JEE Advanced 2024 · Paper 2 · Mathematics Q15

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MathematicsNumerical+3Question 15 of 51

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PARAGRAPH "I"

Let $S = \{1, 2, 3, 4, 5, 6\}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties:

$R$ has exactly 6 elements.
For each $(a, b) \in R$ , we have $|a - b| \ge 2$ .

Let $Y = \{R \in X : \text{The range of } R \text{ has exactly one element}\}$ and $Z = \{R \in X : R \text{ is a function from } S \text{ to } S\}$ .

Let $n(A)$ denote the number of elements in a set $A$ .

If the value of $n(Y) + n(Z)$ is $k^2$ , then $|k|$ is _____.

Your answer

Answer and worked solution

Official answer: 35.99

For each value $v$ , count the $a\in S$ with $|a-v|\ge2$ :

$v$	1	2	3	4	5	6
allowed $a$	4	3	3	3	3	4

Y. A relation with range $\{v\}$ and 6 elements would need $6$ allowed $a$ 's for that $v$ . No $v$ has more than $4$ , so $n(Y)=0$ .

Z. A function $S\to S$ in $X$ assigns each $a$ a value $f(a)$ with $|a-f(a)|\ge2$ . By symmetry the counts per $a$ are the same as in the table, so
$n(Z)=4\cdot3\cdot3\cdot3\cdot3\cdot4=1296.$

$n(Y)+n(Z)=1296=36^2$ , so $|k|=36$ .

Topics:Relations Fundamental principle of counting Types of functions

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JEE Advanced 2024 · Paper 2, Mathematics question 15