Question

178. The range of f(theta) = 3cos²theta - 8 root 3 cos theta.sin theta + 5sin²theta - 7 is given by
i) [-7 , 7]
ii)[-10 , 4]
iii)[-4 , 4]
iv)[-10 , 7]

Answer 1

$$\begin{gathered} \mathrm{Dear}\mathrm{Student}, \\ \mathrm{f}\left(\mathrm{\theta }\right)=3\left[{\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }\right]+2{\sin }^2\mathrm{\theta }-8\sqrt{3}\mathrm{sin\theta cos\theta }-7 \\ =3-7+2{\sin }^2\mathrm{\theta }-8\sqrt{3}\mathrm{sin\theta cos\theta } \\ =-4+2{\sin }^2\mathrm{\theta }-8\sqrt{3}\mathrm{sin\theta cos\theta } \\ =-2-2\left(1-{\sin }^2\mathrm{\theta }\right)-4\sqrt{3}\sin 2\mathrm{\theta } \\ =-2-2{\cos }^2\mathrm{\theta }-4\sqrt{3}\sin 2\mathrm{\theta } \\ =-3+1-2{\cos }^2\mathrm{\theta }-4\sqrt{3}\sin 2\mathrm{\theta } \\ =-3-\cos 2\mathrm{\theta }-4\sqrt{3}\sin 2\mathrm{\theta } \\ =-3-\left(\cos 2\mathrm{\theta }+4\sqrt{3}\sin 2\mathrm{\theta }\right) \\ \mathrm{Now}\mathrm{acosx}+\mathrm{bsinx}\mathrm{lies}\mathrm{from}-\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}\mathrm{to}\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2} \\ \mathrm{so}\mathrm{here}\mathrm{for}\cos 2\mathrm{\theta }+4\sqrt{3}\sin 2\mathrm{\theta },\mathrm{a}=1,\mathrm{b}=4\sqrt{3} \\ \mathrm{so}\mathrm{it}\mathrm{will}\mathrm{lie}\mathrm{from}-\sqrt{1+48}\mathrm{to}\sqrt{1+48} \\ =-7\mathrm{to}7 \\ \mathrm{so}\mathrm{putting}\mathrm{in}\mathrm{equation}1 \\ \mathrm{f}\left(\mathrm{\theta }\right)\mathrm{will}\mathrm{lie}\mathrm{from}-3-7\mathrm{to}-3+7 \\ =\left[-10,4\right] \end{gathered}$$