A) infinite number of real roots
B) no real roots
C) exactly one real root
D) exactly four real roots
Correct Answer: B
Solution :
Let \[{{e}^{\sin x}}=t\] \[\Rightarrow \,\,{{t}^{2}}-4t-1=0\] \[\Rightarrow \,\,t=\frac{4\pm \sqrt{16+4}}{2}\] \[\Rightarrow \,\,t={{e}^{\sin x}}=2\pm \sqrt{5}\] \[\Rightarrow \,\,\,{{e}^{\sin x}}=2-\sqrt{5}\], \[{{e}^{\sin x}}=2+\sqrt{5}\] \[{{e}^{\sin x}}=2-\sqrt{5}<0\]\[\Rightarrow \sin x=In(2+\sqrt{5})>1\] so rejected so rejected hence no solution
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