MCQ
If the vectors $4\hat{\text{i}}+11\hat{\text{j}}+\text{m}\hat{\text{k}},7\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}$ and $\hat{\text{i}}+5\hat{\text{j}}+4\hat{\text{k}}$ are coplanar, then $m =$
- A$0$
- B$38$
- C$-10$
- ✓$10$
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$(i)$ for $p \geqslant 0$ , $f(x) = 0$ has one negative root and $f(x)$ is monotonic
$(ii)$ for $-1 < p < 0$ , $f(x)$ = $0$ has one negative root and $f(x)$ is nonmonotonic
$(iii)$ for $p < 0$ , $f(x)$ = $0$ has three real and distinct roots.
$\left[\begin{array}{lll}a & b & c\end{array}\right]\left[\begin{array}{lll}1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0\end{array}\right]$................$(E)$
$1.$ 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 $3.$