Sample QuestionsINVERSE TRIGNOMETRIC FUNCTIONS questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The principal value of $\tan^{-1}\Big(\tan\frac{3\pi}{5}\Big)$ is:
- A
$\frac{2\pi}{5}$
- ✓
$\frac{-2\pi}{5}$
- C
$\frac{3\pi}{5}$
- D
$\frac{-3\pi}{5}$
Answer: B.
View full solution →$\sin\Big\{2\cos^{-1}\Big(\frac{-3}{5}\Big)\Big\}$ is equal to:
- A
$\frac{6}{25}$
- B
$\frac{24}{25}$
- C
$\frac{4}{5}$
- ✓
$-\frac{24}{25}$
Answer: D.
View full solution →$\tan^{-1}(\sqrt{3})$
- A
$\frac{\pi}{6}$
- ✓
$\frac{\pi}{3}$
- C
$\frac{2\pi}{3}$
- D
$\frac{5\pi}{6}$
Answer: B.
View full solution →If $\cos^{-1}\text{x}>\sin^{-1}\text{x},$ then:
- ✓
$\frac{1}{\sqrt2}<\text{x}\leq1$
- B
$0\leq\text{x}\leq\frac{1}{\sqrt2}$
- C
$-1\leq\text{x}<\frac{1}{\sqrt2}$
- D
$\text{x}>0$
Answer: A.
View full solution →Choose the correct answer from the given four options. The value of the expression $\tan\Big(\frac{1}{2}\cos^{-1}\frac{2}{\sqrt{5}}\Big)$ is: Hint: $\bigg[\tan\frac{\theta}{2}=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}\bigg]$
- A
$2+\sqrt{5}$
- ✓
$\sqrt{5}-2$
- C
$\frac{\sqrt{5}+2}{2}$
- D
$5+\sqrt{2}$
Answer: B.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\tan^{-1}\big(\frac{2}{5}\big)+\tan^{-1}\big(\frac{3}{7}\big)=\frac{\pi}{4}.$
Reason: $\tan^{-1}\big(\frac{\text{x}}{\text{y}}\big)+\tan^{-1}\big(\frac{\text{y}-\text{x}}{\text{y}+\text{x}}\big)=\frac{\pi}{4}.$
- ✓
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
Answer: A.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: If $\text{x}=\frac{1}{5\sqrt{2}}$ then $\{\text{x}\cos(\cot^{-1}\text{x})+\sin(\cot^{-1}\text{x})\}^{2}=\frac{51}{50}.$
Reason: $\tan\Big[\cos^{-1}\Big(\frac{1}{5\sqrt{2}}\Big)-\sin^{-1}\Big(\frac{4}{\sqrt{17}}\Big)\Big]=\frac{29}{3}.$
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- ✓
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
Answer: B.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\tan^{-1}\big(\frac{3}{4}\big)+\tan^{-1}\big(\frac{1}{7}\big)=\frac{\pi}{4}.$
Reason: For $x > 0, y > 0, xy < 1, \tan^{-1}\text{x}+\tan^{-1}\text{y}=\tan^{-1}\Big(\frac{\text{x}+\text{y}}{1-\text{xy}}\Big).$
- ✓
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
Answer: A.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: If $0<\text{x}\leq\frac{\pi}{2},$ then $\sin^{-1}(\cos\text{x})+\cos^{-1}(\sin\text{x})=\pi-2\text{x}.$
Reason: $\cos^{-1}\text{x}=\frac{\pi}{2}-\sin^{-1}\text{x} $ for all $\text{x}\in[-1,1].$
- ✓
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
Answer: A.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\tan^{-1}\big[\text{x}+\sqrt{1+\text{x}^{2}}\big]=\frac{\pi}{2}-\frac{1}{2}\cot^{-1}.$
Reason: $\sin^{2}\Big[2\tan^{-1}\sqrt{\frac{1+\text{x}}{1-\text{x}}}\Big]=1-\text{x}^{2}.$
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- ✓
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
Answer: B.
View full solution →If $\tan^{-1}\text{x} +\tan^{-1}\text{y} = \frac{\pi}{4},\text{xy} < 1,$then write the value of x + y + xy.
