JEE Main 2026 Apr 6 Shift 1, Mathematics Q1: Inverse Trigonometric Functions

JEE Main2026Apr 6, Shift 1Mathematics

Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f (x) = sin^{- 1} (\frac{x + [ x ]}{3})$ is $[α, β)$ , then $α^{2} + β^{2}$ is equal to:

A
$2$
B
$5$
C
$10$
D
$13$

Solution

The $sin^{- 1}$ argument must lie in $[- 1, 1]$ , so $- 3 \leq x + [x] \leq 3$ , i.e. $- 3 - [x] \leq x \leq 3 - [x]$ .

Test integer values of $[x]$ on the intervals where they apply:

$∙$ If $[x] = - 1$ , then $x \in [- 1, 0)$ and the constraint $- 2 \leq x \leq 4$ is satisfied.

$∙$ If $[x] = 0$ , then $x \in [0, 1)$ and $- 3 \leq x \leq 3$ holds.

$∙$ If $[x] = 1$ , then $x \in [1, 2)$ and $- 4 \leq x \leq 2$ holds.

$∙$ For $[x] \geq 2$ the upper bound $x \leq 3 - [x]$ fails, and similarly for $[x] \leq - 2$ .

Combining the surviving pieces, the domain is $[- 1, 2)$ , so $α = - 1$ and $β = 2$ .

$∴ α^{2} + β^{2} = 1 + 4 = 5$ .

Concept behind this question

Inverse Trigonometric FunctionsNotes, formulas and examples →

Practice more Mathematics

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →

JEE Main 2026 Apr 6 Shift 1, Mathematics Q1: Inverse Trigonometric Functions

Concept behind this question

More previous year questions on Inverse Trigonometric Functions

Practice more Mathematics