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ISI MStat 2018 PSA Problem 12 | Sequence of positive numbers

This is a problem from ISI MStat 2018 PSA Problem 12 based on Sequence of positive numbers

Sequence of positive numbers - ISI MStat Year 2018 PSA Question 12


Let a_n ,n \ge 1 be a sequence of positive numbers such that a_{n+1} \leq a_{n} for all n, and \lim {n \rightarrow \infty} a{n}=a . Let p_{n}(x) be the polynomial p_{n}(x)=x^{2}+a_{n} x+1 and suppose p_{n}(x) has no real roots for every n . Let \alpha and \beta be the roots of the polynomial p(x)=x^{2}+a x+1 . What can you say about (\alpha, \beta)?

  • (A) \alpha=\beta, \alpha and \beta are not real
  • (B) \alpha=\beta, \alpha and \beta are real.
  • (C) \alpha \neq \beta, \alpha and \beta are real.
  • (D) \alpha \neq \beta, \alpha and \beta are not real

Key Concepts


Sequence

Quadratic equation

Discriminant

Check the Answer


Answer: is (D)

ISI MStat 2018 PSA Problem 12

Introduction to Real Analysis by Bertle Sherbert

Try with Hints


Write the discriminant. Use the properties of the sequence a_n .

Note that as <img decoding="> has no real root so discriminant is <img decoding="> so <img decoding="> and <img decoding=">'s are positive and decreasing so <img decoding="> . So , what can we say about a ?

Therefore we can say that 0 \le a < 2 hence discriminant of P  , a^2-4 must be strictly negative so option D.

Similar Problems and Solutions



ISI MStat 2018 PSA Problem 12
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


This is a problem from ISI MStat 2018 PSA Problem 12 based on Sequence of positive numbers

Sequence of positive numbers - ISI MStat Year 2018 PSA Question 12


Let a_n ,n \ge 1 be a sequence of positive numbers such that a_{n+1} \leq a_{n} for all n, and \lim {n \rightarrow \infty} a{n}=a . Let p_{n}(x) be the polynomial p_{n}(x)=x^{2}+a_{n} x+1 and suppose p_{n}(x) has no real roots for every n . Let \alpha and \beta be the roots of the polynomial p(x)=x^{2}+a x+1 . What can you say about (\alpha, \beta)?

  • (A) \alpha=\beta, \alpha and \beta are not real
  • (B) \alpha=\beta, \alpha and \beta are real.
  • (C) \alpha \neq \beta, \alpha and \beta are real.
  • (D) \alpha \neq \beta, \alpha and \beta are not real

Key Concepts


Sequence

Quadratic equation

Discriminant

Check the Answer


Answer: is (D)

ISI MStat 2018 PSA Problem 12

Introduction to Real Analysis by Bertle Sherbert

Try with Hints


Write the discriminant. Use the properties of the sequence a_n .

Note that as <img decoding="> has no real root so discriminant is <img decoding="> so <img decoding="> and <img decoding=">'s are positive and decreasing so <img decoding="> . So , what can we say about a ?

Therefore we can say that 0 \le a < 2 hence discriminant of P  , a^2-4 must be strictly negative so option D.

Similar Problems and Solutions



ISI MStat 2018 PSA Problem 12
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


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