JEE 2023 — Mathematics PYQ

Solution

1. Expressing the numerator:

   We need to find $\{\frac{4^{2022}}{15}\}$. Let's write $4^{2022}$ in terms of $4^2$ because $16$ is close to $15$.

   $$4^{2022}=(4^2{)}^{1011}=16^{1011}$$

2. Using Binomial Expansion:

   Write $16$ as $(15+1)$:

   $$16^{1011}=(15+1{)}^{1011}$$

   By expansion:

   $$(15+1{)}^{1011}=(\genfrac{}{}{0ex}{}{1011}{0})15^{1011}+(\genfrac{}{}{0ex}{}{1011}{1})15^{1010}+\cdots +(\genfrac{}{}{0ex}{}{1011}{1010})15^1+1$$

   Every term except the last one is a multiple of $15$.

   $$16^{1011}=15K+1$$

3. Calculating Fractional Part:

   $$\frac{4^{2022}}{15}=\frac{15K+1}{15}=K+\frac{1}{15}$$

   Since $K$ is an integer, the fractional part is $\frac{1}{15}$.

Final Answer:

The fractional part is $\frac{1}{15}$.