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question_answer1) If \[5\cos 2\theta +2{{\cos }^{2}}\frac{\theta }{2}+1=0,-\pi <\theta <\pi \] then find number of solutions.
question_answer2) Find the number of solutions of the equation \[{{\sin }^{4}}x=1+{{\tan }^{4}}x\] in \[\left( 0,4\pi \right).\]
question_answer3) Find the number of solutions of the equation \[{{\sin }^{2}}x+3\operatorname{sinx}+2=0\] in the interval \[\left( -\pi ,\pi \right)\].
question_answer4) Find the total number of solutions of \[{{16}^{{{\sin }^{2}}x}}+{{16}^{{{\cos }^{2}}x}}=10\] is \[x\in \left[ 0,\,\,3\pi \right]\].
question_answer5) If \[{{\sin }^{2}}x-\cos x=1/4,\] if the sum of values of x between 0 and \[2\pi \] is \[k\pi \] then find k.
question_answer6) If \[\exp \left[ \left( {{\sin }^{2}}x+{{\sin }^{4}}x+{{\sin }^{6}}x+.... \right)ln\,\,2 \right]\] satisfies the equation \[{{y}^{2}}-9y+8=0,\] and the value of \[\frac{\cos x}{\cos x+\sin x},0<x<\frac{\pi }{2},\]is \[\frac{a-1}{2}\] then find a.
question_answer7) Find the number of integral values of k for which the equation \[7\text{ }cosx\text{ }+\text{ }5sin\text{ }x=2k+1\] has a solution.
question_answer8) Find the total number of solutions to \[\cot x|=\cot x+\frac{1}{\sin x},x\in \left[ 0,3\pi \right].\]
question_answer9) If \[\sin \,\,\theta =\frac{1}{2}\] and \[\cos \theta =-\frac{\sqrt{3}}{2}\] and the general value of \[\theta \] is \[2n\pi +\frac{k\pi }{\lambda }\left( n\in Z \right)\] then find \[k+\lambda .\]
question_answer10) If \[{{\sin }^{4}}x-{{\cos }^{2}}x\,\,\sin x+2{{\sin }^{2}}x+\sin x=0,\] \[\forall \,\,x\,\,\in \,\,\left[ 0,3\pi \right]\] then find number of solutions.
question_answer11) If the most general solution of \[{{2}^{1+\left| \cos x \right|+{{\cos }^{2}}x+{{\left| \cos x \right|}^{3}}+.....}}^{\infty }=4,\] is \[n\,\pi \pm \frac{\pi }{k}\] then find k.
question_answer12) The values of x between 0 and \[2\pi \] which satisfy the equation \[\sin x\sqrt{8{{\cos }^{2}}x}=1\] are in AP. If the common difference of the AP is \[\frac{\pi }{k}\] then find k.
question_answer13) Find the total number of solutions to \[\tan x+\cot x=2\text{cosec}x\] in \[\left[ -2\pi ,\,\,2\pi \right].\]
question_answer14) If general solution of \[\cot \theta -\tan \theta =\sec \theta \] is \[n\pi +{{\left( -1 \right)}^{n}}\frac{\pi }{k},\,\,n\in \,Z\] then find k.
question_answer15) If general solution of \[7\,\,{{\cos }^{2}}\theta +3\,{{\sin }^{2}}\theta =4\] is \[n\pi \,\,\pm \,\,\frac{\pi }{3},\,\,n\in \,Z\] then find k.
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