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NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Inverse Trigonometric Functions

NTA Abhyas 2020 · Mathematics · Inverse Trigonometric Functions

If f(x) = tan^ - 1 x^ 2 + 4x + sin^ - 1 x^ 2 + 4x + 1 , then

If

f\left(x\right) = tan^{- 1}\sqrt{x^{2} + 4x} + sin^{- 1}\sqrt{x^{2} + 4x + 1}

, then

Options

Adomain of $f$ \left(x\right) $contains$ 3 integers only
Brange of $f \left(x\right) has two elements only$
Cf $\left(x\right)$ is a constant function $\forall x \in \mathbb{R}$
D $f \left(x\right) contains only two elements in its domain$

Correct answer

D. $f \left(x\right) contains only two elements in its domain$

Step-by-step solution

∵ x^{2} + 4x \ge 0

for the first term
and

0 \le x^{2} + 4x + 1 \le 1

for the second term

∴

by both the results, there is only one possibility

x^{2} + 4x = 0

⇒ x = 0, - 4

∴ f\left(x\right) = \tan^{-1} \left(0\right) + \sin^{-1} \left(1\right) = \dfrac{\pi}{2}

&

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