If $\left.A=\mid x \in R: x^2+6 x-7 \lt 0\right\}$ and $B=\left\{x \in R: x^2+9 x+14 \gt 0\right\}$, then which of the following is/ are correct? 1. $(A \cap B)=(-2,1)$ 2. $(A / B)=(-7,-2)$ Select the correct answer using the code given below:

• Correct answer
  A

  1 only
• B

  2 only
• C

  Both 1 and 2
• D

  Neither 1 nor 2

Solution

$\begin{aligned} & \mathrm{x}^2+6 \mathrm{x}-7 \lt 0 \\ & \Rightarrow(\mathrm{x}+7)(\mathrm{x}-1) \lt 0 \\ & \Rightarrow \mathrm{x}=(-7,1) \\ & \Rightarrow \mathrm{A}=\{-6,-5,-4,-3,-2,-1,0\} \\ & \Rightarrow \mathrm{x}^2+9 \mathrm{x}+14 \gt 0 \\ & \Rightarrow(\mathrm{x}+7)(\mathrm{x}+2) \gt 0 \\ & \Rightarrow \mathrm{x}=(-\infty,-7) \cup(-2, \infty) \\ & \Rightarrow \mathrm{B}=\mathrm{R}-\{-7,-6,-5,-4,-3,-2\}\end{aligned}$
So $A \cap B=(-2,1)$

Hence, the correct answer is option: 'A'.