If A - x- x - 2k- k - and k - 100 - B - x- x - 2k- k - and k 11 and C - x- x - k3- k - and k 11 - then the value of nA B-nB C-nA C is

A = {x: x = 2k, k ∈ℕ and k ≤ 100 } ⇒ A = {2, 4, 6, 8, …, 200} ⇒ n(A) = 100 B = {x: x = 2^k, k ∈𝕎 and k 11} nbsp; ⇒ B = {2^0, 2^1, 2^2, 2^3, …, 2^10} ⇒ n(B) = ...

286

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Byju's Answer

Standard XI

Mathematics

Intersection

If A = x: x =...

Question

If $A = {x : x = 2 k, k \in N and k \leq 100}$ , $B = {x : x = 2^{k}, k \in W and k < 11}$ and $C = {x : x = k^{3}, k \in N and k < 11}$ , then the value of $n (A △ B) + n (B △ C) + n (A △ C)$ is

220

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216

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203

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207

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Solution

The correct option is B $216$
$A = {x : x = 2 k, k \in N and k \leq 100} \Rightarrow A = {2, 4, 6, 8, \dots, 200} \Rightarrow n (A) = 100$
$B = {x : x = 2^{k}, k \in W and k < 11}$
$\Rightarrow B = {2^{0}, 2^{1}, 2^{2}, 2^{3}, \dots, 2^{10}} \Rightarrow n (B) = 11$
$C = {x : x = k^{3}, k \in N and k < 11} \Rightarrow C = {1^{3}, 2^{3}, 3^{3}, \dots, 10^{3}} \Rightarrow n (C) = 10$

Now, $A \cap B = {2^{1}, 2^{2}, 2^{3}, 2^{4}, 2^{5}, 2^{6}, 2^{7}}$
$\Rightarrow n (A \cap B) = 7 \Rightarrow n (A △ B) = n (A) + n (B) - 2 n (A \cap B) \Rightarrow n (A △ B) = 100 + 11 - 14 \Rightarrow n (A △ B) = 97 \dots (1)$

$B \cap C = {1, 2^{3}, 4^{3}, 8^{3}} \Rightarrow n (B \cap C) = 4 \Rightarrow n (B △ C) = n (B) + n (C) - 2 n (B \cap C) \Rightarrow n (B △ C) = 11 + 10 - 8 \Rightarrow n (B △ C) = 13 \dots (2)$

$A \cap C = {2^{3}, 4^{3}} \Rightarrow n (A \cap C) = 2 \Rightarrow n (A △ C) = n (A) + n (C) - 2 n (A \cap C) \Rightarrow n (A △ C) = 100 + 10 - 4 \Rightarrow n (A △ C) = 106 \dots (3)$

From equations $(1), (2)$ and $(3)$ ,
$n (A △ B) + n (B △ C) + n (A △ C) = 97 + 13 + 106 = 216$

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If A={x:x=2k,k∈N and k≤100}, B={x:x=2k,k∈W and k<11} and C={x:x=k3,k∈N and k<11}, then the value of n(A△B)+n(B△C)+n(A△C) is

If $A = {x : x = 2 k, k \in N and k \leq 100}$ , $B = {x : x = 2^{k}, k \in W and k < 11}$ and $C = {x : x = k^{3}, k \in N and k < 11}$ , then the value of $n (A △ B) + n (B △ C) + n (A △ C)$ is