -34 × 16 + 51a - 3b = 0
= 51a - 3b = 34 × 16 = 544
= 51a - 3b = 544 ....(i)
Only option number (2) satisfies the equation number (i).
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