Question

Match the LIST-I with LIST-II

LIST-I (Expressions)		LIST-II (Values)
A.	\( i^{49} \)	I.	1
B.	\( i^{38} \)	II.	\(-i\)
C.	\( i^{103} \)	III.	\(i\)
D.	\( i^{92} \)	IV.	\(-1\)

Choose the correct answer from the options given below:

• A. A - I, B - II, C - III, D - IV
• B. A - I, B - III, C - II, D - IV
• C. A - III, B - IV, C - II, D - I
• D. A - III, B - IV, C - I, D - II

Hint:

To quickly find the remainder when dividing a number by 4, you only need to look at its last two digits. For example, for \(i^{103}\), you just need the remainder of 03 divided by 4, which is 3.

Answer 1

The Correct Option is C

Step 1: Understand the cyclic nature of powers of \(i\). The sequence \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\) repeats every four powers. To calculate \(i^n\), find the remainder \(k\) when \(n\) is divided by 4, and then \(i^n = i^k\). If the remainder is 0, it is equivalent to \(i^4 = 1\).
Step 2: Calculate each expression based on the remainder.- A. \(i^{49}\): \(49 \div 4\) has a remainder of 1. Therefore, \(i^{49} = i^1 = i\). This corresponds to III.- B. \(i^{38}\): \(38 \div 4\) has a remainder of 2. Therefore, \(i^{38} = i^2 = -1\). This corresponds to IV.- C. \(i^{103}\): \(103 \div 4\) has a remainder of 3. Therefore, \(i^{103} = i^3 = -i\). This corresponds to II.- D. \(i^{92}\): \(92 \div 4\) has a remainder of 0. Therefore, \(i^{92} = i^4 = 1\). This corresponds to I.
Step 3: Match the expressions to their values.The pairings are: A \(\rightarrow\) III, B \(\rightarrow\) IV, C \(\rightarrow\) II, D \(\rightarrow\) I. This matches option (3).