csat-suneung 2022 Q9

csat-suneung · South-Korea · csat__math 4 marks Exponential Equations & Modelling Geometric Properties of Exponential/Logarithmic Curves

The line $y = 2 x + k$ meets the graphs of the two functions $$y = \left( \frac { 2 } { 3 } \right) ^ { x + 3 } + 1 , \quad y = \left( \frac { 2 } { 3 } \right) ^ { x + 1 } + \frac { 8 } { 3 }$$ at points $\mathrm { P }$ and $\mathrm { Q }$ respectively. When $\overline { \mathrm { PQ } } = \sqrt { 5 }$, what is the value of the constant $k$? [4 points]
(1) $\frac { 31 } { 6 }$
(2) $\frac { 16 } { 3 }$
(3) $\frac { 11 } { 2 }$
(4) $\frac { 17 } { 3 }$
(5) $\frac { 35 } { 6 }$

The line $y = 2 x + k$ meets the graphs of the two functions
$$y = \left( \frac { 2 } { 3 } \right) ^ { x + 3 } + 1 , \quad y = \left( \frac { 2 } { 3 } \right) ^ { x + 1 } + \frac { 8 } { 3 }$$
at points $\mathrm { P }$ and $\mathrm { Q }$ respectively. When $\overline { \mathrm { PQ } } = \sqrt { 5 }$, what is the value of the constant $k$? [4 points]\\
(1) $\frac { 31 } { 6 }$\\
(2) $\frac { 16 } { 3 }$\\
(3) $\frac { 11 } { 2 }$\\
(4) $\frac { 17 } { 3 }$\\
(5) $\frac { 35 } { 6 }$

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