If \( \alpha \) is imaginary \( n^{\text {th }}(n \geq 3) \) root o...
If \( \alpha \) is imaginary \( n^{\text {th }}(n \geq 3) \) root of unity. Which of the following are true.
(A) \( \sum_{r=1}^{n-1}(n-r) \alpha^{r}=\frac{n \alpha}{1-\alpha} \)
(B) \( \sum_{r=1}^{n-1}(n-r) \sin \frac{2 r \pi}{n}=\frac{n}{2} \cot \frac{\pi}{n} \)
\( \mathrm{P} \)
(C) \( \sum_{r=1}^{n-1}(n-r) \cos \frac{2 r \pi}{n}=-\frac{n}{2} \)
(D) \( \sum_{r=1}^{n-1}(n-r) \alpha^{r}=\frac{n}{1-\alpha} \)
📲 PW App Link - https://bit.ly/YTAI_PWAP
🌐 PW Website - https://www.pw.live/
Other Videos By PW Solutions
| 2023-02-25 | Let \( f(x)=\int_{0}^{x}|2 t-3| d t \), then \( f \) is
(A) continu... |
| 2023-02-25 | Let \( A=\int_{1}^{e^{2}} \frac{\ell n x}{\sqrt{x}} d x \), then
(A... |
| 2023-02-25 | Given \( f \) is an odd function defined everywhere, periodic with ... |
| 2023-02-25 | Let \( f(x) \) be a continuous function and \( I=\int_{1}^{9} \sqrt... |
| 2023-02-25 | Column - I
Column - II
(A) Area bounded by region \( 0 \leq y \leq ... |
| 2023-02-25 | If \( t(x)=2^{[x \mid} \), where \( \{x\} \) denotes the fractional... |
| 2023-02-25 | Let \( f(x)=2 x^{3}-15 x^{2}+24 x \) and \( g(x)=\int_{0}^{x} f(t) ... |
| 2023-02-25 | Which of the following is true?
(A) Numbur of roots of the equation... |
| 2023-02-25 | The value of integral \( \int_{a}^{b} \frac{|x|}{x} d x, ab \) is:
... |
| 2023-02-25 | Find area bounded by \( y=f-1(x), x=10, x=4 \) and \( x \)-axis giv... |
| 2023-02-25 | If \( \alpha \) is imaginary \( n^{\text {th }}(n \geq 3) \) root o... |
| 2023-02-25 | If \( \alpha, \beta, \gamma \) are distinct roots of \( x^{3}-3 x^{... |
| 2023-02-25 | Let \( z_{1}, z_{2}, z_{3} \) be three distinct complex numbers sat... |
| 2023-02-25 | If \( \int_{1}^{x} \frac{d t}{|t| \sqrt{t^{2}-1}}=\frac{\pi}{6} \),... |
| 2023-02-25 | The locus of equation \( \operatorname{Arg}\left(\frac{z-1-2 i}{z+3... |
| 2023-02-25 | A uniform unbiased die is constructed in the shape of a regular tet... |
| 2023-02-25 | If \( z \) is a complex number then the equation \( z^{2}+z|z|+|z|^... |
| 2023-02-25 | The equation \( |z+i|-|z-i|=k \) represents
(A) a hyperbela if \( 0... |
| 2023-02-25 | If \( z \) satisfies the inequality \( |z-1-2 i| \leq 1 \), then wh... |
| 2023-02-25 | The value of \( i^{n}+i^{-n} \), for \( i=\sqrt{-1} \) and \( n \in... |
| 2023-02-25 | The quadratic equation \( z^{2}+\left(p+i p^{\prime}\right) z+q+i q... |
0