MCQ 11 Mark
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 21 Mark
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.
AnswerCorrect option: B. Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
View full question & answer→MCQ 31 Mark
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 41 Mark
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 51 Mark
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.
AnswerCorrect option: B. Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
View full question & answer→MCQ 61 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\text{cosec}^{-1}\Big(\frac{1}{2}+\frac{1}{\sqrt{2}}\Big)>\sec^{-1}\Big(\frac{1}{2}+\frac{1}{\sqrt{2}}\Big).$
Reason: $\text{cosec}^{-1}(\text{x})>\sec^{-1}(\text{x})$ if $1<\text{x}<\sqrt{2}.$
- ✓
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 71 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\text{cosec}^{-1}\text{x}>\sin^{-1}\text{x},$ for all $\text{x}\in[-1,1].$
Reason: $\text{cosec}^{-1}\text{x}$ is decreasing function in $[-1, 1].$
- 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.
AnswerCorrect option: B. Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
View full question & answer→MCQ 81 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\cos^{-1}\text{x}-\sin^{-1}\text{x}=0,$ then $\text{x}=\frac{1}{\sqrt{2}}.$
Reason: $\cot^{-1}\text{x}+\sin^{-1}\text{x}=\frac{\pi}{2}.$
- ✓
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 91 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: The solution of $\sin^{-1}(6\text{x})+\sin^{-1}(6\sqrt{3}\text{x})=\frac{-\pi}{2}$ is $\text{x}=\frac{1}{12}.$
Reason: $\cot^{-1}\text{x}$ is increasing function for $0\leq\text{x}\leq1.$
- A
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.
- ✓
$A$ is false but $R$ is true.
AnswerCorrect option: D. $A$ is false but $R$ is true.
View full question & answer→MCQ 101 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\cot\big[\frac{\pi}{2}-2\cot^{-1}3\big]=7.$
Reason: $\sin^{-1}\big(\frac{4}{5}\big)+2\tan^{-1}\big(\frac{1}{3}\big)=\frac{\pi}{2}.$
- 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.
AnswerCorrect option: B. Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
View full question & answer→MCQ 111 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\tan^{-1}\text{x}+\tan^{-1}\big(\frac{1}{\text{x}}\big)=\frac{\pi}{2}.$
Reason: $\tan^{-1}\text{x}+\cot^{-1}\text{x}=\frac{\pi}{2}$ for all $\text{x}\in\text{R}.$
- ✓
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 121 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: The value of $\cos^{-1}\text{x}+\cos^{-1}\Big\{\frac{\text{x}}{2}+\frac{\sqrt{2-3\text{x}^{2}}}{2}\Big\}=\frac{\pi}{3}$ when $\frac{1}{2}\leq\text{x}\leq1.$
Reason: $\cot^{-1}\text{x}$ is increasing function for $0\leq\text{x}\leq1.$
- A
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 $A.$
- ✓
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
AnswerCorrect option: C. $A$ is true but $R$ is false.
View full question & answer→MCQ 131 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\sin^{-1}\frac{8}{17}+\sin^{-1}\frac{3}{5}=\sin^{-1}\frac{77}{85}.$
Reason: $\sin^{-1}\text{x}+\sin^{-1}\text{y}=\sin^{-1}\big(\text{x}\sqrt{1-\text{y}^{2}}+\text{y}\sqrt{1-\text{x}^{2}}\big)$ for $-1\leq\text{x},\text{y}\leq1,\text{x}^{2}+\text{y}^{2}\leq1.$
- ✓
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 141 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $\cos^{-1}\text{x}=2\sin^{-1}\sqrt{\frac{1-\text{x}}{2}}=2\cos^{-1}\sqrt{\frac{1+\text{x}}{2}}.$
Reason: $1+\cos\text{A}=2\cos^{2}\big(\frac{\text{A}}{2}\big)$ and $1-\cos\text{A}=2\sin^{2}\big(\frac{\text{A}}{2}\big).$
- ✓
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 151 Mark
Assertion (A): Domain of $y=\cos ^{-1}(x)$ is $[-1,1]$.
