The maximum value of the expression $\frac{1}{\sin^2 \theta + 3 \sin \theta \cos \theta + 5 \cos^2 \theta}$ is

Let $x, y, z \in [0, 1]$. Then the maximum value of $\sqrt{|x-y|} + \sqrt{|y-z|} + \sqrt{|z-x|}$ is

If the equation $\cos^4 \theta + \sin^4 \theta + \lambda = 0$ has real solutions for $\theta$,then $\lambda$ lies in the interval

If $y = 3 \sin x + 4 \cos x$,then find the maximum value of $y$.

$a, b, c$ are the sides of a scalene triangle $ABC$. If angles $\alpha, \beta, \gamma$ lie between $0$ and $\pi$ such that $\cos \alpha = \frac{a}{b+c}, \cos \beta = \frac{b}{c+a}$ and $\cos \gamma = \frac{c}{a+b}$,then $\tan^2 \frac{\alpha}{2} + \tan^2 \frac{\beta}{2} + \tan^2 \frac{\gamma}{2} =$

If $P+Q+R=\frac{\pi}{4}$,then $\cos \left(\frac{\pi}{8}-P\right)+\cos \left(\frac{\pi}{8}-Q\right)+\cos \left(\frac{\pi}{8}-R\right)=$