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Question

The number of values of cN for which the the quadratic equation x213x+4c=0 has distinct integer roots are:

A
3
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B
1
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C
2
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D
4
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Solution

The correct option is D 4
Given equation is x213x+6c=0

On comparing with the standard quadratic equation ax2+bx+c=0,a0
we get, a=1; b=13; c=6c
Using discriminant method,
x=b±D2a
If roots are distinct integers, then discriminant should be perfect square of an integer.
Now, D=b24ac=(13)24.1.6c
D=16924c
Also, D>016924c>0c<16924
Since, cNc must be an element of set {1,2,3,4,5,6,7}
So, If c=1, D=16924(1)=145, which is not a perfect square.
c=2, D=16924(2)=121, it is a perfect square .
c=3, D=16924(3)=97, which is not a perfect square.
c=4, D=16924(4)=73,which is not a perfect square.
c=5, D=16924(5)=49 it is a perfect square .
c=6, D=16924(6)=25, it is a perfect square .
c=7, D=16924(7)=1, it is a perfect square .

Hence, the possible values are 2,5,6,7. The number of possible values of c are 4.

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