IIT JAM Exam > IIT JAM Questions > Let f(x) = (x-1) (x-2) (x-3) (x-4) and let &a...
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Let f(x) = (x-1) (x-2) (x-3) (x-4) and let α = f(3/2), β = f(5/2) and γ = f(7/2). Which of the following is/are CORRECT?
- a)α and β have the same sign
- b)α and γ have the same sign
- c)β and γ have the same sign
- d)αβ and βγ have the same sign
Correct answer is option 'B,D'. Can you explain this answer?
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Most Upvoted Answer
Let f(x) = (x-1) (x-2) (x-3) (x-4) and let α = f(3/2), β = ...
Understanding the Function f(x)
The function f(x) = (x-1)(x-2)(x-3)(x-4) is a polynomial of degree 4, with roots at x = 1, 2, 3, and 4.
Evaluating f at Specific Points
- Calculate α = f(3/2)
f(3/2) = (3/2 - 1)(3/2 - 2)(3/2 - 3)(3/2 - 4)
= (1/2)(-1/2)(-3/2)(-5/2)
= (1/2) * (-1/2) * (3/2) * (5/2)
= Positive value.
- Calculate β = f(5/2)
f(5/2) = (5/2 - 1)(5/2 - 2)(5/2 - 3)(5/2 - 4)
= (3/2)(-1/2)(-1/2)(-3/2)
= Negative value.
- Calculate γ = f(7/2)
f(7/2) = (7/2 - 1)(7/2 - 2)(7/2 - 3)(7/2 - 4)
= (5/2)(3/2)(1/2)(-1/2)
= Negative value.
Analyzing Signs of α, β, and γ
- Comparing α and β
α is positive, β is negative.
Hence, they do NOT have the same sign.
- Comparing α and γ
α is positive, γ is negative.
Hence, they do NOT have the same sign.
- Comparing β and γ
Both β and γ are negative.
Hence, they have the same sign.
- Comparing pairs αβ and βγ
α is positive while β and γ are negative.
Therefore, α and β do NOT have the same sign, but β and γ do.
Conclusion
Based on the evaluations, the correct options are:
- Option B: α and γ have the same sign.
- Option D: β and γ have the same sign.
Hence, the final answer is B and D.
The function f(x) = (x-1)(x-2)(x-3)(x-4) is a polynomial of degree 4, with roots at x = 1, 2, 3, and 4.
Evaluating f at Specific Points
- Calculate α = f(3/2)
f(3/2) = (3/2 - 1)(3/2 - 2)(3/2 - 3)(3/2 - 4)
= (1/2)(-1/2)(-3/2)(-5/2)
= (1/2) * (-1/2) * (3/2) * (5/2)
= Positive value.
- Calculate β = f(5/2)
f(5/2) = (5/2 - 1)(5/2 - 2)(5/2 - 3)(5/2 - 4)
= (3/2)(-1/2)(-1/2)(-3/2)
= Negative value.
- Calculate γ = f(7/2)
f(7/2) = (7/2 - 1)(7/2 - 2)(7/2 - 3)(7/2 - 4)
= (5/2)(3/2)(1/2)(-1/2)
= Negative value.
Analyzing Signs of α, β, and γ
- Comparing α and β
α is positive, β is negative.
Hence, they do NOT have the same sign.
- Comparing α and γ
α is positive, γ is negative.
Hence, they do NOT have the same sign.
- Comparing β and γ
Both β and γ are negative.
Hence, they have the same sign.
- Comparing pairs αβ and βγ
α is positive while β and γ are negative.
Therefore, α and β do NOT have the same sign, but β and γ do.
Conclusion
Based on the evaluations, the correct options are:
- Option B: α and γ have the same sign.
- Option D: β and γ have the same sign.
Hence, the final answer is B and D.
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Let f(x) = (x-1) (x-2) (x-3) (x-4) and let α = f(3/2), β = f(5/2) and γ = f(7/2). Which of the following is/are CORRECT?a)α andβ have the same signb)αandγ have the same signc)βandγ have the same signd)αβandβγ have the same signCorrect answer is option 'B,D'. Can you explain this answer?
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Let f(x) = (x-1) (x-2) (x-3) (x-4) and let α = f(3/2), β = f(5/2) and γ = f(7/2). Which of the following is/are CORRECT?a)α andβ have the same signb)αandγ have the same signc)βandγ have the same signd)αβandβγ have the same signCorrect answer is option 'B,D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let f(x) = (x-1) (x-2) (x-3) (x-4) and let α = f(3/2), β = f(5/2) and γ = f(7/2). Which of the following is/are CORRECT?a)α andβ have the same signb)αandγ have the same signc)βandγ have the same signd)αβandβγ have the same signCorrect answer is option 'B,D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f(x) = (x-1) (x-2) (x-3) (x-4) and let α = f(3/2), β = f(5/2) and γ = f(7/2). Which of the following is/are CORRECT?a)α andβ have the same signb)αandγ have the same signc)βandγ have the same signd)αβandβγ have the same signCorrect answer is option 'B,D'. Can you explain this answer?.
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