Question

The number of positive integral values of ${}^{\prime }{a}^{\prime }$ for which there is no solution of the equation $acosx+cotx+1=cosecx,$ where $x\ne \frac{n\pi }{2},n\in Z,$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (D)

$sinx+cosx=1-asinxcosx$
$\Rightarrow a^{2}si{n}^{2}xco{s}^{2}x-2\left(a+1\right)sinxcosx=0$
$\Rightarrow \sin 2x\left(\frac{a^2}{2}\sin 2x-2(a+1)\right)=0$ either $\sin 2x=0$ 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, ${}^{\prime }{a}^{\prime }$ can't be equal to $1,2,3,4$