ISI MStat 2018 PSA Problem 12 | Sequence of positive numbers

Content

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

But try the problem first...

Answer: is (D)

Source

Try with Hints

First hint

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

Second Hint

Note that as has no real root so discriminant is so and 's are positive and decreasing so $0\leq a_n<2$ . So , what can we say about a ?

Final Step

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

Outstanding Statistics Program with Applications

Learn More

How to Prepare ?

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) $?