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Question

If $$3\cos x +2\cos 3\mathrm{x}= \cos y,3 \sin x +2\sin 3\mathrm{x}= \sin y$$, then the value of $$\cos 2\mathrm{x}$$ is


A
1
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B
1
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C
0
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D
12
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Solution

The correct option is A $$-1$$
$$\Rightarrow3cosx+2cos3x=cosy$$

Squaring

$$\Rightarrow9{ cos }^{ 2 }x+4{ cos }^{ 2 }3x+12cosxcos3x={ cos }^{ 2 }y$$   ...(1)

$$\Rightarrow3sinx+2sin3x=siny$$

Squaring

$$\Rightarrow9{ sin }^{ 2 }x+4{ sin }^{ 2 }3x+12sinxsin3x={ sin }^{ 2 }y$$   ...(2)

Adding (1) and (2)

$$\Rightarrow9({ cos }^{ 2 }x+{ sin }^{ 2 }x)+4({ cos }^{ 2 }3x+{ sin }^{ 2 }3x)+12(cosxcos3x+sinxsin3x)={ cos }^{ 2 }y+{ sin }^{ 2 }y$$

$$\Rightarrow9+4+12cos(3x-x)=1$$

$$\Rightarrow 12cos(2x)=-12$$

$$\Rightarrow cos2x=-1$$


Mathematics

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