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Question

If x satisfies the inequality (x2−x−1)(x2−x−7)<−5, then which of the following statements is (are) TRUE?

A
Number of integral values of x satisfying the given inequality is 2
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B
x2+1(2,5), if x<0
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C
x21(3,8), if x>0
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D
Number of integral values of x satisfying the given inequality is 0
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Solution

The correct options are B x2+1∈(2,5), if x<0 C x2−1∈(3,8), if x>0 D Number of integral values of x satisfying the given inequality is 0(x2−x−1)(x2−x−7)<−5 Let x2−x=t Then, (t−1)(t−7)<−5 ⇒t2−8t+12<0⇒(t−2)(t−6)<0⇒t∈(2,6)⇒2<x2−x<6 Case I:x2−x>2 ∴x2−x−2>0⇒(x+1)(x−2)>0∴x∈(−∞,−1)∪(2,∞) ⋯(1) Case II:x2−x<6 ∴x2−x−6<0⇒(x+2)(x−3)<0∴x∈(−2,3) ⋯(2) From (1) and (2), x∈(−2,−1)∪(2,3) There are no integral values of x in the solution set. For x<0, x∈(−2,−1) ⇒x2∈(1,4) ⇒x2+1∈(2,5) For x>0, x∈(2,3) ⇒x2∈(4,9) ⇒x2−1∈(3,8)

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