Reason $(R)$ : The range of the principal value branch of $y=\cos ^{-1}(x)$ is $[0, \pi]-\left\{\frac{\pi}{2}\right\}$.
- A
Both Assertion (A) and Reason (R) are true and Reason $(R)$ is the correct explanation of the Assertion (A).
- B
Both Assertion (A) and Reason (R) are true, but Reason $(R)$ is not the correct explanation of the Assertion (A).
- C
Assertion (A) is true, but Reason (R) is false.
- D
Assertion (A) is false, but Reason (R) is true.
AnswerDomain of $\cos ^{-1}(x)$ is $[-1,1]$.
$\therefore \quad$ Assertion $( A )$ is true.
The range of principal value branch of $\cos ^{-1}(x)$ is $[0, \pi]$.
$\therefore \quad$ Reason $(R)$ is false.
View full question & answer→MCQ 161 Mark
Assertion $(A)$ : The range of the function $f(x)=2 \sin ^{-1} x+\frac{3 \pi}{2}$, where $x \in[-1,1]$, is $\left[\frac{\pi}{2}, \frac{5 \pi}{2}\right]$.
Reason $(R)$ : The range of the principal value branch of $\sin ^{-1}(x)$ is $[0, \pi]$
AnswerGiven, $f(x)=2 \sin ^{-1} x+\frac{3 \pi}{2}, x \in[-1,1]$
As, $-\frac{\pi}{2} \leq \sin ^{-1} x \leq \frac{\pi}{2}$
$\Rightarrow-2 \times \frac{\pi}{2} \leq 2 \sin ^{-1} x \leq 2 \times \frac{\pi}{2} \Rightarrow-\pi \leq 2 \sin ^{-1} x \leq \pi$
$\Rightarrow \quad-\pi+\frac{3 \pi}{2} \leq 2 \sin ^{-1} x+\frac{3 \pi}{2} \leq \pi+\frac{3 \pi}{2}$
$\Rightarrow \frac{\pi}{2} \leq 2 \sin ^{-1} x+\frac{3 \pi}{2} \leq \frac{5 \pi}{2}$
$\therefore \quad$ The range of $f(x)$ is $\left[\frac{\pi}{2}, \frac{5 \pi}{2}\right]$
$\therefore \quad$ Assertion $(A)$ is true.
Now, the range of the principal branch of $\sin ^{-1}(x)$ is $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$
$\therefore \quad$ Reason (R) false.
View full question & answer→MCQ 171 Mark
Assertion (A): The principal value of $\cot ^{-1}(\sqrt{3})$ is $\frac{\pi}{6}$.
Reason $( R )$ : Domain of $\cot ^{-1} x$ is $R -\{-1,1\}$.
- A
Both Assertion (A) and Reason (R) are true and Reason $(R)$ is the correct explanation of the Assertion (A).
- B
Both Assertion (A) and Reason (R) are true, but Reason $(R)$ is not the correct explanation of the Assertion (A).
- C
Assertion (A) is true and Reason (R) is false.
- D
Assertion (A) is false and Reason (R) is true.
AnswerWe know that, $\cot ^{-1}(x), x \in(0, \pi)$
$\cot ^{-1}(\sqrt{3})=\cot ^{-1}\left(\cot \frac{\pi}{6}\right)=\frac{\pi}{6}$
$\left[\because \cot ^{-1}(\cot \theta)=\theta\right]$
$\therefore \quad$ Assertion $( A )$ is true.
Domain of $\cot ^{-1} x$ is $R$.
So, reason (R) is false.
View full question & answer→MCQ 181 Mark
Assertion (A) : All trigonometric functions have their inverses over their respective domains.
Reason $( R )$ : The inverse of $\tan ^{-1} x$ exists for some $x \in R$.
