EgmatQuantExpert wrote:
Detailed Solution
Step-I: Given Info:
We are given that \(a\), \(b\), \(n >0\) and are integers. Also \(n =3a – b^3\) and we are asked to find if \((n^2 + 3)\) is divisible by 2.
Step-II: Interpreting the Question Statement:
Let’s start from our expression i.e. \((n^2 + 3)\), this expression is divisible by 2 only if it’s even, since 3 is odd, for \(n^2 + 3\) to be even \(n^2\) has to be odd ( as odd + odd =even) and \(n^2\) can be odd only when \(n\) is odd.
Now, we know that \(n = 3a – b^3\) , for \(n\) to be odd, one of the \(3a\) or \(b^3\) has to be odd and other has to be even as the difference of an even and an odd number will always be odd. The even/odd nature of \(3a\) would depend on the even/odd nature of \(a\) and similarly the even odd nature of \(b^3\) would depend on the even/odd nature of \(b\). So, if we can establish that the even/odd nature of \(a\) and \(b\) are either similar or opposite, we will find our answer.
Step-III: Statement-I
Statement-I tells us that \(a^2 – 4b^3 = 5\), it tells us that difference of two numbers is odd. Since \(4b^3\) would always be even, for the difference of \(a^2\) and \(4b^3\) to be odd, \(a^2\) would have to be odd. For \(a^2\) to be odd, \(a\) has to be odd. But St-I does not tell us anything about the even/odd nature of \(b\).
So, Statement-I alone is insufficient.
Step-IV: Statement-II
Statement-II tells us that \(a^2 - 3b^3= 6\), it tells us that difference of two numbers is even. This is only possible in two cases:
a) When both \(a^2\) and \(3b^3\) are odd, for this to happen both \(a\) and \(b\) have to be odd or
b) When both \(a^2\) and \(3b^3\) are even, for this to happen both \(a\) and \(b\) have to be even
But, we know that for n to be odd, both \(a\) and \(b\) have to of opposite even/odd natures. We see that in St-II, in both the cases \(a\) and \(b\) are of the same nature, thus in both the cases, \(n\) would be even.
Hence, Statement-II is sufficient to answer our question.
Step-V: Combining Statements I & II
Since, we have a unique answer from Statement-II alone, we don’t need to combine the information from Statement-I and II.
Thus, the answer is Option B.
Key Takeaways
1. In even-odd questions, simplify complex expressions into simpler expressions using the properties of even-odd combinations.
2. Know the properties of even/odd combinations to save the time spent deriving them in the test
3. The even/odd nature of some expressions can be determined without knowing the exact even/odd nature of the variables of the expressions by using the even/odd combination property
Regards
Harsh
I read the expression a^2-4*b^3-5=0 as (a^2-4)*(b^3-5)=0
By this, a=2 and b^3=5. Subst. this is n=3a-b^3 which makes n= (3*2)-5=1
thus n^2+3 becomes, 4 which is divisible by 2!!!
thus making each statement alone sufficient (opt. D) as the answer!!
Pl use proper parenthesis..No parenthesis are need there.
Mathematically \(a^2 - 4∗b^3-5 = 0\) can ONLY mean \((a^2) -(4∗b^3)-(5) = 0\) and nothing else.