If the roots of the equation x²-n x+m=0 differ by 1, then

Correct Answer is : n2−4m−1=0n²-4m-1=0n2−4m−1=0

Correct Answer is : n2−4m−1=0n²-4m-1=0n2−4m−1=0

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UPSC NDA Math Model Paper 3

Question : 39 of 120

Marks: +1, -0

x^{2}-n x+m=0

1,

then

Solution:

Let

\alpha

and

\beta

be roots of the equation

x^{2}-n x+m=0

Given, the roots of the equation

x^{2}-n x+m=0

differ by 1

\Rightarrow \alpha-\beta=1

\Rightarrow \beta=\alpha+1

Now,

The sum of the roots

=\alpha+\beta=-(-n)=n

\Rightarrow \alpha+\alpha+1=n

\Rightarrow 2\alpha+1=n

\Rightarrow \alpha=\frac{n-1}{2}

The product of the roots

=\alpha\beta=m

\Rightarrow \alpha(\alpha+1)=m

\Rightarrow \left(\frac{n-1}{2}\right)\left(\frac{n-1}{2}+1\right)=m

\Rightarrow \left(\frac{n-1}{2}\right)\left(\frac{n-1+2}{2}\right)=m

\Rightarrow (n-1)(n+1)=4m

\Rightarrow n^{2}-1=4m

\Rightarrow n^{2}-4m-1=0

herefore

Option 1 is correct.

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