If the coefficients of x2 and x3 are both zero, in the expansion of the expression (1 + ax…

Question

If the coefficients of x2
 and x3
 are both zero, in the expansion of the expression (1 + ax + bx2
) (1 -- 3x)15 in
powers of x, then the ordered pair (a,b) is equal to :

Accepted Answer

(1 + ax + bx2)(1 -- 3x)15

Co-eff. of x2 = 1.15C2(--3)2 + a.15C1(--3) + b.15C0

$$ = {{15 \times 14} \over 2} \times 9 - 15 \times 3a + b = 0$$ (given)

$$ \Rightarrow $$ 945 -- 45a + b = 0 ...(i)

Now co-eff. of x3 = 0

$$ \Rightarrow $$ 15C3(--3)3 + a.15C2(--3)2 + b.15C1(--3) = 0

$$ \Rightarrow {{15 \times 14 \times 13} \over {3 \times 2}} \times ( - 3 \times 3 \times 3) + a \times {{15 \times 14 \times 9} \over 2 } -- b \times  3 \times  15 = 0$$

$$ \Rightarrow $$ 15 \times  3[--3 \times  7 \times  13 + a \times  7 \times  3 -- b] = 0

$$ \Rightarrow $$ 21a -- b = 273 ...(ii)

From (i) and (ii)

a = +28, b = 315 $$ \equiv $$ (a, b) $$ \equiv $$ (28, 315)

Solutions · 1