Consider the following statements : I. ({u}+{v}+{f}) is an integer. II. ({f}+{v}) is an integer. Which of the statements given above is/are correct ?

Correct Answer is : Both I and II

Correct Answer is : Both I and II

Test Index

UPSC NDA I 2026 Mathematics Paper

Question Numbers: 46-50

For the next five (05) items that follow:

Let

u

be a positive integer and

f

be a real number lying between 0 and 1 .

Further,

(\sqrt{2}+1)^{10}=\mathrm{u}+\mathrm{f}

and

(\sqrt{2}-1)^{10}=v

Question : 46 of 120

Marks: +1, -0

(\mathrm{u}+\mathrm{v}+\mathrm{f})

(\mathrm{f}+\mathrm{v})

Which of the statements given above is/are correct ?

Solution:

Concept:

For conjugate surds,

(\sqrt2+1)^n+(\sqrt2-1)^n

is always an integer.

Explanation:

Let

(\sqrt2+1)^{10}=u+f

and

(\sqrt2-1)^{10}=v

Since

\sqrt2-1

is the conjugate of

\sqrt2+1

(u+f)+v=(\sqrt2+1)^{10}+(\sqrt2-1)^{10}=\text{integer}.

Thus

u+v+f

is an integer, so Statement I is correct.

Also,

f+v=(u+f)+v-u=\text{integer}-u=\text{integer}

Thus Statement II is also correct.

Answer:

Both I and II are correct.

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