A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question: What is the remainder when $x^{2n}-y^{2n} + 1$ is divided by $x^n + y^n$, where $n$ is a natural number? Statement-I : $n$ is odd. Statement-II : $n$ is even. Which one of the following is correct in respect of the above Question and the Statements?

Question

CDS - 2025

A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
What is the remainder when $x^{2n}-y^{2n} + 1$ is divided by $x^n + y^n$, where $n$ is a natural number?
Statement-I :
$n$ is odd.
Statement-II :
$n$ is even.
Which one of the following is correct in respect of the above Question and the Statements?

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CDS 2 2025 Maths Question Paper (14-Sep-2025)

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The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone

The Question can be answered by using either Statement alone

The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

The Question can be answered even without using any of the Statements

The correct answer is

The Question can be answered even without using any of the Statements

Remainder Calculation

The question asks for the remainder when the expression $x^{2n}-y^{2n} + 1$ is divided by $x^n + y^n$, where $n$ is a natural number.

Applying Algebraic Identities

We can simplify the expression $x^{2n}-y^{2n}$ using the difference of squares identity, $a^2 - b^2 = (a-b)(a+b)$.

Let $a = x^n$ and $b = y^n$. Then $a^2 = (x^n)^2 = x^{2n}$ and $b^2 = (y^n)^2 = y^{2n}$.

Applying the identity, we get:

$x^{2n}-y^{2n} = (x^n)^2 - (y^n)^2 = (x^n - y^n)(x^n + y^n)$

This shows that $x^{2n}-y^{2n}$ is exactly divisible by $x^n + y^n$, with a remainder of 0.

Determining the Final Remainder

Now, let's consider the full expression $x^{2n}-y^{2n} + 1$. We can rewrite it as:

$x^{2n}-y^{2n} + 1 = (x^{2n}-y^{2n}) + 1$

Substituting the factored form from the previous step:

$x^{2n}-y^{2n} + 1 = (x^n - y^n)(x^n + y^n) + 1$

According to the division algorithm, Dividend = Divisor $\times$ Quotient + Remainder.

In this case:

Dividend is $x^{2n}-y^{2n} + 1$
Divisor is $x^n + y^n$
Quotient is $x^n - y^n$
Remainder is $1$

Therefore, the remainder is 1.

Evaluating the Statements

The result that the remainder is 1 holds true for any natural number $n$. Let's examine the given statements:

Statement-I: $n$ is odd. If $n$ is odd, the remainder is still 1.
Statement-II: $n$ is even. If $n$ is even, the remainder is also 1.

Since the remainder is 1 regardless of whether $n$ is odd or even, we don't need any specific information about $n$ to answer the question (as long as $n$ is a natural number). The algebraic simplification itself is sufficient.

Conclusion on Statement Sufficiency

The question can be answered using basic algebraic principles without relying on the specific conditions given in Statement-I or Statement-II. Hence, the question can be answered even without using any of the statements.

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