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 :

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 : (A) (28, 315) (B) (-54, 315) (C) (-21, 714) (D) (24, 861)

Tardigrade Admin · Accepted Answer

Coefiicient of x2= 15 C2 × 9-3 a( 15 C1)+b=0 ⇒ -45 a+b+ 15 C2 × 9=0 Also, -27 × 15 C 3+9 a × 15 C 2-3 b × 15 C 1=0 ⇒ 9 × 15 C 2 a -45 b -27 × 15 C 3=0 ⇒ 21 a - b -273=0 ldots (i) + (ii) -24 a +672=0 ⇒ a =28 So , b =315

Q. 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 :

(28, 315)

(-54, 315)

(-21, 714)

(24, 861)

Coefiicient of x2=15C2​×9−3a(15C1​)+b=0 ⇒−45a+b+15C2​×9=0 Also, −27×15C3​+9a×15C2​−3b×15C1​=0 ⇒9×15C2​a−45b−27×15C3​=0 ⇒21a−b−273=0… (i) + (ii) −24a+672=0 ⇒a=28 So,b=315

Q. If the coefficients of $x^{2}$ and $x^{3}$ are both zero, in the expansion of the expression $(1 + a x + b x^{2}) (1 - 3 x)^{15}$ in powers of $x$ , then the ordered pair $(a, b)$ is equal to :