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JEE MainMathematicsEasyInequalities

Number of integral values of m for which the inequality
x2−mx−2x2−3x+4>−1\frac{x^{2} - mx - 2}{x^{2} - 3x + 4} > - 1 is satisfied by all real
values of x.

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Given  x2−mx−2x2−3x+4\frac{x^{2} - mx - 2}{x^{2} - 3x + 4}> - 1 for
all x∈Rx \in R
⇒\Rightarrow x^2 - mx - 2 > - (x^2 - 3x + 4)  \ (as x^2 -
3x + 4 > 0 ∀\forallx ∈\inR)
⇒2x2−(m+3)x+2>0\Rightarrow 2x^{2} - (m + 3)x + 2 > 0 for all x ∈\in R
⇒\Rightarrow disc. = (m + 3)^2 - 16 < 0
⇒\Rightarrow m^2 + 6x - 7 < 0
⇒\Rightarrow (m + 7) (m - 1) < 0
⇒\Rightarrow m ϵ\epsilon (-7, 1)
⇒\Rightarrow no. of integral values of m is 7

Source: Grass-G / M-23 Mathematics Paper 31 Dec FST.docx

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