JEE MATHEMATICS MCQ: TRIGONOMETRIC EQUATIONS

1 Easy

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Analyzing +4 | -1

The number of solutions of

2\,\,{\cos ^2}\left( {\frac{x}{2}} \right){\sin ^2}\,\,x = {x^2} + \frac{1}{{{x^2}}},\,\,0 \le \,\,\,x\,\,\, \le \frac{\pi }{2}

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Correct option: A

Option A (0) is correct because the left-hand side (LHS) of the equation is less than 2, while the right-hand side (RHS) is greater than or equal to 2, implying no solution.

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The number of solutions of the equation

\sin \,\,x\,\,\;cos\,\,3x = \sin \,\,3x\,\,\cos \,\,5x

\left[ {0,\;\,\frac{\pi }{2}} \right]

is The number of ordered pairs

\left( {x,\,y} \right)

where

x,\,\,y \in \left[ {0,10} \right]

satisfying

\left( {\sqrt {{{\sin }^2} - \sin \,\,x + \frac{1}{2}} } \right){2^{{{\sec }^2}y}}\,\, \cdot \,\, \le 1

is Statement I: If slope of the tangent line is zero, then tangent line is perpendicular to the

\mathrm{X}

-axis.

Statement II: If

\theta \rightarrow \frac{\pi}{2}

, then tangent line is parallel to the Y-axis.
If

\sin x+\cos x=\frac{\sqrt{7}}{2}

, where

x \in\left[0, \frac{\pi}{4}\right]

, then

\tan \frac{x}{2}

is equal to The value of

\tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5}

is If

0 < x < \pi

and

\cos x+\sin x=\frac{1}{2}

, then

\tan x

is equal to