MCQ
The value of $3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\ldots \infty}}}}$ is equal to
- ✓$1.5+\sqrt{3}$
- B$2+\sqrt{3}$
- C$3+2 \sqrt{3}$
- D$4+\sqrt{3}$
So, $x=3+\frac{1}{4+\frac{1}{x}}=3+\frac{1}{\frac{4 x+1}{x}}$
$\Rightarrow(x-3)=\frac{x}{(4 x+1)}$
$\Rightarrow(4 x+1)(x-3)=x$
$\Rightarrow 4 x^{2}-12 x+x-3=x$
$\Rightarrow 4 x^{2}-12 x-3=0$
$x=\frac{12 \pm \sqrt{(12)^{2}+12 \times 4}}{2 \times 4}=\frac{12 \pm \sqrt{12(16)}}{8}$
$=\frac{12 \pm 4 \times 2 \sqrt{3}}{8}=\frac{3 \pm 2 \sqrt{3}}{2}$
$x=\frac{3}{2} \pm \sqrt{3}=1.5 \pm \sqrt{3}$
But only positive value is accepted
So, $x=1.5+\sqrt{3}$
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$(A)$ $\arg (-1- i )=\frac{\pi}{4}$, where $i =\sqrt{-1}$
$(B)$ The function $f: R \rightarrow(-\pi, \pi]$, defined by $f(t)=\arg (-1+i t)$ for all $t \in R$, is continuous at all points of $R$, where $i=\sqrt{-1}$
$(C)$ For any two non-zero complex numbers $z_1$ and $z_2$, $\arg \left(\left(\frac{z_1}{z_2}\right)-\arg \left(z_1\right)+\arg \left(z_2\right)\right.$ is an integer multiple of $2 \pi$.
$(D)$ For any three given distinct complex numbers, $z_1, z_2$ and $z_3$, the locus of the point $z$ satisfying the condition $\arg \left(\frac{\left( z - z _1\right)\left( z _2- z _3\right)}{\left( z - z _3\right)\left( z _2- z _1\right)}\right)=\pi$, lies on a straight line
Statement $1:$ If arg $Z+$ arg $W = \pi ,$ then $Z = -\overline W $.
Statement $2:$ $\left| Z \right| = \left| W \right|,$ implies arg $Z-$ arg $\overline W = \pi .$