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Q.
The number of positive integral values of $'a'$ for which there is no solution of the equation $acosx+cotx+1=cosecx,$ where $x\neq \frac{n \pi }{2},n\in Z,$ is
NTA AbhyasNTA Abhyas 2022
Solution:
$sinx+cosx=1-asinxcosx$
$\Rightarrow a^{2}sin^{2}xcos^{2}x-2\left(a + 1\right)sinxcosx=0$
$\Rightarrow \sin 2 x\left(\frac{a^2}{2} \sin 2 x-2(a+1)\right)=0$ either $\sin 2 x=0$ or
$sin2x=\frac{4 \left(a + 1\right)}{a^{2}}$
For no solution $\left|\frac{4 \left(a + 1\right)}{a^{2}}\right|>1$ $\Rightarrow a\in \left(2 - 2 \sqrt{2} , 2 + 2 \sqrt{2}\right).$
Hence, $'a'$ can't be equal to $1,2,3,4$