Xavier Aptitude Test 2011 Paper

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Question : 50 of 101

Marks: +1, -0

If

x

=

(9 + 4 √5) ^{48}

, =

[x] \ f

where

[x]

is defined as integral part of

x

and

f

is a fraction, then

x\ (1– f)

equals

1
Less than 1
More than 1
Between 1 and 2
None of the above

Solution:

(a) Let

a, b, c

be natural numbers :

\ (a+\√{a^{2}-1}\ )^{b}

will have the form

\ (c+√{c^{2}-1}\ )

which is slightly less than.

c+\√{c^{2}}=c+c=2 c

When

a=9,

and

b=48

(a+\√{a^{2}-1}\ )^{b}=x, \∴

integer part of

c+\√{c^{2}-1}=

greatest integer less than

2 \ {c}=2 \ {c}-1

∴

Fractional part of

c+\√{c^{2}-1}=c+\√{c^{2}-1}-(2 c-1)

=\√{c^{2}-1}-c+1

(1-f)=c-√{c^{2}-1}

x(1-f)= (c+\√{c^{2}-1}\ )\ (c-\√{c^{2}-1}\ )

=1.

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