UPSC NDA Math Model Paper 6

Option A : $\{ x: x \in N$ and $x^{2} < 36\}$ As we can see that, elements of $\{ x: x \in N$ and $x^{2} < 36\}$ are: $1,2,3,4,5$ So, there are 5 elements in the set $\{ x: x \in N$ and $x^{2} < 36\}$ Hence, the set $\{ x: x \in N$ and $x^{2} < 36\}$ is not a singleton set. Option B: $\{ x \in z: 0 < x < 10 \}$ As we can see that, elements of {x ∈ z: 0 < x < 10} are: 1, 2, 3, ........, 9 So, there are 9 elements in the set {x ∈ z: 0 < x < 10} Hence, the set {x ∈ z: 0 < x < 10} is not a singleton set OptionC: $\{ x: x \in N$ and $x^{2}=x\}$ ∵ $x^2 = x$ $\Rightarrow x^2 - x = 0$ ⇒ x (x - 1) = 0 ⇒ x = 0 or 1 But since x ∈ N. So, x = 0 ∉ {x: x ∈ N and $x^2$ = x} So, only x = 1 ∈ {x: x ∈ N and $x^2$ = x} Hence, {x: x ∈ N and $x^2$ = x} is a singleton set Option D: {x ∈ N: x is even} As we can see that, elements of {x ∈ N: x is even} are: 2, 4, 6, 8, ........ So, there are infinitely many elements in the set {x ∈ N: x is even} Hence, {x ∈ N: x is even} is not a singleton set