S={(x, y, z): x, y, z ∈ Z, x+2 y+3 z=42 mathrm{x}, mathrm{y}, mathrm{z} ≥ 0} ...........

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1.Set Theory

hard

$S=\{(x, y, z): x, y, z \in Z, x+2 y+3 z=42$ $\mathrm{x}, \mathrm{y}, \mathrm{z} \geq 0\}$ ...........

$167$

$169$

$168$

$165$

(JEE MAIN-2024)

Solution

$x+2 y+3 z=42$, $x, y, z \geq 0$

$z=0$ $x+2 y=42 \Rightarrow 22$

$z=1$ $x+2 y=39 \Rightarrow 20$

$z=2$ $x+2 y=36 \Rightarrow 19$

$z=3$ $x+2 y=33 \Rightarrow 17$

$z=4$ $x+2 y=30 \Rightarrow 16$

$z=5$ $x+2 y=27 \Rightarrow 14$

$z=6$ $x+2 y=24 \Rightarrow 13$

$z=7$ $x+2 y=21 \Rightarrow 11$

$z=8$ $x+2 y=18 \Rightarrow 10$

$z=9$ $x+2 y=15 \Rightarrow 8$

$z=10$ $x+2 y=12 \Rightarrow 7$

$z=11$ $x+2 y=9 \Rightarrow 5$

$z=12$ $x+2 y=6 \Rightarrow 4$

$z=13$ $x+2 y=3 \Rightarrow 2$

$z=14$ $x+2 y=0 \Rightarrow 1$

Total : $169$

Standard 11

Mathematics

Let $A_1, A_2, \ldots \ldots, A_m$ be non-empty subsets of $\{1,2,3, \ldots, 100\}$ satisfying the following conditions:

$1.$ The numbers $\left|A_1\right|,\left|A_2\right|, \ldots,\left|A_m\right|$ are distinct.

$2.$ $A_1, A_2, \ldots, A_m$ are pairwise disjoint.(Here $|A|$ donotes the number of elements in the set $A$ )Then, the maximum possible value of $m$ is

normal

(KVPY-2016)

Let $\bigcup \limits_{i=1}^{50} X_{i}=\bigcup \limits_{i=1}^{n} Y_{i}=T$ where each $X_{i}$ contains $10$ elements and each $Y_{i}$ contains $5$ elements. If each element of the set $T$ is an element of exactly $20$ of sets $X_{i}$ 's and exactly $6$ of sets $Y_{i}$ 's, then $n$ is equal to

medium

(JEE MAIN-2020)

Let $A = \{x:x \in R,\,|x|\, < 1\}\,;$ $B = \{x:x \in R,\,|x – 1| \ge 1\}$ and $A \cup B = R – D,$then the set $D$ is

hard

Let $A=\{n \in N: H . C . F .(n, 45)=1\}$ and Let $B=\{2 k: k \in\{1,2, \ldots, 100\}\}$. Then the sum of all the elements of $A \cap B$ is

medium

(JEE MAIN-2022)

If $X = \{ {4^n} – 3n – 1:n \in N\} $ and $Y = \{ 9(n – 1):n \in N\} ,$ then $X \cup Y$ is equal to

hard

$S=\{(x, y, z): x, y, z \in Z, x+2 y+3 z=42$ $\mathrm{x}, \mathrm{y}, \mathrm{z} \geq 0\}$ ...........

Solution

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