Let $f:[0,2] \rightarrow R$ be the function defined by
$f ( x )=(3-\sin (2 \pi x )) \sin \left(\pi x -\frac{\pi}{4}\right)-\sin \left(3 \pi x +\frac{\pi}{4}\right)$
If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$, then the value of $\beta-\alpha$ is. . . . . . . . .
$0$
$1$
$5$
$6$
If $0 \le x < 2\pi $ , then the number of real values of $x,$ which satisfy the equation $\cos x + \cos 2x + \cos 3x + \cos 4x = 0$ is . . .
If $\sin \,\theta + \sqrt 3 \cos \,\theta = 6x - {x^2} - 11,x \in R$ , $0 \le \theta \le 2\pi $ , then the equation has solution for
If equation in variable $\theta, 3 tan(\theta -\alpha) = tan(\theta + \alpha)$, (where $\alpha$ is constant) has no real solution, then $\alpha$ can be (wherever $tan(\theta - \alpha)$ & $tan(\theta + \alpha)$ both are defined)
If $5{\cos ^2}\theta + 7{\sin ^2}\theta - 6 = 0$, then the general value of $\theta $ is
If $\mathrm{n}$ is the number of solutions of the equation
$2 \cos x\left(4 \sin \left(\frac{\pi}{4}+x\right) \sin \left(\frac{\pi}{4}-x\right)-1\right)=1, x \in[0, \pi]$
and $S$ is the sum of all these solutions, then the ordered pair $(\mathrm{n}, \mathrm{S})$ is :