csat-suneung 2012 Q30

csat-suneung · South-Korea · csat__math-humanities 4 marks Exponential Functions Intersection and Distance between Curves

For natural numbers $a , b$, let P and Q be the points where the curve $y = a ^ { x + 1 }$ and the curve $y = b ^ { x }$ meet the line $x = t ( t \geq 1 )$, respectively. Find the number of all ordered pairs $( a , b )$ of $a , b$ satisfying the following condition. For example, $a = 4 , b = 5$ satisfies the following condition. [4 points]
(A) $2 \leq a \leq 10, 2 \leq b \leq 10$
(B) For some real number $t \geq 1$, $\overline { \mathrm { PQ } } \leq 10$.

For natural numbers $a , b$, let P and Q be the points where the curve $y = a ^ { x + 1 }$ and the curve $y = b ^ { x }$ meet the line $x = t ( t \geq 1 )$, respectively.\\
Find the number of all ordered pairs $( a , b )$ of $a , b$ satisfying the following condition. For example, $a = 4 , b = 5$ satisfies the following condition. [4 points]\\
(A) $2 \leq a \leq 10, 2 \leq b \leq 10$\\
(B) For some real number $t \geq 1$, $\overline { \mathrm { PQ } } \leq 10$.

Paper Questions

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