JEE Main 2026 — Jan 28, Shift 2 — Mathematics solution

JEE Main2026Jan 28, Shift 2Mathematics

The sum of all the elements in the range of $f (x) = Sgn (sin x) + Sgn (cos x) + Sgn (tan x) + Sgn (cot x), x \neq = \frac{nπ}{2}, n \in Z$ , where $Sgn (t) = {1, if t > 0 - 1, if t < 0$ , is

A
4
B
2
C
$- 2$
D
0

Solution

Evaluate signs of $sin x, cos x, tan x, cot x$ in each open quadrant (avoiding multiples of $π /2$ ):

$x \in (0, π /2)$ : all positive $\Rightarrow f = 1 + 1 + 1 + 1 = 4$ .

$x \in (π /2, π)$ : $sin x > 0$ , $cos x < 0$ , $tan x < 0$ , $cot x < 0 \Rightarrow f = 1 - 1 - 1 - 1 = - 2$ .

$x \in (π, 3 π /2)$ : $sin x < 0$ , $cos x < 0$ , $tan x > 0$ , $cot x > 0 \Rightarrow f = - 1 - 1 + 1 + 1 = 0$ .

$x \in (3 π /2, 2 π)$ : $sin x < 0$ , $cos x > 0$ , $tan x < 0$ , $cot x < 0 \Rightarrow f = - 1 + 1 - 1 - 1 = - 2$ .

Range $= {- 2, 0, 4}$ . Sum $= - 2 + 0 + 4 = 2$ .

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