NIMCET 2021 Question Paper with solutions

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Question : 8 of 120

Marks: +1, -0

If

|{k}|=5,0°≤ \θ \≤ 360°

, then total number of different solutions of

3 \cos \θ+

4 \sin \θ=k

is

0
1
2
infinite

Solution:

Given equation

3 \cos \θ+4 \sin \θ=k

Also

(k)=5 \⇒k=\±5

3 \cos \θ+4 \sin \θ=\±5

{3}/{5} \cos \θ+{4}/{5} \sin \θ=\± 1 \......

(1)
Let

{3}/{5}=r \sin \α

and

{4}/{5}=r \cos \α

Where

r=1

and

=\α=\tan ^{-1}({3}/{4})

By eq. (1)

{\Sin} \α\cos \θ+\cos \α\sin \θ=\±1

{\Sin}(\θ+\α)=\±1

Since

0° < \θ < 360°

Hence two solution exists.

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