JEE Main | 2014Math miscellaneousMedium
Question
If a ∈ R and the equation
-3(x - [x])2 + 2 (x - [x]) + a2 = 0
(where [x] denotes the greatest integer ≤ x) has no integral solution, then all possible values of a lie in the interval
-3(x - [x])2 + 2 (x - [x]) + a2 = 0
(where [x] denotes the greatest integer ≤ x) has no integral solution, then all possible values of a lie in the interval
Options
A.(-2, -1)
B.(-∞, - 2) ∪ (2, ∞)
C.(-1, 0) ∪ (0, 1)
D.(1, 2)
Solution
-3(x- [x])2 + 2[x - [x]) + a2 = 0
3 {x}2 - 2{x} - a2 = 0
a ≠ 0, 3
= a2
a2 = 3
0 ≤ {x} < 1 and -
≤ {x} - 
0 ≤ 3
For non-integral solution
0 < a2 < 1 and a ∈ (-1, 0) υ (0, 1)
Alternative
-3{x}2 + 2{x} + a2 = 0
Now, -3{x}2 + 2{x}
to have no integral roots 0 < a2 < 1
∴ a ∈ (-1, 0) υ(0, 1)
3 {x}2 - 2{x} - a2 = 0
a ≠ 0, 3
= a2a2 = 3
0 ≤ {x} < 1 and -
≤ {x} - 0 ≤ 3
For non-integral solution
0 < a2 < 1 and a ∈ (-1, 0) υ (0, 1)
Alternative
-3{x}2 + 2{x} + a2 = 0
Now, -3{x}2 + 2{x}
to have no integral roots 0 < a2 < 1
∴ a ∈ (-1, 0) υ(0, 1)
Create a free account to view solution
View Solution FreeMore Math miscellaneous Questions
Number of solutions of the equation ex(ex - 2) = 5|ex - 1| - 7 is -...If A = and I = then which one of the following holds for all n ≥ 1, by the principle of mathematical indunction...If f(y) is differentiable at y = 5 and f(5) = 1, f′(5) = , then the value of (y + t)etdt is -...The number of even divisors of the number N = 12600 is :-...If a, b, c are in A.P., b, c, d are in G.P. & c, d, e are in H.P., then -...