WBJEE · Maths · Complex Number
If \(\cos \theta+i \sin \theta, \theta \in \mathbb{R}\), is the root of the equation
\(a_0 x^n+a_1 \cdot x^{n-1}+\ldots .+a_{n-1} x+a_n=0, a_0, a_1, \ldots \ldots . a_n \in \mathbb{R}, a_0 \neq 0\)
then the value of \(a_1 \sin \theta+a_2 \sin 2 \theta+\ldots .+a_n \sin n \theta\) is
- A 2n
- B n
- C 0
- D n+1
Answer & Solution
Correct Answer
(C) 0
Step-by-step Solution
Detailed explanation
Hint: \(a_0 x^n+a_1 \cdot x^{n-1}+\ldots .+a_{n-1} x+a_n=0\) divided by \(x^n\) \(a_0+a_1 \frac{1}{x}+\ldots \ldots+a_{n-1} \frac{1}{x^{n-1}}+a_n \frac{1}{x^n}=0\) put \(x=\cos \theta+i \sin \theta\) and equating imaginary part…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If a hyperbola passes through the point \(\mathrm{P}(\sqrt{2}, \sqrt{3})\) and foci at \(( \pm 2,0)\) then the tangent to this hyperbola at \(\mathrm{P}\) isWBJEE 2023 Easy
- \(\lim _{x \rightarrow 1}\left(\frac{1}{\ln x}-\frac{1}{(x-1)}\right)\)WBJEE 2020 Medium
- Let the tangent and normal at any point \(P\left(at^2\right.\), 2at), \((a > 0)\), on the parabola \(y^2=4 a x\) meet the axis of the parabola at \(T\) and \(\mathrm{G}\) respectively. Then the radius of the circle through \(\mathrm{P}, \mathrm{T}\) and \(\mathrm{G}\) isWBJEE 2022 Medium
- If the tangent to the curve \(y^{2}=x^{3}\) at \(\left(m^{2}, m^{3}\right)\) is also a normal to the curve at \(\left(M^{2}, M^{3}\right)\), then the value of \(m M\) isWBJEE 2020 Medium
- If the matrix \(\left(\begin{array}{ccc}0 & a & a \\ 2 b & b & -b \\ c & -c & c\end{array}\right)\) is orthogonal, then the values of \(a, b, c\) areWBJEE 2025 Medium
- Let \(x_{n}=\left(1-\frac{1}{3}\right)^{2}\left(1-\frac{1}{6}\right)^{2}\left(1-\frac{1}{10}\right)^{2} \ldots\) \(\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^{2}, n \geq 2\) Then, the value of \(\lim _{n \rightarrow \infty} x_{n}\) isWBJEE 2015 Hard
More PYQs from WBJEE
- Consider a region in free space bounded by the surfaces of an imaginary cube having sides of length \(a\) as shown in the figure. A charge \(+Q\) is placed at the centre \(O\) of the cube. \(P\) is such a point out side the cube that the line \(O P\) perpendicularly intersects the surface \(A B C D\) at \(R\) and also \(O R=R P=a / 2\). A charge \(+Q\) is placed at point \(P\) also. What is the total electric flux through the five faces of the cube other than \(A B C D ?\)
WBJEE 2018 Easy - Given the standard half-cell potentials \(\left(E^{\circ}\right)\) of
the following as
\[
\begin{array}{l}
\text { Zn } \longrightarrow \mathrm{Zn}^{2+}+2 e^{-} ; \quad E^{\circ}=+0.76 \mathrm{V} \\
\mathrm{Fe} \longrightarrow \mathrm{Fe}^{2+}+2 e^{-} ; \quad E^{\circ}=0.41 \mathrm{V}
\end{array}
\]
Then the standard e.m.f. of the cell with the reaction \(\mathrm{Fe}^{2+}+\mathrm{Zn} \longrightarrow \mathrm{Zn}^{2+}+\mathrm{Fe}\) isWBJEE 2018 Medium - The differential equation of the family of curves \(y=e^{x}(A \cos x+B \sin x)\) where \(A, B\) are arbitrary constants isWBJEE 2020 Easy
- The following figure shows the variation of potential energy V(x) of a particle with distance x. The particle has
WBJEE 2024 Easy - Which of the following has the largest number of atoms?WBJEE 2020 Medium
- Let \(A, B, C\) be three non-void subsets of set \(S\). Let \((A \cap C) \cup\left(B \cap C^{\prime}\right)=\phi\) where \(C^{\prime}\) denote the complement of set \(C\) in S. ThenWBJEE 2021 Easy