First slide

Operations on Sets

Question

Let P = $P = {θ : s i n θ - c o s θ = \sqrt{2} c o s θ} a n d {Q = θ : s i n θ + c o s θ = \sqrt{2} s i n θ}$ be two sets, then

Moderate

A

$P \subset Q a n d Q ~ P \neq ϕ$
B

$Q ⊄ P$
C

$P ⊄ Q$
D

P =Q

Solution

$\sin θ - \cos θ = \sqrt{2} \cos θ$
$\Leftrightarrow \sin θ = (\sqrt{2 + 1}) \cos θ$
$\Leftrightarrow \cos θ = (\sqrt{2 - 1}) \sin θ$
$\Leftrightarrow \sin θ + \cos θ = \sqrt{2} \sin θ$
$\Rightarrow P = Q$

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