JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $A = [a_{ij}], a_{ij} \in \mathbb{Z} \cap [0, 4], 1 \le i, j \le 2$ . The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is

Choose the correct answer:

Explanation

1. Jab Sum $S = 3$ ho

Yahan formula $\binom{n+r-1}{r-1}$ lagega:

N_3 = \binom{3+4-1}{4-1} = \binom{6}{3} = \mathbf{20}

2. Jab Sum $S = 5$ ho

Yahan ek entry 5 nahi ho sakti, isliye $\binom{5+3}{3}$ mein se wo cases minus karenge jahan koi ek entry $\ge 5$ ho:

N_5 = \binom{8}{3} - 4\binom{0+3}{3} = 56 - 4(1) = \mathbf{52}

3. Jab Sum $S = 7$ ho

Yahan bhi wahi logic (Total - Cases where any $x_i \ge 5$ ):

N_7 = \binom{7+3}{3} - 4\binom{2+3}{3} = \binom{10}{3} - 4\binom{5}{3} = 120 - 4(10) = \mathbf{80}

4. Jab Sum $S = 11$ ho

Isme "Inclusion-Exclusion" use hoga:

N_{11} = \binom{11+3}{3} - 4\binom{6+3}{3} + 6\binom{1+3}{3}

N_{11} = 364 - 4(84) + 6(4) = 364 - 336 + 24 = \mathbf{52}

Final Total Calculation

Total = N_3 + N_5 + N_7 + N_{11}

Total = 20 + 52 + 80 + 52 = \mathbf{204}

Answer: 204

Choose the correct answer:

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