Question

Consider the following properties which are related to equations and their roots:
A. If \(1,\alpha,\beta,\gamma\) are roots of \(x^4-1=0\), then \((1-\alpha)(1-\beta)(1-\gamma)=0\),
B. If \(1,\alpha,\beta,\gamma\) are roots of \(x^4-1=0\), then \((1-\alpha)(1-\beta)(1-\gamma)=4\),
C. If \(\omega,\omega^2,\omega^3,\ldots,\omega^{n-1}\) are the imaginary roots of unity, then \((1-\omega)(1-\omega^2)\cdots(1-\omega^{n-1})=n-1\),
D. If \(\omega,\omega^2,\omega^3,\ldots,\omega^{n-1}\) are the imaginary roots of unity, then \((1-\omega)(1-\omega^2)\cdots(1-\omega^{n-1})=n\).

• A. A, C
• B. B, D
• C. A, D
• D. B, C

Hint:

For \(n\)th roots of unity, \((1-\omega)(1-\omega^2)\cdots(1-\omega^{n-1})=n\).

Answer 1

The Correct Option is B

Concept:
For roots of unity: \[ x^n-1=(x-1)(x-\omega)(x-\omega^2)\cdots(x-\omega^{n-1}) \]

Step 1: Use derivative idea for \(x^4-1\).
Let: \[ f(x)=x^4-1 \] The roots are: \[ 1,\alpha,\beta,\gamma \] Then: \[ f(x)=(x-1)(x-\alpha)(x-\beta)(x-\gamma) \] Differentiate and put \(x=1\): \[ f'(1)=(1-\alpha)(1-\beta)(1-\gamma) \] Now: \[ f'(x)=4x^3 \] \[ f'(1)=4 \] So: \[ (1-\alpha)(1-\beta)(1-\gamma)=4 \] Thus, B is correct and A is incorrect.

Step 2: Use general roots of unity result. \[ (1-\omega)(1-\omega^2)\cdots(1-\omega^{n-1})=n \] Thus, D is correct and C is incorrect. \[ \therefore \text{Correct Answer is (B)} \]