Advertisements
Advertisements
Question
The domain of \[\cos^{- 1} \left( x^2 - 4 \right)\] is
Options
[3, 5]
[−1, 1]
\[\left[ - \sqrt{5}, - \sqrt{3} \right] \cup \left[ \sqrt{3}, \sqrt{5} \right]\]
\[\left[ - \sqrt{5}, - \sqrt{3} \right] \cap \left[ \sqrt{3}, \sqrt{5} \right]\]
Solution
The domain of \[\cos^{- 1} \left( x \right)\] is [-1, 1]
\[\therefore - 1 \leq x^2 - 4 \leq 1\]
\[ \Rightarrow - 1 + 4 \leq x^2 - 4 + 4 \leq 1 + 4\]
\[ \Rightarrow 3 \leq x^2 \leq 5\]
\[ \Rightarrow \pm \sqrt{3} \leq x \leq \pm \sqrt{5}\]
\[ \Rightarrow x \in \left[ - \sqrt{5}, - \sqrt{3} \right] \cup \left[ \sqrt{3}, \sqrt{5} \right]\]
Hence, the correct answer is option (c).
APPEARS IN
RELATED QUESTIONS
Solve the equation for x:sin−1x+sin−1(1−x)=cos−1x
Find the domain of definition of `f(x)=cos^-1(x^2-4)`
Find the principal values of the following:
`cos^-1(-sqrt3/2)`
Evaluate the following:
`cos^-1{cos (13pi)/6}`
Evaluate the following:
`cos^-1(cos4)`
Evaluate the following:
`tan^-1(tan (7pi)/6)`
Evaluate the following:
`tan^-1(tan4)`
Evaluate the following:
`cot^-1{cot (-(8pi)/3)}`
Evaluate the following:
`cot^-1{cot ((21pi)/4)}`
Write the following in the simplest form:
`tan^-1{sqrt(1+x^2)-x},x in R `
Evaluate the following:
`sin(tan^-1 24/7)`
Evaluate:
`sec{cot^-1(-5/12)}`
Evaluate:
`cot(sin^-1 3/4+sec^-1 4/3)`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x > 0
Solve the following equation for x:
tan−1(x −1) + tan−1x tan−1(x + 1) = tan−13x
If `cos^-1 x/2+cos^-1 y/3=alpha,` then prove that `9x^2-12xy cosa+4y^2=36sin^2a.`
`tan^-1 2/3=1/2tan^-1 12/5`
`tan^-1 1/7+2tan^-1 1/3=pi/4`
`sin^-1 4/5+2tan^-1 1/3=pi/2`
`2tan^-1 3/4-tan^-1 17/31=pi/4`
Solve the following equation for x:
`cos^-1((x^2-1)/(x^2+1))+1/2tan^-1((2x)/(1-x^2))=(2x)/3`
Prove that `2tan^-1(sqrt((a-b)/(a+b))tan theta/2)=cos^-1((a costheta+b)/(a+b costheta))`
Write the value of sin (cot−1 x).
Evaluate: \[\sin^{- 1} \left( \sin\frac{3\pi}{5} \right)\]
If 4 sin−1 x + cos−1 x = π, then what is the value of x?
Wnte the value of the expression \[\tan\left( \frac{\sin^{- 1} x + \cos^{- 1} x}{2} \right), \text { when } x = \frac{\sqrt{3}}{2}\]
Wnte the value of\[\cos\left( \frac{\tan^{- 1} x + \cot^{- 1} x}{3} \right), \text{ when } x = - \frac{1}{\sqrt{3}}\]
If \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{\sqrt{1 + x^2} + \sqrt{1 - x^2}} \right)\] = α, then x2 =
If sin−1 x − cos−1 x = `pi/6` , then x =
If x < 0, y < 0 such that xy = 1, then tan−1 x + tan−1 y equals
If α = \[\tan^{- 1} \left( \frac{\sqrt{3}x}{2y - x} \right), \beta = \tan^{- 1} \left( \frac{2x - y}{\sqrt{3}y} \right),\]
then α − β =
The value of \[\sin^{- 1} \left( \cos\frac{33\pi}{5} \right)\] is
If tan−1 (cot θ) = 2 θ, then θ =
The value of \[\sin\left( 2\left( \tan^{- 1} 0 . 75 \right) \right)\] is equal to
The value of \[\tan\left( \cos^{- 1} \frac{3}{5} + \tan^{- 1} \frac{1}{4} \right)\]
If y = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \frac{dy}{dx} + y \cos^2 x = 0 .\]
Find the domain of `sec^(-1)(3x-1)`.
