Question

Let $$P = \left\{ {\theta :\sin \theta - \cos \theta = \sqrt 2 \cos \theta } \right\}$$       and $$Q = \left\{ {\theta :\sin \theta + \cos \theta = \sqrt 2 \cos \theta } \right\}$$       be two sets. Then

A. $$P \subset Q\,\,{\text{and }}Q - P \ne \phi $$
B. $$Q \not\subset P\,$$
C. $$P \not\subset Q\,$$
D. $$P = Q$$  
Answer :   $$P = Q$$
Solution :
$$\eqalign{ & P = \left\{ {\theta :\sin \theta - \cos \theta = \sqrt 2 \cos \theta } \right\} \cr & \sin \theta = \left( {\sqrt 2 + 1} \right)\cos \theta ,\tan \theta = \sqrt 2 + 1 \cr & Q = \left\{ {\theta :\sin \theta + \cos \theta = \sqrt 2 \cos \theta } \right\} \cr & \cos \theta = \left( {\sqrt 2 - 1} \right)\sin \theta \,\,{\text{or}}\tan \theta = \sqrt 2 + 1 \cr & \therefore \,\,P = Q \cr} $$

Releted MCQ Question on
Calculus >> Sets and Relations

Releted Question 1

If $$X$$ and $$Y$$ are two sets, then $$X \cap {\left( {X \cup Y} \right)^c}$$   equals.

A. $$X$$
B. $$Y$$
C. $$\phi $$
D. None of these
View Answer
Releted Question 2

The expression $$\frac{{12}}{{3 + \sqrt 5 + 2\sqrt 2 }}$$    is equal to

A. $$1 - \sqrt 5 + \sqrt 2 + \sqrt {10} $$
B. $$1 + \sqrt 5 + \sqrt 2 - \sqrt {10} $$
C. $$1 + \sqrt 5 - \sqrt 2 + \sqrt {10} $$
D. $$1 - \sqrt 5 - \sqrt 2 + \sqrt {10} $$
View Answer
Releted Question 3

If $${x_1},{x_2},.....,{x_n}$$    are any real numbers and $$n$$ is any positive integer, then

A. $$n\sum\limits_{i = 1}^n {{x_i}^2 < {{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^2}} $$
B. $$\sum\limits_{i = 1}^n {{x_i}^2 \geqslant {{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^2}} $$
C. $$\sum\limits_{i = 1}^n {{x_i}^2 \geqslant n{{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^2}} $$
D. none of these
View Answer
Releted Question 4

Let $$S$$ = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of $$S$$ is equal to

A. 25
B. 34
C. 42
D. 41
View Answer

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Sets and Relations

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