tmua.me Tau Paper 1 · Question 12

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tmua.me Tau Paper 1 · Q12

Paper 1TrigonometryFunctions & graphs

For $a>1$ and real $x$ , define

F_a(x)=\frac{\sin x}{a+\sin x}+\frac{\cos x}{a+\cos x}, \qquad H_a(x)=F_a(x)F_a(x+\pi).

For which values of $a$ is the minimum of $H_a(x)$ on $0\le x<2\pi$ attained at exactly four distinct values of $x$ ?

1<a<\sqrt{\dfrac32}

a>\sqrt{\dfrac52}

1<a<\sqrt{\dfrac52}

1<a\le\sqrt{\dfrac52}

\sqrt{\dfrac32}<a<\sqrt{\dfrac52}

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