The number of solutions of the equation | tan -1| x||=√(x2+1)2-4 x2 is

Question

The number of solutions of the equation | tan -1| x||=√(x2+1)2-4 x2 is (A) 2 (B) 3 (C) 4 (D) none of these

Tardigrade Admin · Accepted Answer

√(x2+1)2-4 x2=√(x2-1)2=|x2-1| ⇒ | tan -1| x |=|x2-1| Draw the graphs of y =| tan -1| x | and y =| x 2-1| <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/193d1a3f20bd58d5a99d67ca25ae21d7-.png" /> From the graph, it is clear that equation has four roots.

Q. The number of solutions of the equation $∣ ∣ tan^{- 1} ∣ ∣ x ∣∣ = (x^{2} + 1)^{2} - 4 x^{2}$ is

2

3

4

none of these

$(x^{2} + 1)^{2} - 4 x^{2} = (x^{2} - 1)^{2} = ∣ ∣ x^{2} - 1 ∣ ∣$
$\Rightarrow ∣ ∣ tan^{- 1} ∣ ∣ x ∥ = ∣ ∣ x^{2} - 1 ∣ ∣$
Draw the graphs of $y = ∣ ∣ tan^{- 1} ∣ ∣ x ∥$ and $y = ∣ ∣ x^{2} - 1 ∣ ∣$

From the graph, it is clear that equation has four roots.

Q. The number of solutions of the equation ∣∣​tan−1∣∣​x∣∣=(x2+1)2−4x2​ is

2

3

4

none of these

(x2+1)2−4x2​=(x2−1)2​=∣∣​x2−1∣∣​ ⇒∣∣​tan−1∣∣​x∥=∣∣​x2−1∣∣​ Draw the graphs of y=∣∣​tan−1∣∣​x∥ and y=∣∣​x2−1∣∣​ From the graph, it is clear that equation has four roots.

Q. The number of solutions of the equation $∣ ∣ tan^{- 1} ∣ ∣ x ∣∣ = (x^{2} + 1)^{2} - 4 x^{2}$ is

$(x^{2} + 1)^{2} - 4 x^{2} = (x^{2} - 1)^{2} = ∣ ∣ x^{2} - 1 ∣ ∣$
$\Rightarrow ∣ ∣ tan^{- 1} ∣ ∣ x ∥ = ∣ ∣ x^{2} - 1 ∣ ∣$
Draw the graphs of $y = ∣ ∣ tan^{- 1} ∣ ∣ x ∥$ and $y = ∣ ∣ x^{2} - 1 ∣ ∣$

From the graph, it is clear that equation has four roots.