Q41 / 160 · AP EAMCET 2024 (20 May Shift 1) · Mathematics

Let $\mathrm{f}(\mathrm{x})=3+2 \mathrm{x}$ and $\mathrm{gn}(\mathrm{x})=($ fo fo fo..... n times $)(\mathrm{x}) . \forall n \in \mathrm{~N}$ if all the lines $y=g_{\mathrm{n}}(x)$ pass through a fixed point $(\alpha, \beta)$, then $\alpha+\beta=$

  • A
    $-5$
  • B
    $-4$
  • C
    $-3$
  • Correct answer
    D
    $-6$

Solution

$f(x)=3+2 x \Rightarrow g_1(x)=3+2 x$ $g_2(x)=f \circ f(x)=9+4 x$ $\mathrm{g}_3(x)=f \circ f \circ f(x)=21+8 x$ $g_n(x)=($ fofof $\ldots . . n$ times $)(x)=3\left(2^n-1\right)+2^n x$ For fixed point $x=y$ $\Rightarrow x=3\left(2^{\mathrm{n}}-1\right)+2^{\mathrm{n}} x \Rightarrow x=-3$ and $y=3\left(2^{\mathrm{n}}-1\right)+2^{\mathrm{n}} x \Rightarrow y=-3$ So, $(-3,-3)$ is a fixed point $\forall n \in N$ $\Rightarrow \alpha+\beta=-3-3=-6$

Hence, the correct answer is option: 'D'.

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