Let X= x: x=n3+2 n+1, n ∈ N and Y= x: x=3 n2+7, n ∈ N then

Question

Let X= x: x=n3+2 n+1, n ∈ N and Y= x: x=3 n2+7, n ∈ N then (A) X ∩ Y is a subset of x: x=3 n+5, n ∈ N  (B) X ∩ Y ⊆ x: x=n2+n+1, n ∈ N  (C) 34 ∈ X ∩ Y (D) none of these

Tardigrade Admin · Accepted Answer

If n3+2 n+1=3 n2+7 ⇒ n3-3 n2+2 n-6=0 ⇒ (n-3)(n2+2)=0 ⇒ n=3 text as n ∈ N text So, x=3 × 32+7=34 ∈ X ∩ Y In (a) and (b) x ≠ 34, for any n ∈ N.

Q. Let $X = {x : x = n^{3} + 2 n + 1, n \in N}$ and $Y = {x : x = 3 n^{2} + 7, n \in N}$ then

$X \cap Y$ is a subset of ${x : x = 3 n + 5, n \in N}$

$X \cap Y \subseteq {x : x = n^{2} + n + 1, n \in N}$

$34 \in X \cap Y$

none of these

If $n^{3} + 2 n + 1 = 3 n^{2} + 7$
$\Rightarrow n^{3} - 3 n^{2} + 2 n - 6 = 0$
$\Rightarrow (n - 3) (n^{2} + 2) = 0$
$\Rightarrow n = 3 as n \in N$
$So, x = 3 \times 3^{2} + 7 = 34 \in X \cap Y$
In (a) and (b) $x \neq = 34$ , for any $n \in N$ .

Q. Let X={x:x=n3+2n+1,n∈N} and Y={x:x=3n2+7,n∈N} then

X∩Y is a subset of {x:x=3n+5,n∈N}

X∩Y⊆{x:x=n2+n+1,n∈N}

34∈X∩Y