JEE 2023 — Mathematics PYQ

Matrix $M=\left[\begin{array}{cc}a& amp;b\\ c& amp;d\end{array}\right]$ ke 4 positions hain, aur har position ke liye 3 choices $\{0,1,2\}$ hain.

2. Non-Invertible Matrices $n(A^c)$ nikalna

Invertible hone ke liye $|M|\ne 0$ hona chahiye. Hum pehle wo cases nikalenge jahan $|M|=ad-bc=0$, yani $ad=bc$.

• Case 1: $ad=bc=0$

  • $ad=0$ ke liye 5 pairs hain: $(0,0),(0,1),(0,2),(1,0),(2,0)$.

  • $bc=0$ ke liye bhi 5 pairs hain.

  • Total cases $=5\times 5=25$.

• Case 2: $ad=bc=1$

  • $ad=1$ tabhi hoga jab $(1,1)$ ho (sirf 1 pair).

  • $bc=1$ ke liye bhi sirf 1 pair $(1,1)$.

  • Total cases $=1\times 1=1$.

• Case 3: $ad=bc=2$

  • $ad=2$ ke liye 2 pairs: $(1,2),(2,1)$.

  • $bc=2$ ke liye bhi 2 pairs: $(1,2),(2,1)$.

  • Total cases $=2\times 2=4$.

• Case 4: $ad=bc=4$

  • $ad=4$ ke liye sirf 1 pair: $(2,2)$.

  • $bc=4$ ke liye sirf 1 pair: $(2,2)$.

  • Total cases $=1\times 1=1$.

3. Probability Calculation

Non-invertible matrices ki total sankhya $n(A^c)=25+1+4+1=31$.

Invertible matrices ki sankhya $n(A)=n(S)-n(A^c)=81-31=50$.

Ab probability $P(A)$ hogi: