Ques. 1JEE Main 2026, 24 January (Evening Shift)
The number of elements in the set \x [0, 180 ] : (x + 100 ) = (x + 50 ) x (x - 50 )\ is .
Answer
4
Solution
(x + 100 ) x = (x + 50 ) (x - 50 )
(x + 100 ) x (x + 100 ) x = (x + 50 ) (x - 50 ) (x + 50 ) (x - 50 ) = 100 - 2x 100 + 2x
Applying componendo–dividendo,
(2x + 100 ) 100 = 2 100 -2 2x
2 (2x + 100 ) 2x = -2 100 100
(4x + 100 ) + 100 = - 200
(4x + 100 ) = -( 100 + 200 ) = -2 150 50 = - 40
4x + 100 = 360 k - 40 or 4x + 100 = 360 k + 220
x = 90 k - 35 or x = 90 k + 30
In [0, 180 ] this gives x = 30 , 55 , 120 , 145 .
none of these makes x, (x 50 ) or (x + 100 ) undefined or zero, all four are valid.
Number of elements = 4
Answer: 4