CAT 2025 Question Paper | Quant Slot 3

CAT Questions

CAT 2025 Quant was dominated by Algebra followed by Arithmetic. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 9 : If \( 12 ^ { 12 x } \times 4 ^ { 24 x + 12 } \times 5 ^ { 2 y } = 8 ^ { 4 z } \times 20 ^ { 12 x } \times 243 ^ { 3 x - 6 } \), where \( x , y \) and \( z \) are natural numbers, then \( x + y + z \) equals

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Video Explanation

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Explanatory Answer

Given:
\[ 12^{12x}\times4^{24x+12}\times5^{2y} =8^{4z}\times20^{12x}\times243^{3x-6} \]

Express all numbers in prime factors.

Left-Hand Side:
\[ 12^{12x}=2^{24x}\times3^{12x} \]
\[ 4^{24x+12}=2^{48x+24} \]
Therefore: \[ LHS=2^{72x+24}\times3^{12x}\times5^{2y} \]

Right-Hand Side:
\[ 8^{4z}=2^{12z} \]
\[ 20^{12x}=2^{24x}\times5^{12x} \]
\[ 243^{3x-6}=3^{15x-30} \]
Therefore: \[ RHS=2^{12z+24x}\times3^{15x-30}\times5^{12x} \]

Equating powers of 3:
\[ 12x=15x-30\Rightarrow x=10 \]
Equating powers of 5:
\[ 2y=12x\Rightarrow y=6x=60 \]
Equating powers of 2:
\[ 72x+24=12z+24x \]
\[ 48x+24=12z \]
\[ z=4x+2=42 \]

Therefore:
\[ x+y+z=10+60+42=112 \]

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The answer is '112'

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