A) 1
B) 2
C) 3
D) 4
Correct Answer: C
Solution :
Let \[p(x)=12{{x}^{3}}-13{{x}^{2}}-5x+9\] and \[q(x)=(3x+2)=3\,\left( x+\frac{2}{3} \right)\] \[=3\,\left[ x-\left( \frac{-2}{3} \right) \right]\] When \[P(x)\] is divided by \[(3x+2)\] and the remainder is \[P\,\left( -\frac{2}{3} \right)\] Now, \[P\left( -\frac{2}{3} \right)=12\,{{\left( -\frac{2}{3} \right)}^{3}}-13\,{{\left( -\frac{2}{3} \right)}^{2}}-5\,\left( -\frac{2}{3} \right)+9\] \[=12\times \left( \frac{-8}{27} \right)-13\times \frac{4}{9}+\frac{10}{3}+9\] \[=-\frac{96}{27}-\frac{52}{9}+\frac{10}{3}+9\] \[=\frac{-96-156+90+243}{27}=\frac{81}{27}=3\] \[\therefore \] Required remainder = 3
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