The number of integral solutions of (x + 2/x2 + 1) (1/2) is

Question

The number of integral solutions of (x + 2/x2 + 1) > (1/2) is (A) 4 (B) 5 (C) 3 (D) none of these

Tardigrade Admin · Accepted Answer

(x + 2/x2 + 1) > (1/2) : :: ⇒ (-x2 + 2x + 3/ x2 + 1) > 0 ⇒ : : - x2 +2x + 3 > 0 [ ∵ x2 + 1 > 0, ∀ x ∈ R] ⇒ : : x2 - 2x - 3 < 0 ⇒ - 1 < x < 3 ∵ x is integer ∴ x = 0,1, 2, hence three values.

Q. The number of integral solutions of $\frac{x + 2}{x ^{2} + 1} > \frac{1}{2}$ is

4

5

3

none of these

$\frac{x + 2}{x ^{2} + 1} > \frac{1}{2} \Rightarrow \frac{- x ^{2} + 2 x + 3}{x ^{2} + 1} > 0$
$\Rightarrow - x^{2} + 2 x + 3 > 0 [∵ x^{2} + 1 > 0, \forall x \in R]$
$\Rightarrow x^{2} - 2 x - 3 < 0 \Rightarrow - 1 < x < 3$
$∵$ x is integer $∴$ x = 0,1, 2, hence three values.

Q. The number of integral solutions of x2+1x+2​>21​ is

4

5

3

none of these

x2+1x+2​>21​⇒x2+1−x2+2x+3​>0 ⇒−x2+2x+3>0[∵x2+1>0,∀x∈R] ⇒x2−2x−3<0⇒−1<x<3 ∵ x is integer ∴ x = 0,1, 2, hence three values.

Q. The number of integral solutions of $\frac{x + 2}{x ^{2} + 1} > \frac{1}{2}$ is

$\frac{x + 2}{x ^{2} + 1} > \frac{1}{2} \Rightarrow \frac{- x ^{2} + 2 x + 3}{x ^{2} + 1} > 0$
$\Rightarrow - x^{2} + 2 x + 3 > 0 [∵ x^{2} + 1 > 0, \forall x \in R]$
$\Rightarrow x^{2} - 2 x - 3 < 0 \Rightarrow - 1 < x < 3$
$∵$ x is integer $∴$ x = 0,1, 2, hence three values.