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JEE MATHEMATICS MCQ: TRIGONOMETRIC EQUATIONS

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The equation (cos⁔ pāˆ’1)x2+(cos⁔ p)x+sin⁔p=0\left( {\cos \,p - 1} \right){x^2} + \left( {\cos \,p} \right)x + \sin p = 0, where xx is a variable, has real roots. Then, the interval of pp may be any one of the following:
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Correct option: D
(0,ā€…ā€Šā€‰Ļ€)\left( {0,\;\,\pi } \right)
Option D ((0,ā€…ā€ŠĻ€)\left( {0,\;\pi } \right)) is correct because the condition for real roots of the quadratic equation is satisfied within this interval.

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The number of solutions of the equation sin⁔  xā€‰ā€‰ā€…ā€Šcos  3x=sin⁔  3x  cos⁔  5x\sin \,\,x\,\,\;cos\,\,3x = \sin \,\,3x\,\,\cos \,\,5x in [0,ā€…ā€Šā€‰Ļ€2]\left[ {0,\;\,\frac{\pi }{2}} \right] is The number of ordered pairs (x, y)\left( {x,\,y} \right) where x,  y∈[0,10]x,\,\,y \in \left[ {0,10} \right] satisfying (sin⁔2āˆ’sin⁔  x+12)2sec⁔2y  ⋅  ≤1\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 X\mathrm{X}-axis.

Statement II: If θ→π2\theta \rightarrow \frac{\pi}{2}, then tangent line is parallel to the Y-axis.
If sin⁔x+cos⁔x=72\sin x+\cos x=\frac{\sqrt{7}}{2}, where x∈[0,Ļ€4]x \in\left[0, \frac{\pi}{4}\right], then tan⁔x2\tan \frac{x}{2} is equal to The value of tan⁔π5+2tan⁔2Ļ€5+4cot⁔4Ļ€5\tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5} is If 0<x<Ļ€0 < x < \pi and cos⁔x+sin⁔x=12\cos x+\sin x=\frac{1}{2}, then tan⁔x\tan x is equal to
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