True False[1 Marks ]

Question 11 Mark

State True or False for the statement. The value of the expression $(\cos^{-1}x)^2$ is equal to $\sec^2x.$

Answer

$\because\ (\cos^{-1}\text{x})^2$
$=\Big(\sec^{-1}\frac{1}{\text{x}}\Big)^2\neq\sec^2\text{x}$

Question 21 Mark

State True or False for the statement.
The domain of trigonometric functions can be restricted to any one of their branch (not necessarily principal value) in order to obtain their inverse functions.

Answer

True.Solution:
Yes, the domain of trigonometric functions can be restricted in their domain to obtain their inverse functions.

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Question 31 Mark

State True or False for the statement.
The minimum value of n for which $\tan^{-1}\frac{\text{n}}{\pi}>\frac{\pi}{4},\ \text{n}\in\text{N},$ is valid is 5.

Answer

False.Solution:
$\tan^{-1}\frac{\text{n}}{\pi}>\frac{\pi}{4}$
$\Rightarrow\ \frac{\text{n}}{\pi}>\tan\frac{\pi}{4}$
$\Rightarrow\ \frac{\text{n}}{\pi}>1$
$\Rightarrow\ \text{n}>\pi$
So, the minimum value of n is 4.

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Question 41 Mark

State True or False for the statement.
The principal value of $\sin^{-1}\Big[\cos\Big(\sin^{-1}\frac{1}{2}\Big)\Big]$ is $\frac{\pi}{3}.$

Answer

True.Solution:
$\sin^{-1}\Big[\cos\Big(\sin^{-1}\frac{1}{2}\Big)\Big]=\sin^{-1}\Big[\cos\frac{\pi}{6}\Big]$
$=\sin^{-1}\frac{\sqrt{3}}{2}$
$=\sin^{-1}\sin\frac{\pi}{3}=\frac{\pi}{3}$

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Question 51 Mark

State True or False for the statement.
All trigonometric functions have inverse over their respective domains.

Answer

False.Solution:
We know that, all trigonometric functions have inverse over their restricted domains.

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Question 61 Mark

State True or False for the statement.
The graph of inverse trigonometric function can be obtained from the graph of their corresponding trigonometric function by interchanging x and y axes.

Answer

True.Solution:
If we interchange the coordinates of all the points in the graph of any trigonometric function, then we will get the graph of an inverse function of the respective function.
In other words, the graph of inverse trigonometric function is a mirror image (i.e., reflection) along the line y = x of the corresponding trigonometric function.

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Question 71 Mark

State True or False for the statement.
The least numerical value, either positive or negative of angle $\theta$ is called principal value of the inverse trigonometric function.

Answer

True.Solution:
We know that, the smallest numerical value, either positive or negative of $\theta$ is called the principal value of the function.

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MATHS True False[1 Marks ]

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