CAT 2024 Question Paper | Quant Slot 2

CAT Questions

CAT 2024 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 4 A function \(f\) maps the set of natural numbers to whole numbers, such that \(f(x y)=f(x) f(y)+f(x)+f(y)\) for all \(x, y\) and \(f(p)=1\) for every prime number \(p\). Then, the value of \(f(160000)\) is

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Video Explanation

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Explanatory Answer

\(\begin{array}{l}f(2)=1 \\ f(4)=f\left(2^2\right)=f(2) f(2)+f(2)+f(2)=(1)(1)+1+1=3=2^2-1 \\ f(8)=f\left(2^3\right)=f\left(2^* 4\right)=f(2) f(4)+f(2)+f(4)=(1)(3)+1+3=7=2^3-1 \\ f(5)=1 \\ f\left(5^2\right)=f(5) f(5)+f(5)+f(5)=(1)(1)+1+1=3=2^2-1 \\ f\left(x^n\right)=2^n-1 \text { where } x \text { is a prime number. } \\ \begin{aligned} f(160000) & =f\left(2^{8 *} 5^4\right)=f\left(2^8\right) f\left(5^4\right)+f\left(2^8\right)+f\left(5^4\right) \\ & =\left(255^* 15\right)+255+15 \\ & =4095\end{aligned}\end{array}\)

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The question is "A function \(f\) maps the set of natural numbers to whole numbers, such that \(f(x y)=f(x) f(y)+f(x)+f(y)\) for all \(x, y\) and \(f(p)=1\) for every prime number \(p\). Then, the value of \(f(160000)\) is"

Hence, the answer is '4095'

Choice B is the correct answer.

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