Question
Let = -1+i 3 2 and = -1-i 3 2 , i= -1 . If (7-7 +9 )^ 20 +(9+7 -7 )^ 20 +(-7+9 +7 )^ 20 +(14+7 +7 )^ 20 =m^ 10 , then m is \_\_\_\_\_
JEE Main
Mathematics
Complex Number
2026
JEE Main 2026 (22 January Shift 1)
JEE Main Mathematics Question (2026) — Solution
Options
- A. A
- B. B
- C. C
- D. D
Answer
A. A
Step-by-step solution
= , = ^2 (cube roots of unity). Let z_1 = 7-7 +9 ^2, z_2 = 9+7 -7 ^2, z_3 = -7+9 +7 ^2. Calculating: z_1 = 6-8i 3 , z_2 = 9+7i 3 , z_3 = -15+i 3 . All have |z_k| = 228 . Verify: z_2 = z_1 , z_3 = z_1 ^2. z_1^ 20 +z_2^ 20 +z_3^ 20 = z_1^ 20 (1+ ^ 20 + ^ 40 ) = z_1^ 20 (1+ ^2+ ) = 0. Fourth term: 14+7 +7 ^2 = 14+7(-1) = 7. So 7^ 20 = 49^ 10 . Total = 0 + 49^ 10 = m^ 10 m = 49.
Related: Mathematics — Complex Number · All PYQ Banks