KVPY SX SB Exam 20141 Question Paper

Question : 88 of 120

Marks: +1, -0

For a real number x let [x] denote the largest integer less than or equal to x . The smallest positive integer n for which the integral

∫↖{n}↙{1}[x][√x]\dx

exceeds 60 is

Solution:

Let I =

∫↖{n}↙{1}[x][√x]\dx

1 ≤ x < 4 \;\;\;[√x]=1

4 ≤ x < 9 \;\;\;[√x]=2

4 ≤ x < 9 \;\;\;[√x]=3

Now,

I=∫↖{2}↙{1}\dx+∫↖{3}↙{2}2\dx+∫↖{4}↙{3}3\dx+∫↖{5}↙{4}8\dx

+∫↖{6}↙{5}10\dx +∫↖{7}↙{6}12\dx+∫↖{8}↙{7}14\dx +∫↖{9}↙{8}16\dx

+∫↖{10}↙{9}27\dx+∫↖{11}↙{10}30\dx....

I = 1 + 2 + 3 + 8 + 10 + 12 + 14 + 16 = 66
So n = 9

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