If _ - a ^a a - x a + x ,dx = k , then k =

$\left[\begin{array}{lll}a & b & c\end{array}\right]\left[\begin{array}{lll}1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0\end{array}\right]$................$(E)$

$1.$ 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 $3.$

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If the area of the region $\left\{(x, y): 0 \leq y \leq \min \left\{2 x, 6 x-x^2\right\}\right\}$ is $A$, then $12 A$ is equal to

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If $q_1$ , $q_2$ , $q_3$ are roots of the equation $x^3 + 64$ = $0$ , then the value of $\left| {\begin{array}{*{20}{c}}
{{q_1}}&{{q_2}}&{{q_3}} \\
{{q_2}}&{{q_3}}&{{q_1}} \\
{{q_3}}&{{q_1}}&{{q_2}}
\end{array}} \right|$ is

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A company has two plants $\mathrm{A}$ and $\mathrm{B}$ to manufacture motorcycles. $60 \%$ motorcycles are manufactured at plant $\mathrm{A}$ and the remaining are manufactured at plant B. $80 \%$ of the motorcycles manufactured at plant $\mathrm{A}$ are rated of the standard quality, while $90 \%$ of the motorcycles manufactured at plant $B$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the probability that it was manufactured at plant $\mathrm{B}$, then $126 \mathrm{p}$ is

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