For each positive real number λ, let \A_\\lambda be the set of all natural numbers n such that |\sin( \sqrt{n +1} - \sin ( \sqrtn ) |

Correct Answer is : A12c,A13c,A25c are all finite sets

A_{1/3} is a finite set but A_{1/2} ,A_{2/5} are infinite sets

A_{1/2}^c,A_{1/3}^c,A_{2/5}^c are all finite sets

A_{1/3},A_{2/5} are finite sets and A_{1/2} is an infinite set

KVPY SX SB Exam 2016 Question Paper

Question : 87 of 120

Marks: +1, -0

For each positive real number λ, let

\A_\λ

be the set of all natural numbers n such that

|\sin( √{n +1} - \sin ( √n ) | < \λ

.Let

A_\λ^c

be the complement of

\A_\λ

in the set of all natural numbers. Then -

Solution:

As n → ∞

|\sin √{x+1}-\sin√x|→0

∴ There exist infinite natural numbers for which

|\sin√{x+1}-\sin√x|< λ ∀ λ > 0

Hence

A_{1/2},A_{1/3},A_{2/5}

are all finite sets

Go to Question:

Prev Question

Next Question