- A
Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
- B
Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
- C
Assertion (A) is true but Reason (R) is false.
- D
Assertion (A) is false but Reason (R) is true.
AnswerAll trigonometric functions are periodic and hence not invertible over their respective domains but all trigonometric functions have inverse over their restricted domains.
Inverse of $\tan ^{-1} x$ is $\tan x$ which is defined for
$x \in R-(2 n+1) \frac{\pi}{2}, n \in Z$
$\therefore \quad$ Assertion is false and reason is true
View full question & answer→MCQ 191 Mark
Assertion (A): The domain of the function $\sec ^{-1} 2 x$ is $\left(-\infty,-\frac{1}{2}\right] \cup\left[\frac{1}{2}, \infty\right) .$ Reason $(R): \sec ^{-1}(-2)=-\frac{\pi}{4}$
- A
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$\sec ^{-1} x$ is defined if $x \leq-1$ or $x \geq 1$.
Hence, $\sec ^{-1} 2 x$ will be defined if $x \leq-\frac{1}{2}$ or $x \geq \frac{1}{2}$
The range of the function $\sec ^{-1} x$ is $[0, \pi]-\left\{\frac{\pi}{2}\right\}$
Hence, $A$ is true and $R$ is false.
View full question & answer→MCQ 201 Mark
Assertion (A) : The domain of the function
$\sec ^{-1} 2 x$ is $\left(-\infty,-\frac{1}{2}\right] \cup\left[\frac{1}{2}, \infty\right)$.
Reason (R): $\sec ^{-1}(-2)=-\frac{\pi}{4}$.
- A
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).
- ✓
(A) is true but (R) is false.
- D
(A) is false but (R) is true.
AnswerCorrect option: C. (A) is true but (R) is false.
(c) : $\sec ^{-1} x$ is defined if $x \leq-1$ or $x \geq 1$.
Hence, $\sec ^{-1} 2 x$ will be defined if $x \leq-\frac{1}{2}$ or $x \geq \frac{1}{2}$
The range of the function $\sec ^{-1} x$ is $[0, \pi]-\left\{\frac{\pi}{2}\right\}$.
Hence, Assertion is true and Reason is false.
View full question & answer→MCQ 211 Mark
Assertion (A) : The domain for
$f(x)=\sin ^{-1}\left(\frac{1+x^2}{2 x}\right)$ is $\{0,1\}$.
Reason (R) : $\sin ^{-1} x$ is defined only if $x \in[-1,1]$.
- A
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.
- ✓
(A) is false but (R) is true.
AnswerCorrect option: D. (A) is false but (R) is true.
(d) : $f(x)=\sin ^{-1}\left(\frac{1+x^2}{2 x}\right)$ is defined for
$
-1 \leq \frac{1+x^2}{2 x} \leq 1
$
or $\left|\frac{1+x^2}{2 x}\right| \leq 1$
$\Rightarrow\left|1+x^2\right| \leq|2 x|$, for all $x \in R$
$\Rightarrow 1+x^2 \leq|2 x|$, for all $x\left(\right.$ as $1+x^2>0$ )
$\Rightarrow x^2-2|x|+1 \leq 0$
$\Rightarrow|x|^2-2|x|+1 \leq 0\left(\right.$ as $\left.x^2=|x|^2\right)$
$\Rightarrow(|x|-1)^2 \leq 0$
But $(|x|-1)^2$ is always either positive or zero. Thus,
$
(|x|-1)^2=0
$
or $\quad|x|=1$ or $x= \pm 1$
Hence, domain for $f(x)$ is $\{-1,1\}$.
View full question & answer→MCQ 221 Mark
Assertion (A) : Range of $f(x)=\cot ^{-1}\left(2 x-x^2\right)$ is $(0, \pi)$.
Reason (R) : $\cot ^{-1} x$ is defined for all $x \in R$.
- A
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.
- ✓
$(A)$ is false but $(R)$ is true.
