1
Let R denote the set of all real numbers and let i= -1. Consider the matrices S= bmatrix 0 & -1 \ 1 & 0 bmatrix and T= bmatrix 1 & 1 \ 0 & 1 bmatrix. Let a,b,c,d be real numbers such that ST= bmatrix a & b \ c & d bmatrix. Let H=\x+iy: x,y R and y>0\. Then which of the following statements is (are) TRUE ?
- b+iad+ic=i
- If = -1+i 32, then a +bc +d=
- If m is an integer greater than 2 such that (ST)2=(ST)m, then m is an integer multiple of 8
- If z H, then az+bcz+d H
Answer
(2) If = -1+i 32, then a +bc +d=, (4) If z H, then az+bcz+d H
Solution
ST= bmatrix 0 & -1 \ 1 & 1 bmatrix, so a=0, b=-1, c=1, d=1.
(A) -11+i= -1+i2 i. False.
(B) -1 +1= -1- 2= 1 2= . True.
(C) M=ST has trace 1, det 1, so M2-M+I=O M3=-I, M6=I and the order of M is 6. Mm=M2 6 (m-2), e.g. m=14 is not a multiple of 8. False.
(D) w= -1z+1= -( z+1)|z+1|2 Imw= Imz|z+1|2>0. True.
Answer: (B), (D)
JEE Advanced 2026 Paper 2