Question:
If \( 1, \omega, \omega^2, \ldots, \omega^{n-1} \) are \( n \)th roots of unity, then the value of \( (9-\omega)(9-\omega^2)\cdots(9-\omega^{n-1}) \) is
If \( 1, \omega, \omega^2, \ldots, \omega^{n-1} \) are \( n \)th roots of unity, then the value of \( (9-\omega)(9-\omega^2)\cdots(9-\omega^{n-1}) \) is
Show Hint
Always use $x^n - 1$ identity for roots of unity products.
Updated On: Apr 23, 2026
- $\frac{9^n + 1}{8}$
- $9^n - 1$
- $\frac{9^n - 1}{8}$
- $9^n + 1$
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Concept:
Product of $(x - \omega^k)$ identity
Step 1: \[ (x-1)(x-\omega)\cdots(x-\omega^{n-1}) = x^n - 1 \]
Step 2: Put $x = 9$: \[ (9-1)(9-\omega)\cdots(9-\omega^{n-1}) = 9^n - 1 \]
Step 3: Divide by $(9-1)=8$: \[ (9-\omega)(9-\omega^2)\cdots = \frac{9^n - 1}{8} \] Conclusion:
Answer = $\frac{9^n - 1}{8}$
Step 1: \[ (x-1)(x-\omega)\cdots(x-\omega^{n-1}) = x^n - 1 \]
Step 2: Put $x = 9$: \[ (9-1)(9-\omega)\cdots(9-\omega^{n-1}) = 9^n - 1 \]
Step 3: Divide by $(9-1)=8$: \[ (9-\omega)(9-\omega^2)\cdots = \frac{9^n - 1}{8} \] Conclusion:
Answer = $\frac{9^n - 1}{8}$
Was this answer helpful?
0
0
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
View More Questions
Top MET Complex Numbers and Quadratic Equations Questions
- The number of real solutions of \( x^{2} - 3|x| + 2 = 0 \) is:
- The roots of $(x-a)(x-a-1)+(x-a-1)(x-a-2)+(x-a)(x-a-2)=0, a \in R$ are always
- Let \( f(x) = x^{2} + ax + b \), where \( a, b \in \mathbb{R} \). If \( f(x)=0 \) has all its roots imaginary, then the roots of \( f(x) + f'(x) + f''(x) = 0 \) are
- If \( \alpha, \beta, \gamma \) are the roots of \( x^{3} + 4x + 1 = 0 \), then the equation whose roots are \[ \frac{\alpha^{2}}{\beta+\gamma}, \quad \frac{\beta^{2}}{\gamma+\alpha}, \quad \frac{\gamma^{2}}{\alpha+\beta} \] is
- If \( f(x) = 2x^{4} - 13x^{2} + ax + b \) is divisible by \( x^{2} - 3x + 2 \), then \( (a, b) \) is equal to
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions