MCQ
The value of $0.\mathop {234}\limits^{\,\,\, \bullet \,\, \bullet } $ is
- ✓$\frac{{232}}{{990}}$
- B$\frac{{232}}{{9990}}$
- C$\frac{{232}}{{900}}$
- D$\frac{{232}}{{9909}}$
= $0.2 + 0.034 + 0.00034 + 0.0000034 +.............$
= $0.2 + \frac{{34}}{{1000}} + \frac{{34}}{{100000}} + \frac{{34}}{{10000000}} + .....\infty $
$ = \frac{2}{{10}} + 34\left[ {\frac{1}{{{{10}^3}}} + \frac{1}{{{{10}^5}}} + \frac{1}{{{{10}^7}}} + ........\infty } \right]$
$ = \frac{2}{{10}} + 34\left[ {\frac{{1/{{10}^3}}}{{1 - 1/1000}}} \right] = \frac{2}{{10}} + 34 \times \frac{1}{{1000}} \times \frac{{100}}{{99}}$
$ = \frac{2}{{10}} + \frac{{34}}{{990}} = \frac{{232}}{{990}}$.
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$x+y+\alpha z=2$
$3 x+y+z=4$
$x+2 z=1$
have a unique solution $\left(x^{*}, y^{*}, z^{*}\right)$. If $\left(\alpha, x^{*}\right),\left(y^{*}, \alpha\right)$ and $\left(x^{*},-y^{*}\right)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is