SATHEE: Binomial Theorem 1 Question 2

Binomial Theorem 1 Question 2

2. 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

(2019 Main, 10 April I)

(a) $(28, 315)$

(b) $(- 21, 714)$

(d) $(- 54, 315)$

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Answer:

Correct Answer: 2. (c)

Solution:

Given expression is $(1 + a x + b x^{2}) (1 - 3 x)^{15}$ . In the expansion of binomial $(1 - 3 x)^{15}$ , the $(r + 1)$ th term is

$T_{r + 1} =^{15} C_{r} (- 3 x)^{r} =^{15} C_{r} (- 3)^{r} x^{r}$

Now, coefficient of $x^{2}$ , in the expansion of $(1 + a x + b x^{2}) (1 - 3 x)^{15}$ is

$^{15} C_{2} (- 3)^{2} + a^{15} C_{1} (- 3)^{1} + b^{15} C_{0} (- 3)^{0} = 0$ (given)

$\Rightarrow (105 \times 9) - 45 a + b = 0$

$\Rightarrow 45 a - b = 945$

Similarly, the coefficient of $x^{3}$ , in the expansion of $(1 + a x + b x^{2}) (1 - 3 x)^{15}$ is

$^{15} C_{3} (- 3)^{3} + a^{15} C_{2} (- 3)^{2} + b^{15} C_{1} (- 3)^{1} = 0$ (given)

$\Rightarrow - 12285 + 945 a - 45 b = 0$

$\Rightarrow 63 a - 3 b = 819$

$\Rightarrow 21 a - b = 273$

From Eqs. (i) and (ii), we get

$24 a = 672 \Rightarrow a = 28$

So, $b = 315$

$\Rightarrow (a, b) = (28, 315)$

Key Idea Use the general term (or $(r + 1)$ th term) in the expansion of binomial $(a + b)^{n}$

i.e.

$T_{r + 1} =^{n} C_{r} a^{n - r} b^{r}$

Let a binomial $2 x^{2} - {\frac{3}{x^{2}}}^{6}$ , it’s $(r + 1)$ th term

$\begin{aligned} = T_{r + 1} =^{6} C_{r} {(2 x^{2})}^{6 - r} - \frac{3}{x^{2}} \\ =^{6} C_{r} (- 3)^{r} (2)^{6 - r} x^{12 - 2 r - 2 r} \\ =^{6} C_{r} (- 3)^{r} (2)^{6 - r} x^{12 - 4 r} \end{aligned}$

Now, the term independent of $x$ in the expansion of $\frac{1}{60} - \frac{x^{8}}{81} 2 x^{2} - \frac{3}{x^{2}}$

$=$ the term independent of $x$ in the expansion of $\frac{1}{60} 2 x^{2} - {\frac{3}{x^{2}}}^{6} +$ the term independent of $x$ in the expansion of $- \frac{x^{8}}{81} 2 x^{2} - {\frac{3}{x^{2}}}^{6}$

$\begin{array}{rrr} = \frac{^{6} C_{3}}{60} (- 3)^{3} (2)^{6 - 3} x^{12 - 4 (3)} & [put r = 3] + - \frac{1}{81} & ^{6} C_{5} (- 3)^{5} (2)^{6 - 5} x^{12 - 4 (5)} x^{8} & [put r = 5] \end{array}$