MCQ
Choose the correct option from given four options$:\ \int\limits^{\frac{\pi}{4}}_{\frac{-\pi}{4}}\frac{\text{dx}}{1+\cos2\text{x}}$ is equal to:
- ✓$1$
- B$2$
- C$3$
- D$4$
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$\left[\begin{array}{lll}a & b & c\end{array}\right]\left[\begin{array}{lll}1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0\end{array}\right]$................$(E)$
$1.$ If the point $P(a, b, c)$, with reference to $( E )$, lies on the plane $2 x+y+z=1$, then the value of $7 a+b+c$ is
$(A)$ $0$ $(B)$ $12$ $(C)$ $7$ $(D)$ $6$
$2.$ Let $\omega$ be a solution of $x^3-1=0$ with $\operatorname{Im}(\omega)>0$. If $a=2$ with $b$ and $c$ satisfying $( E )$, then the value of $\frac{3}{\omega^a}+\frac{1}{\omega^b}+\frac{3}{\omega^c}$ is equal to
$(A)$ $-2$ $(B)$ $2$ $(C)$ $3$ $(D)$ $-3$
$3.$ Let $b=6$, with $a$ and $c$ satisfying (E). If $\alpha$ and $\beta$ are the roots of the quadratic equation $a x^2+b x+c=0$, then
$\sum_{n=0}^{\infty}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)^n$ is
$(A)$ $6$ $(B)$ $7$ $(C)$ $\frac{6}{7}$ $(D)$ $\infty$
Give the answer question $1,2$ and $3.$
$(A)$ $y\left(\frac{\pi}{4}\right)=\frac{\pi^2}{8 \sqrt{2}}$
$(B)$ $y^{\prime}\left(\frac{\pi}{4}\right)=\frac{\pi^2}{18}$
$(C)$ $y\left(\frac{\pi}{3}\right)=\frac{\pi^2}{9}$
$(D)$ $y ^{\prime}\left(\frac{\pi}{3}\right)=\frac{4 \pi}{3}+\frac{2 \pi^2}{3 \sqrt{3}}$