KVPY SX SB Exam 19-Nov-2017 Question Paper

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Question : 2 of 120

Marks: +1, -0

Among the inequalities below, which ones aretrue for all natural numbers n greater than 1000 ?
I.

n ! ≤ n^n

II.

(n!)^2≤ n^n

III.

10^n≤n!

IV.

n^n≤ (2n)!

I and IV only
I, III and IV only
II and IV only
I, II, III and IV

Solution:

(A)

n^n/n!=(n/n)(n/{n-1})(n/{n-2}).....(n/1)

≥1

n^n≥ n !

(correct)
(C)

n!/10^n={(n)(n -1)(n -2)......}/{(10) (10).......\n \times (10)}

given that n > 1000
clearly

n!/10^n≥1

n! ≥ 10^n

(D)

{(1.2.3.4........n)(n +1)(n + 2)(n + 3)......(n + n)}/n^n

(n !) (1+1/n)(1+2/n)(1+3/n)

.....(1+n/n)

clearly ≥ 1.

2n! ≥ n^n

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