Question

If the coefficient of \( x \) in the expansion of \( (1+a x)^{8}(1+3 x)^{4}-(1+x)^{3}(1+2 x)^{4} \) is zero, then \( a= \) 

1) \( 1 / 4 \) 

2) \( -1 / 4 \) 

3) \( 1 / 8 \) 

4) \( -1 / 8 \)

Answer 1

Correct option is (4) -1/8

Term of x in (1 + ax)⁸ (1 + 3x)⁴

= 1 x 4(3x) + 8(ax) x 1

Coefficient of x in (1 + ax)⁸ (1 + 3x)⁴ = 12 + x

Coefficient of x in (1 + x)³ (1 + 2x)⁴ = 1 x 3 + 1 x 8 = 11

\(\therefore\) Coefficient of x in (1 + ax)⁸ (1 + 3x)⁴ -  (1 + x)³ (1 + 2x)⁴

= 12 + 8 -11

= 8a + 1

\(\because\) 8a + 1 = 0

⇒ \(a = \frac{-1}8\)