UPSC CDS 1 2022 Math Paper

Concept: Natural number: All the positive integers from 1 to infinity. Even number: Whole numbers that have the digits $0,2,4,6$, or 8 in their one's place are even numbers. Odd number: Any number which cannot be divided by two. Calculation: Statement: 1 Let $n, n+1$, and $n+2$ be three consecutive natural numbers. Let, $P(n)=n^{3}+(n+1)^{3}+(n+2)^{3}$ is divisible by 9 . For $n=1$, $\Rightarrow 1^{3}+2^{3}+3^{3}$ $\Rightarrow 1+8+27=36 \;\text{ is divisible by }\; 9$ $\Rightarrow \;\text{ For }\; n=4$ $\Rightarrow 4^{3}+5^{3}+6^{3}$ $\Rightarrow 64+125+215=405$ is divisible by 9 Hence, statement 1 is correct. Statement: 2 Take $3^{2}, 5^{2}, 7^{2}$ we get $\Rightarrow 9,25,49$ which is divisible by 8 and gives 1 as remainder. Similarly, we can check for any other value as well. Hence, statement 2 is correct. $\therefore$ Both 1 and 2 are correct.