AnswerCorrect option: D. $(A)$ is false but $(R)$ is true.
Let $\theta=\cot ^{-1}\left(2 x-x^2\right) \text {, where } \theta \in(0, \pi)$
$\Rightarrow \cot \theta=2 x-x^2 \text {, where } \theta \in(0, \pi)$
$=1-\left(1-2 x+x^2\right) \text {, where } \theta \in(0, \pi)$
$=1-(1-x)^2 \text {, where } \theta \in(0, \pi)$
$\Rightarrow \cot \theta \leq 1$, where $\theta \in(0, \pi)$
$\Rightarrow \frac{\pi}{4} \leq \theta<\pi \Rightarrow$ Range of $f(x)$ is $\left[\frac{\pi}{4}, \pi\right)$.
View full question & answer→MCQ 231 Mark
Assertion (A) : Number of roots of the equation $\cot ^{-1} x+\cos ^{-1} 2 x+\pi=0$ is zero.
Reason (R) : Range of $\cot ^{-1} x$ and $\cos ^{-1} x$ is $(0, \pi)$ and $[0, \pi]$, respectively.
- ✓
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.
AnswerCorrect option: A. Both (A) and (R) are true and (R) is the correct explanation of (A).
(a): Reason is correct, from which we can say $\cot ^{-1} x+\cos ^{-1} 2 x=-\pi$ is not possible. Hence, both the assertion, reason are correct, and reason is the correct explanation of assertion.
View full question & answer→MCQ 241 Mark
Assertion (A) : Range of $f(x)=\sin ^{-1} x$ $+\tan ^{-1} x+\sec ^{-1} x$ is $\left\{\frac{\pi}{4}, \frac{3 \pi}{4}\right\}$.
Reason (R) : $f(x)=\sin ^{-1} x+\tan ^{-1} x+\sec ^{-1} x$ is defined for all $x \in[-1,1]$.
- A
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).
- ✓
(A) is true but (R) is false.
- D
(A) is false but (R) is true.
AnswerCorrect option: C. (A) is true but (R) is false.
(c) : $f(x)=\sin ^{-1} x+\tan ^{-1} x+\sec ^{-1} x$;
clearly, domain of $f(x)$ is $x= \pm 1$.
Thus, the range is $\{f(1), f(-1)\}$, i.e., $\left\{\frac{\pi}{4}, \frac{3 \pi}{4}\right\}$.
View full question & answer→MCQ 251 Mark
Assertion (A) : Principal value of $\sin ^{-1}\left(\sin \left(\frac{2 \pi}{3}\right)\right)$ is $\frac{\pi}{3}$.
Reason (R) : Principal value branch of $\sin ^{-1}$ function is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.
- ✓
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.
AnswerCorrect option: A. Both (A) and (R) are true and (R) is the correct explanation of (A).
(a) : Let $y=\sin ^{-1}\left(\frac{2 \pi}{3}\right) \sin ^{-1}(\sin (\pi-\pi / 3))$
$=\sin ^{-1}\left(\sin \frac{\pi}{3}\right)=\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)$
$=\frac{\pi}{3} \quad\left[\because \sin ^{-1}:[-1,1] \rightarrow\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\right]$
View full question & answer→MCQ 261 Mark
Assertion (A) : All trigonometric functions have their inverses over their respective domains.
Reason (R) : The inverse of $\tan ^{-1} x$ exists for some $x \in R$.
- A
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.
- ✓
(A) is false but (R) is true.
AnswerCorrect option: D. (A) is false but (R) is true.
(d) : All trigonometric functions are periodic and hence not invertible over their respective domains but all trigonometric functions have inverse over their restricted domains.
Inverse of $\tan ^{-1} x$ is $\tan x$ which is defined for
$
x \in R-(2 n+1) \frac{\pi}{2}, n \in Z
$
$\therefore \quad$ Assertion is false and reason is true.
View full question & answer→