View full solution →Write the value of $\tan^{-1}\bigg[2\sin\bigg(2\cos^{-1}\frac{\sqrt{3}}{2}\bigg)\bigg].$
View full solution →Write the principal value of $\tan^{-1}(\sqrt{3}) - \cot^{-1}( - \sqrt{3}).$
View full solution →Find the principal value of $\tan^{–1} \sqrt{3} – \sec^{–1}(– 2)$.
View full solution →What is the principal value of $\cot^{-1} \Bigg(\cos\frac{2\pi}{3}\Bigg)+\sin^{-1}\Bigg(\sin\frac{2\pi}{3}\Bigg)?$
View full solution →Prove that:
$3\sin^{-1}\text{x}=\sin^{-1}(3\text{x}-4\text{x}^3),\text{x}\in\Big[-\frac{1}{2},\frac{1}{2}\Big]$
View full solution →If $\cot\Big(\cos^{-1}\frac{3}{5}+\sin^{-1}\text{x}\Big)=0,$ find the values of x.
View full solution →If x > 1, then write the value of $\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$ in terms of $\tan^{-1}\text{x.}$
View full solution →Find the set values of $\text{cosec}^{-1}\Big(\frac{\sqrt3}{2}\Big)$
View full solution →Evaluate the following:
$\cot^{-1}\Big\{\cot\Big(-\frac{8\pi}{3}\Big)\Big\}$
View full solution →Evaluate:
$\tan\Bigg\{ 2\tan ^{-1} \bigg(\frac{1}{5}\bigg) + \frac{\pi}{4}\Bigg\}$
View full solution →If $\tan^{-1} \frac{\text{x - 3}}{\text{x - 4}} + \tan^{-1} \frac{\text{x + 3}}{\text{x + 4}} = \frac{\pi}{4},$ then find the value of x.
View full solution →Solve the equation for $x: \sin^{-1}x + \sin^{-1}(1 - x) = \cos^{-1}x$
View full solution →Solve the following equation:$\cos(\tan^{-1}\text{x}) = \sin\bigg(\cot^{-1}\frac{3}{4}\bigg).$
View full solution →Show that: $\tan\bigg(\frac{1}{2}\sin^{-1}\frac{3}{4}\bigg) = \frac{4-\sqrt{7}}{3}.$
View full solution →Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean and variance of the number of kings.
Prove the following results:
$\sin^{-1}\frac{5}{13}+\cos^{-1}\frac{3}{5}=\tan^{-1}\frac{63}{16}$
View full solution →Show that $2\tan^{-1}\text{x}+\sin^{-1}\frac{2\text{x}}{1+\text{x}^2}$ is constant for $\text{x}\geq1,$ find that constant.
View full solution →Prove the following results:
$2\sin^{-1}\frac{3}{5}-\tan^{-1}\frac{17}{31}=\frac{\pi}{4}$
View full solution →Solve the following equation for x:
$\cot^{-1}\text{x}-\cot^{-1}(\text{x}+2)=\frac{\pi}{12},\text{x}>0$
View full solution →Fill in the blank.
The set of values of $\sec^{-1}\Big(\frac{1}{2}\Big)$ is __________.
View full solution →Fill in the blank.If $\text{y}=2\tan^{-1}\text{x}+\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big),$ then ____ < y < ____.
View full solution →Fill in the blank.
The principal value of $\tan^{-1}\sqrt{3}$ is __________.
View full solution →Fill in the blank.
The value of $\cos(\sin^{-1}\text{x}+\cos^{-1}\text{x}),$ where $|\text{x}|\leq1,$ is ______.
View full solution →Fill in the blank.The principal value of $\cos^{-1}\Big(-\frac{1}{2}\Big)$ is __________.
View full solution →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.
View full solution →State True or False for the statement.
The principal value of $\sin^{-1}\Big[\cos\Big(\sin^{-1}\frac{1}{2}\Big)\Big]$ is $\frac{\pi}{3}.$
View full solution →State True or False for the statement.
All trigonometric functions have inverse over their respective domains.
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.
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.
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