Find the value of K for which the inequality x2-2(4K-1)x+15K​2-2K-7>0 is valid for any x.
 

Dear Student,

Please find below the solution to the asked query:

We have : x 2 - 2 ( 4 k - 1 ) x + 15 k2 - 2 k - 7 > 0  is valid for any x

So,

That is true if a > 0 and D  < 0  , Here

a =  1 , So  a > 0

And D  =  b2 - 4 a c  , So

[ - 2 ( 4 k - 1 ) ]2  - 4 ( 1 ) ( 15 k2 - 2 k - 7 ) < 0

4 ( 16 k2  + 1 - 8 k)   -  60 k2  + 8 k + 28 < 0

64 k2  + 4 - 32 k -  60 k2  + 8 k + 28 < 0

4 k2  - 24 k -  + 32 < 0

4 ( k2  -  6 k -  + 8 ) < 0

k2  -  6 k + 8 < 0

k2  -  4 k - 2 k  + 8 < 0

k ( k -  4 ) - 2 ( k -  4 ) < 0

( k -  2 )( k -  4 ) < 0

So,
k ( 2 , 4 )     ( As , For ( k -  2 )( k -  4 ) = 0  ,  k  =  2 and  4 )

Therefore,

Integral values of k  = 4 , 3 , 2                                              ( Ans )

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