Solution:
Step 1: Analyse Statement 1:\(a^2 +b^2 + c^2\) is Even
• If \(a^2 +b^2 + c^2\) is Even, it gives rise to two possibilities:
o Either one of \(a/b/c\) is Even and the rest two are Odd
o Or, all the three numbers: \(a, b\) and \(c\)are Even.
Per our conceptual knowledge, let’s make a table to understand this.
It is clear from the table that the even-odd nature of \((a + c)\)varies, and we cannot conclude anything about it.
As we do not know the exact even-odd nature of\((a + c)\),
Statement 1 alone is NOT sufficient to answer the question.
Hence, we can eliminate answer choices A and D.
Step 2: Analyse Statement 2:\(b^2 +c^2 + d^2\) is Even
• If \(b^2 +c^2 + d^2\) is Even, it gives rise to two possibilities:
o Either one of \(b/c/d\) is Even and rest two are Odd
o Or, all the three numbers: \(b, c\) and \(d\) are Even.
Let’s make a table to understand this.
It is clear from the table that the even-odd nature of \((a + c)\) varies, and we cannot conclude anything about it.
As we do not know the exact even-odd nature of \((a + c)\),
Statement 2 alone is NOT sufficient to answer the question.
Hence, we can eliminate answer choice B.
Step 3: Combine both Statements:From the first statement, we have: \(a^2 +b^2 + c^2\) = Even
From the second statement, we have: \(b^2 +c^2 + d^2\) = Even
Combining both the statements, we have: a\(^2 +b^2 + c^2 + b^2 +c^2 + d^2\) = Even + Even = Even
Therefore, \(a^2 +2b^2 + 2c^2 + d^2\) is even
We know that, \(2b^2 + 2c^2\) is always even irrespective of the even-odd nature of \(b\) and \(c\), since they are multiplied by an even number \((2)\)
So, our expression reduces to: \(a^2 + d^2\) is even. This will lead to two possibilities:
• Either both a and d are Odd.
• Or both a and d are Even.
Per our conceptual understanding, let us draw a table combining both the tables from Step 3 and Step 4 to see how will the even odd nature of the numbers affect the even-odd nature of the expression: \((a + c)\):
From the table it is evident that the even-odd nature of the expression \((a + c)\) cannot be determined uniquely.
By combining both statements we did not get a unique answer.
Correct Answer: Option E
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