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If the equation ax² + bx + c = x³ + (1/x) (a, b, c ≤ 0) has only real roots, then
(A) the maximum value of 'a' is (-5)
(B) when 'a' takes it's maximum value a, b, c are in AP
(C) when all the roots are integers the value of a + b + c is (-14)
(D) None of these?
(A) the maximum value of 'a' is (-5)
(B) when 'a' takes it's maximum value a, b, c are in AP
(C) when all the roots are integers the value of a + b + c is (-14)
(D) None of these?
Most Upvoted Answer
If the equation ax² + bx + c = x³ + (1/x) (a, b, c ≤ 0) has only real ...
Solution:
Given, ax² - bx + c = x³ - (a/b)
Let the roots of the given equation be α, β and γ.
Then, by Vieta's formulas,
α + β + γ = 0, αβ + βγ + γα = -b/a, and αβγ = -c/a.
Now, since the given equation has only real roots, we have α, β, γ ≤ 0.
Also, since a, b, c ≤ 0, we have αβγ ≥ 0.
Hence, at least two of the roots must be zero.
Case 1: α = 0
Then, βγ = -c/a and β + γ = 0.
Hence, β and γ are the roots of the quadratic equation ax - b + c/x = 0.
Since ax² - bx + c = 0 has only real roots, it follows that ax - b + c/x = 0 also has only real roots.
Thus, the discriminant of ax - b + c/x = 0 is non-negative, i.e., b² - 4ac ≥ 0.
Substituting α = 0, βγ = -c/a and β + γ = 0 in b² - 4ac ≥ 0, we get:
c²/a² - 4ac ≤ 0, i.e., c/a ≤ 4/a².
Since a ≤ 0, we have a² ≤ 0 and hence, c/a ≤ 0.
Thus, the maximum value of a is (-5).
Hence, option (A) is correct.
Case 2: Two roots are zero
Without loss of generality, assume α = β = 0.
Then, γ = -c/a and αβγ = 0, i.e., c = 0.
Hence, the given equation reduces to ax² - bx = 0.
Since a, b ≤ 0, we have a ≤ 0 and b ≥ 0.
Also, the roots of ax² - bx = 0 are 0 and b/a.
Thus, b/a ≤ 0, i.e., b ≤ 0.
Hence, a, b, c ≤ 0 and a, b are in AP.
Thus, option (B) is correct.
Case 3: All roots are integers
Let α, β, γ be the roots of the given equation.
Then, αβγ = -c/a is an integer.
Also, α + β + γ = 0 is an integer.
Hence, at least one of the roots is 0, and the other two roots are of opposite sign.
Without loss of generality, assume α = 0 and β, γ < />
Then, βγ = -c/a is positive and β + γ = 0 is negative.
Hence, β, γ are integers, and |β|, |γ| ≥ 1.
Thus, |βγ| ≥ 1 and hence, |c/a| ≥ 1.
Since a ≤ 0, we have c/a ≤ 0.
Hence, c ≤ 0.
Substituting α = 0, βγ = -c/a and β + γ = 0 in the equation ax² - bx + c = 0, we get:
aγ
Given, ax² - bx + c = x³ - (a/b)
Let the roots of the given equation be α, β and γ.
Then, by Vieta's formulas,
α + β + γ = 0, αβ + βγ + γα = -b/a, and αβγ = -c/a.
Now, since the given equation has only real roots, we have α, β, γ ≤ 0.
Also, since a, b, c ≤ 0, we have αβγ ≥ 0.
Hence, at least two of the roots must be zero.
Case 1: α = 0
Then, βγ = -c/a and β + γ = 0.
Hence, β and γ are the roots of the quadratic equation ax - b + c/x = 0.
Since ax² - bx + c = 0 has only real roots, it follows that ax - b + c/x = 0 also has only real roots.
Thus, the discriminant of ax - b + c/x = 0 is non-negative, i.e., b² - 4ac ≥ 0.
Substituting α = 0, βγ = -c/a and β + γ = 0 in b² - 4ac ≥ 0, we get:
c²/a² - 4ac ≤ 0, i.e., c/a ≤ 4/a².
Since a ≤ 0, we have a² ≤ 0 and hence, c/a ≤ 0.
Thus, the maximum value of a is (-5).
Hence, option (A) is correct.
Case 2: Two roots are zero
Without loss of generality, assume α = β = 0.
Then, γ = -c/a and αβγ = 0, i.e., c = 0.
Hence, the given equation reduces to ax² - bx = 0.
Since a, b ≤ 0, we have a ≤ 0 and b ≥ 0.
Also, the roots of ax² - bx = 0 are 0 and b/a.
Thus, b/a ≤ 0, i.e., b ≤ 0.
Hence, a, b, c ≤ 0 and a, b are in AP.
Thus, option (B) is correct.
Case 3: All roots are integers
Let α, β, γ be the roots of the given equation.
Then, αβγ = -c/a is an integer.
Also, α + β + γ = 0 is an integer.
Hence, at least one of the roots is 0, and the other two roots are of opposite sign.
Without loss of generality, assume α = 0 and β, γ < />
Then, βγ = -c/a is positive and β + γ = 0 is negative.
Hence, β, γ are integers, and |β|, |γ| ≥ 1.
Thus, |βγ| ≥ 1 and hence, |c/a| ≥ 1.
Since a ≤ 0, we have c/a ≤ 0.
Hence, c ≤ 0.
Substituting α = 0, βγ = -c/a and β + γ = 0 in the equation ax² - bx + c = 0, we get:
aγ
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If the equation ax² + bx + c = x³ + (1/x) (a, b, c ≤ 0) has only real roots, then (A) the maximum value of 'a' is (-5) (B) when 'a' takes it's maximum value a, b, c are in AP (C) when all the roots are integers the value of a + b + c is (-14) (D) None of these?
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If the equation ax² + bx + c = x³ + (1/x) (a, b, c ≤ 0) has only real roots, then (A) the maximum value of 'a' is (-5) (B) when 'a' takes it's maximum value a, b, c are in AP (C) when all the roots are integers the value of a + b + c is (-14) (D) None of these? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the equation ax² + bx + c = x³ + (1/x) (a, b, c ≤ 0) has only real roots, then (A) the maximum value of 'a' is (-5) (B) when 'a' takes it's maximum value a, b, c are in AP (C) when all the roots are integers the value of a + b + c is (-14) (D) None of these? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the equation ax² + bx + c = x³ + (1/x) (a, b, c ≤ 0) has only real roots, then (A) the maximum value of 'a' is (-5) (B) when 'a' takes it's maximum value a, b, c are in AP (C) when all the roots are integers the value of a + b + c is (-14) (D) None of these?.
If the equation ax² + bx + c = x³ + (1/x) (a, b, c ≤ 0) has only real roots, then (A) the maximum value of 'a' is (-5) (B) when 'a' takes it's maximum value a, b, c are in AP (C) when all the roots are integers the value of a + b + c is (-14) (D) None of these? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the equation ax² + bx + c = x³ + (1/x) (a, b, c ≤ 0) has only real roots, then (A) the maximum value of 'a' is (-5) (B) when 'a' takes it's maximum value a, b, c are in AP (C) when all the roots are integers the value of a + b + c is (-14) (D) None of these? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the equation ax² + bx + c = x³ + (1/x) (a, b, c ≤ 0) has only real roots, then (A) the maximum value of 'a' is (-5) (B) when 'a' takes it's maximum value a, b, c are in AP (C) when all the roots are integers the value of a + b + c is (-14) (D) None of these?.
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