Question

If $X+Y+Z=0$ , $|X|=|Y|=|Z|=2$ where $\theta$ is the angle between $Y$ and $Z$ then the value of $2cose{c}^2\theta +3co{t}^2\theta$ is:

• A. $\frac{11}{3}$
• B. $\frac{8}{3}$
• C. $\frac{7}{3}$
• D. None of these

Answer 1

The correct option is A $\frac{11}{3}$

Given $X+Y+Z=0$

$\Rightarrow Y+Z=-X$

By applying modulus , we get $|Y+Z|=|X|$

By squaring on both sides , we get $(|Y+Z|{)}^2=|X{|}^2$

$\Rightarrow |Y{|}^2+|Z{|}^2+2|Y||Z|cos\theta =|X{|}^2$

$\Rightarrow cos\theta =-\frac{1}{2}$

$\Rightarrow {\cos }^2\theta =\frac{1}{4}$

So we get ${\sin }^2\theta =\frac{3}{4}$

$\Rightarrow 2{\csc }^2\theta +3{\cot }^2\theta =\frac{2+3{\cos }^2\theta }{{\sin }^2\theta }=\frac{11}{3}$

Therefore option $A$ is correct