The equation $x^2 + k_1 y^2 + k_2 xy = 0$ represents a pair of perpendicular lines,if

The acute angle between the lines represented by $({x^2} + {y^2})\sqrt{3} = 4xy$ is

The angle between the lines represented by the equation $x^{2}-xy-6y^{2}-7x+31y-18=0$ is

If the angle between the lines given by the equation $x^{2}-3xy+\lambda y^{2}+3x-5y+2=0$,$\lambda \geq 0$,is $\tan^{-1}\left(\frac{1}{3}\right)$,then $\lambda=$

If the angle between the pair of straight lines represented by the equation ${x^2} - 3xy + \lambda {y^2} + 3x - 5y + 2 = 0$ is ${\tan ^{ - 1}}\left( {\frac{1}{3}} \right)$,where $\lambda$ is a non-negative real number,then $\lambda$ is:

Four different pairs of lines are given in List-$I$ and the cosine of the angle between every pair of lines is given in List-$II$. Match the following:

List-$I$	List-$II$
$(A)$ $5x^2 + 2\sqrt{7}xy - y^2 = 0$	$(I)$ $\frac{\sqrt{3}}{2}$
$(B)$ $x^2 + \sqrt{11}xy + 2y^2 = 0$	$(II)$ $\frac{1}{2\sqrt{3}}$
$(C)$ $x^2 + 2\sqrt{2}xy + y^2 = 0$	$(III)$ $\frac{1}{2}$
$(D)$ $3x^2 + 4\sqrt{2}xy + y^2 = 0$	$(IV)$ $\frac{2}{3}$
	$(V)$ $\frac{1}{\sqrt{2}}$

The correct match is: