First slide

Operations on Sets

Question

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

Moderate

A

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

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

$34 \in X \cap Y$
D

none of these

Solution

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

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