MCQ
$\mathop \smallint \limits_{\frac{\pi }{4}}^{\frac{{3\pi }}{4}} \frac{{dx}}{{1 + \cos x}} = $ . . . .
- A$-1$
- B$-2$
- ✓$2$
- D$4$
$I = \int\limits_{\frac{{3\pi }}{4}}^{\frac{{3\pi }}{4}} {\frac{{{\rm{dx}}}}{{1 - \cos x}}} $
Using$\int\limits_a^b {f\left( x \right)dx = \int\limits_a^b {f\left( {a + b - x} \right)} } dx$
Adding $(i)$ and $(ii)$
$2I = \int\limits_{\frac{\pi }{4}}^{\frac{{3\pi }}{4}} {\frac{2}{{{{\sin }^2}x}}dx} $
$I = \int\limits_{\frac{\pi }{4}}^{\frac{{3\pi }}{4}} {{{\csc }^2}xdx} $
${\rm{I}} = - (\cot x)_{\pi /4}^{3\pi /4}$
$ = - \left[ {\cot \frac{{3\pi }}{4} - \cot \frac{\pi }{4}} \right] = 2$
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Then for the objective function $z=-x+2 y$
$(i)$ Maximum value of $z$ has at $\ldots \ldots \ldots . . .$
$(ii)$ Minimum value of $z$ has at $\ldots \ldots \ldots . . .$
$(iii)$ The maximum value of $z$ is $\ldots \ldots \ldots . . .$
$(iv)$ The minimum value of $z$ is $\ldots \ldots \ldots . . .$
$STATEMENT -2$ : A parabola is symmetric about its axis.