Question

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

• A. $(-21,714)$
• B. $(-54,315)$
• C. $(28,861)$
• D. $(28,315)$

Answer 1

The correct option is D $(28,315)$

Coefficient of $x^2$ in $(1+ax+b{x}^2)(1-3x{)}^{15}$
$\Rightarrow {}^{15}{C}_2(-3{)}^2+a{(}^{15}{C}_1)(-3)+b{(}^{15}{C}_0\left)\right(-3{)}^0=0$
$\Rightarrow \frac{15\cdot 14}{2}\cdot 9-3a\cdot 15+b=0\Rightarrow 15\cdot 63-45a+b=0\cdots \left(1\right)$

Coefficient of $x^3$ in $(1+ax+b{x}^2)(1-3x{)}^{15}$
$\Rightarrow {}^{15}{C}_3\cdot (-3{)}^3+a{(}^{15}{C}_2)(-3{)}^2+b{(}^{15}{C}_1)(-3)=0$
$\Rightarrow \frac{15\cdot 14\cdot 13}{3\cdot 2\cdot 1}\cdot 3^2-a\cdot 3\cdot \frac{15\cdot 14}{2}+15\cdot b=0\Rightarrow 21\cdot 13-21a+b=0\cdots \left(2\right)$

On subtracting $\left(1\right)-\left(2\right)$ , we have
$15\cdot 63-21\cdot 13-24a=0\Rightarrow a=28\Rightarrow b=315$