KVPY Sample Paper KVPY Stream-SX Model Paper-20

Let Z be the set of integers. If\[A-\left\{ x\in Z:{{2}^{(x+2)({{x}^{2}}-5x+6)}}=1 \right\}\] and \[B=\left\{ x\in Z:-3<2x-1<9 \right\}\] then the number of subsets of the set \[A\times B,\]is:

A) \[{{2}^{15}}\]

B) \[{{2}^{18}}\]

C) \[{{2}^{12}}\]

D) \[{{2}^{10}}\]

Correct Answer: A

Solution :

\[A=\left\{ x\in z:{{2}^{(x+2)({{x}^{2}}-5x+6)}}=1 \right\}\]

\[{{2}^{(x+2)({{x}^{2}}-5x+6)}}={{2}^{0}}\]

\[\Rightarrow \] \[x=-2,\,\,2,\,\,3\]

\[A=\left\{ -2,\,\,2,\,\,3 \right\}\]

\[B=\left\{ x\in z:-3<2x-1<9 \right\}\]

\[B=\{0,1,2,3,4\}\]

Hence, \[A\times B\]has 15 elements.

So number of subsets of \[A\times B\] is \[{{2}^{15}}.\]