What is the value of θ+θ? I. θ+θ=5 II. 2θ=1/2

Correct Answer is : if the statement II alone is sufficient to answer the question.

if the statement 1 alone is sufficient to answer the question.

if the statement II alone is sufficient to answer the question.

if both the statements I and II are sufficient to answer the question but neither statement alone is sufficient.

if both the statements I and II together are not sufficient to answer the question and additional data is required.

Correct Answer is : if the statement II alone is sufficient to answer the question.

Test Index

AP ICET 2016 Paper

Question : 11 of 200

Marks: +1, -0

What is the value of

\cos\theta+\sin\theta

\sec\theta+\csc\theta=5

\sin 2\theta=\frac{1}{2}

if the statement 1 alone is sufficient to answer the question.
if the statement II alone is sufficient to answer the question.
if both the statements I and II are sufficient to answer the question but neither statement alone is sufficient.
if both the statements I and II together are not sufficient to answer the question and additional data is required.

Solution:

I Given :

\sec\theta+\csc\theta=5

\Rightarrow \frac{1}{\cos\theta}+\frac{1}{\sin\theta}=5

\Rightarrow \sin\theta+\cos\theta=5\sin\theta\cos\theta

\Rightarrow \sin\theta+\cos\theta=\frac{5}{2}\sin 2\theta

Since sin 2θ or sinθ or cosθ is not given, we can’t determine sinθ + cosθ.

∴ I alone is not sufficient

II. Given :

\sin 2\theta=\frac{1}{2}

⇒ 2θ = 30° or 150°[Sine is positive in

Q_I

and

Q_{II}

]

⇒ θ = 15° or 75°

\sin 15^{\circ}+\cos 15^{\circ}=\sin 15^{\circ}+\sin 75^{\circ}

Also,

\sin 75^{\circ}+\cos 75^{\circ}=\sin 75^{\circ}+\sin 15^{\circ}

Hence,

\sin\theta+\cos\theta

can be uniquely determined.

∴ II alone is sufficient.

Aliter :

\sin\theta+\cos\theta=\sqrt{\sin\theta+\cos\theta}^{2}

=\sqrt{1+2\sin\theta\cos\theta}=\sqrt{1+\sin 2\theta}=\sqrt{1+\frac{1}{2}}

\sqrt{\frac{3}{2}}

we discard –

\frac{\sqrt{3}}{2}

as θ lies in the first quadrant.

∴ II alone is sufficient.

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