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Question

Let f(x)=x312x be function such that the equation |f(|x|)|=n(n N) has exactly 6 distinct real roots then number of possible values of n are :

A
15
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B
16
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C
17
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D
14
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Solution

The correct option is A 15
f(x)=x312x=x(x212)

The roots can be obtained by x(x212)=0

x=0,x=±12=±23

|f(|x|)|=n(given)

f(x)=3x212=0

3x2=12

x2=4

x=±2

At x=2
f(2)=2(2212)=2(412)=16

Thus, the slope is (2,16)

from the graph, we have n(0,16)

There are 16 solutions

Since nN we have n takes the values
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15

Hence the number of possible values of n are 15

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