AP EAMCET · Maths · Complex Number
If \(\alpha_1, \alpha_2, \alpha_3 \ldots, \alpha_n\) are real numbers, \(\alpha_1 \neq 0\) and \(z=\cos \theta+i \sin \theta\) is a root of the equation \(\alpha_1+\alpha_2 z+\alpha_3 z^2+\ldots+\alpha_n z^{n-1}+z^n=0\), then \(\alpha_1 \cos n \theta+\alpha_2 \cos (n-1) \theta+\ldots+\alpha_n \cos \theta=\)
- A \(1+i\)
- B 1
- C -1
- D \(1-i\)
Answer & Solution
Correct Answer
(C) -1
Step-by-step Solution
Detailed explanation
Given \\(z=\\cos \\theta+i \\sin \\theta = e^{i\\theta}\\) is a root of \\(\\alpha_1+\\alpha_2 z+\\ldots+\\alpha_n z^{n-1}+z^n=0\\). Substitute \\(z=e^{i\\theta}\\): \\(\\alpha_1+\\alpha_2 e^{i\\theta}+\\ldots+\\alpha_n e^{i(n-1)\\theta}+e^{in\\theta}=0\\) Multiply by…
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 the line \(5 x-2 y-6=0\) is a tangent to the hyperbola \(5 x^2-k y^2=12\). then the equation of the normal to this hyperbola at the point \((\sqrt{6}, p)(p \lt 0)\) isAP EAMCET 2024 Medium
- If \(f:[2,3] \rightarrow R\) is defined by \(f(x)=x^3+3 x-2\), then the range \(f(x)\) is contained in the intervalAP EAMCET 2009 Easy
- If \(D(2,1,0), E(2,0,0)\) and \(F(0,1,0)\) are mid-points of the sides \(B C, C A\) and \(A B\) of \(\triangle A B C\), respectively. Then, the centroid of \(\triangle A B C\) isAP EAMCET 2013 Medium
- In triangle \(A B C\) if \(\cos A \cos B+\sin A \sin B \sin C=1\), then \(\sin A+\sin B+\sin C=\)AP EAMCET 2025 Medium
- If , thenAP EAMCET 2021 Easy
- If the order and degree of the differential equation \(x \frac{d^2 y}{d x^2}=\left(1+\left(\frac{d^2 y}{d x^2}\right)^2\right)^{-1 / 2}\) are k and \(l\) respectively, then \(\mathrm{k}, l\) are the roots ofAP EAMCET 2025 Medium
More PYQs from AP EAMCET
- A force \(\overrightarrow{\mathrm{F}}=4 \hat{\mathrm{i}}-15 \hat{\mathrm{j}} \mathrm{N}\), acts on a body resulting in a displacement of \(6 \hat{i}\). If the body had a kinetic energy of 7 joules at the beginning of the displacement, the kinetic energy at the end of the displacement isAP EAMCET 2022 Medium
- Two oxides of nitrogen, \(\mathrm{NO}\) and \(\mathrm{NO}_2\) react together at \(253 \mathrm{~K}\) and form a compound of nitrogen \(X\). \(X\) reacts with water to yield another compound of nitrogen \(Y\). The shape of the anion of \(Y\) molecule is :AP EAMCET 2003 Easy
- An ideal monoatomic gas is carried along the cycle \(A B C D A\) as shown in the figure. The total heat absorbed during this process is
AP EAMCET 2018 Easy - \(\lim _{x \rightarrow \infty}[x-\log (\cosh x)]=\)AP EAMCET 2025 Hard
- If \(\int_0^{2 \pi}\left(\sin ^4 x+\cos ^4 x\right) d x=K \int_0^\pi \sin ^2 x d x+L \int_0^{\frac{\pi}{2}} \cos ^2 x d x\) and \(K, L \in N\), then the number of possible ordered pairs \((\mathrm{K}, \mathrm{L})\) isAP EAMCET 2024 Hard
- The length of the subtangent at \((2,2)\) to the curve \(x^5=2 y^4\) isAP EAMCET 2008 Easy