Question

The Number of values of x satisfying [ln(1+x)]=ln(1+x)+[(1+x)^2]-3 (Where [.] represents greatest integer function)
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Answer 1

Step-by-step explanation:

0<cot

−1

x<π and −π/2<tan

−1

x<π

⇒ [cot

−1

x]∈{0,1,2,3} and [tan

−1

x]∈{−2,−1,0,1}

For [tan

−1

x]+[cot

−1

x]=2, following cases are possible

Case (i): [cot

−1

x]=[tan

−1

x]=1

⇒ 1≤cot

−1

x<2 and 1≤tan

−1

x<π/2

⇒ x∈(cot2,cot1] and x∈[tan1,∞)

∴ x∈ϕ as cot1<tan1

Case (ii): [cot

−1

x]=2, [tan

−1

x]=0

⇒ 2≤cot

−1

x<3 and 0≤tan

−1

x<1

⇒ x∈(cot3,cot2] and x∈[0,tan1)

∴ x∈ϕ as cot2<0

So no solution.

Case (iii): [cot

−1

x]=3, [tan

−1

x]=−1

⇒ 3≤cot

−1

x<π and −1≤tan

−1

x<0

⇒ x∈(−∞,cot3] and x∈[−tan1,0)

∴ x∈ϕ as cot3<−tan1

Therefore, no such values of x exist.