Question : If $\frac{3}{(x+2)(2x+1)}=\frac{a}{2x+1}+\frac{b}{x+2}$ is an identity, the value of $b$ is:
Option 1: 0
Option 2: –1
Option 3: 2
Option 4: 3
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Correct Answer: –1
Solution :
Given: $\frac{3}{(x+2)(2x+1)}=\frac{a}{2x+1}+\frac{b}{x+2}$
$⇒\frac{3}{(x+2)(2x+1)}=\frac{a(x+2)+b(2x+1)}{(2x+1)(x+2)}$
$⇒3=a(x+2)+b(2x+1)$
To eliminate the term '$a$' we have to put $x = -2$,
$⇒3=a(–2+2)+b(2×(–2)+1)$
$⇒3=–3b$
$⇒b=–1$
Hence, the correct answer is –1.
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