The domain of y = ^(-1)(x^2 - 4) is Option 1: [0, ] Option 2: [-√5, -√3] [√3, √5] Option 3: [-√5, -√3] [-√5, √3] Option 4: [-1, 1]

Answer: [-√5, -√3] [√3, √5]. Solution: The domain consists of all x-values that can be input into the function. For ^(-1) (inverse cosine), the input must be between -1 and 1, inclusive. Since the function is y = ^(-1)(x^2 - 4), the expression (x^2 - 4) must satisfy: -1 ≤ x^2 - 4 ≤ 1 For the left inequality: x^2 - 4 ≥ -1 x^2 ≥ 3, x, ≥ √3 This gives: x ≤ -√3 or x ≥ √3 For the right inequality: x^2 - 4 ≤ 1 x^2 ≤ 5, x, ≤ √5 This gives: -√5 ≤ x ≤ √5 Both conditions must be satisfied simultaneously. On the left side: x ≤ -√3 and x ≥ -√5 gives [-√5, -√3] On the right side: x ≥ √3 and x ≤ √5 gives [√3, √5] Domain: [-√5, -√3] [√3, √5]

Algebra

Relations & Functions

Medium

core

The domain of $y = \cos^{-1}(x^2 - 4)$ is

$[0, \pi]$

$[-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]$

$[-\sqrt{5}, -\sqrt{3}] \cap [-\sqrt{5}, \sqrt{3}]$

$[-1, 1]$

✅ Correct Option: 2

The domain consists of all x-values that can be input into the function.

For $\cos^{-1}$ (inverse cosine), the input must be between $-1$ and $1$ , inclusive.

Since the function is $y = \cos^{-1}(x^2 - 4)$ , the expression $(x^2 - 4)$ must satisfy:

$-1 \leq x^2 - 4 \leq 1$

For the left inequality:

$x^2 - 4 \geq -1$

$x^2 \geq 3$

$|x| \geq \sqrt{3}$

This gives: $x \leq -\sqrt{3}$ or $x \geq \sqrt{3}$

For the right inequality:

$x^2 - 4 \leq 1$

$x^2 \leq 5$

$|x| \leq \sqrt{5}$

This gives: $-\sqrt{5} \leq x \leq \sqrt{5}$

Both conditions must be satisfied simultaneously.

On the left side: $x \leq -\sqrt{3}$ and $x \geq -\sqrt{5}$ gives $[-\sqrt{5}, -\sqrt{3}]$

On the right side: $x \geq \sqrt{3}$ and $x \leq \sqrt{5}$ gives $[\sqrt{3}, \sqrt{5}]$

Domain: $[-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]$

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Which of the following relations on the set $A = \{1, 2, 3\}$ are equivalence ?

(A) $R = \{(1,1), (2,2), (1,2), (2,1)\}$

(B) $R = \{(1,1), (2,2), (3,3)\}$

(D) $R = \{(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2), (1,3), (3,1)\}$

(E) $R = \{(1,1), (2,2), (3,3), (1,2)\}$

Choose the correct answer from the options given below :

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Match List I with List II

List I	List II
A. Range of $\lvert x\rvert$	I. $(-5, \infty)$
B. Range of $9x^2 + 6x - 5$ for all $x \geq 0$	II. $[0, \infty)$
C. Domain of $\dfrac{1}{\sqrt{x+5}}$	III. $\{(1,1), (2,2), (3,3)\}$
D. Smallest equivalence relation on Set $\{1,2,3\}$	IV. $[-5, \infty)$

Choose the correct answer from the options given below:

The domain of y=cos⁡−1(x2−4)y = \cos^{-1}(x^2 - 4)y=cos−1(x2−4) is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The domain of $y = \cos^{-1}(x^2 - 4)$ is