JEE 2023 — Mathematics PYQ

Diye gaye equation $A^{\prime }=\alpha A+I$ ka dono taraf transpose lene par:

Ab pehli equation se $A^{\prime }$ ki value ismein rakhte hain:

Chunki $\alpha \ne 1$, hum likh sakte hain:

Step 2: Determinant ka upyog

Maana $k=\frac{1}{1-\alpha }$, toh $A=kI$.

Ab $\det (A^2-A)=4$ mein value rakhte hain:

Chunki $A$ ek $2\times 2$ matrix hai, $\det (mI)=m^2$ hota hai:

Iska matlab hai:

Step 3: $\alpha$ ki values nikalna

Case 1: $k^2-k-2=0$

• Agar $k=2⟹\frac{1}{1-\alpha }=2⟹1=2-2\alpha ⟹\alpha =\frac{1}{2}$

• Agar $k=-1⟹\frac{1}{1-\alpha }=-1⟹1=-1+\alpha ⟹\alpha =2$

Case 2: $k^2-k+2=0$

Iska discriminant D = (-1)^2 - 4(1)(2) = -7 < 0 hai, isliye yahan se koi real value nahi milegi.

Step 4: Sum of values

$\alpha$ ki possible values hain $\frac{1}{2}$ aur $2$.

Sum $=2+\frac{1}{2}=\frac{5}{2}$