List I		List II
1	Range of  $2{\cos }^{2}x+{\sin }^{2}x$	a	$\left[1,3\right]$
2	Range of $1+8{\sin }^2{x}^2{\cos }^2{x}^2$	b	$\left[\frac{3}{4},1\right]$
3	Range of  $c{\mathrm{os}}^{2}x+{\sin }^{4}x$	c	$\left[3,4\right]$
4	Range of  $3{\sin }^{2}x+4{\cos }^{2}x$	d	$\left[1,2\right]$

$2{\cos }^{2}x+{\sin }^{2}x=1+{\cos }^{2}x\in \left[1,2\right]$

$1+8{\sin }^2{x}^2{\cos }^2{x}^2=1+2{\sin }^2\left(2x^2\right)\in \left[1,3\right]$

$$\begin{gathered} c{\mathrm{os}}^{2}x+{\sin }^{4}x={\cos }^{2}x+{\sin }^{2}x(1-{\cos }^{2}x) \\ ={\cos }^{2}x+{\sin }^{2}x-{\sin }^{2}x{\cos }^{2}x=1-\frac{1}{4}{\sin }^{2}2x\in \left[\frac{3}{4},1\right] \end{gathered}$$

 $3{\sin }^{2}x+4{\cos }^{2}x=3+{\cos }^{2}x\in \left[3,4\right]$