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The equation x3 – 3x + [a] = 0, will have three real and distinct roots if –
(where [ ] denotes the greatest integer function)
(where [ ] denotes the greatest integer function)
- a)a ∈ (–∞, 2)
- b)a ∈ (0,2)
- c)a ∈ (–∞, -2) U (0, ∞)
- d)a ∈ [–1, 2)
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The equation x3– 3x + [a] = 0, will have three real and distinct...
f(x) = x3 – 3x + [a]
Let [a] = t (where t will be an integer)
f(x) = x3 – 3x + t ……….(i)
⇒ f ’(x) = 3x2 – 3
⇒ f ‘(x) = 0 has two real and distinct solution which are x = 1 and x = -1
so f(x) = 0 will have three distinct and real solution when f (1). f(-1) < 0
……………. (ii)
Now,
f(1) = (1)3 -3(1) + t = t – 2
f(–1) = (–1)3 – 3 (–1) + t = t + 2
From equation (ii)
(t –2) (t + 2) < 0
⇒ t ∈ (-2, 2)
Now t = [a]
Hence [a] ∈ (-2, 2)
⇒ a ∈ [-1, 2)
Let [a] = t (where t will be an integer)
f(x) = x3 – 3x + t ……….(i)
⇒ f ’(x) = 3x2 – 3
⇒ f ‘(x) = 0 has two real and distinct solution which are x = 1 and x = -1
so f(x) = 0 will have three distinct and real solution when f (1). f(-1) < 0
……………. (ii)
Now,
f(1) = (1)3 -3(1) + t = t – 2
f(–1) = (–1)3 – 3 (–1) + t = t + 2
From equation (ii)
(t –2) (t + 2) < 0
⇒ t ∈ (-2, 2)
Now t = [a]
Hence [a] ∈ (-2, 2)
⇒ a ∈ [-1, 2)
Most Upvoted Answer
The equation x3– 3x + [a] = 0, will have three real and distinct...
Explanation:
Given Equation:
x^3 - 3x + [a] = 0
Conditions for Three Real and Distinct Roots:
For the given equation to have three real and distinct roots, the discriminant of the equation should be greater than 0.
Discriminant Formula:
The discriminant of a cubic equation ax^3 + bx^2 + cx + d = 0 is given by Δ = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2
Apply the formula to the given equation:
a = 1, b = 0, c = -3, d = [a]
Δ = 18*1*0*[a] - 4*0*[a] + 0 - 4*(-3)*[a] - 27*1*[a]^2
Δ = 0 + 0 + 0 + 12[a] - 27[a]^2
Δ = 12[a] - 27[a]^2
For Three Real and Distinct Roots:
Δ > 0
12[a] - 27[a]^2 > 0
3[a](4 - 9[a]) > 0
Find the Range of a:
3[a] > 0 and 4 - 9[a] > 0
a > 0 and a < />
0 < a="" />< />
Greatest Integer Function:
Since a is an integer, the greatest integer function [a] will be the greatest integer less than or equal to a. So, [a] = 0 when a is in the range (0, 1) and [a] = 1 when a is in the range [1, 2).
Final Range of a:
Combining both ranges, a belongs to the interval [1, 2).
Therefore, the correct answer is option 'D'.
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The equation x3– 3x + [a] = 0, will have three real and distinct roots if –(where [ ] denotes the greatest integer function)a)a ∈ (–∞, 2)b)a ∈ (0,2)c)a ∈ (–∞, -2) U (0, ∞)d)a ∈ [–1, 2)Correct answer is option 'D'. Can you explain this answer?
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The equation x3– 3x + [a] = 0, will have three real and distinct roots if –(where [ ] denotes the greatest integer function)a)a ∈ (–∞, 2)b)a ∈ (0,2)c)a ∈ (–∞, -2) U (0, ∞)d)a ∈ [–1, 2)Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The equation x3– 3x + [a] = 0, will have three real and distinct roots if –(where [ ] denotes the greatest integer function)a)a ∈ (–∞, 2)b)a ∈ (0,2)c)a ∈ (–∞, -2) U (0, ∞)d)a ∈ [–1, 2)Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The equation x3– 3x + [a] = 0, will have three real and distinct roots if –(where [ ] denotes the greatest integer function)a)a ∈ (–∞, 2)b)a ∈ (0,2)c)a ∈ (–∞, -2) U (0, ∞)d)a ∈ [–1, 2)Correct answer is option 'D'. Can you explain this answer?.
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