Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^{2} \sin \theta-x(\sin \theta \cos \theta+1)+\cos \theta=0$
$\left(0<\theta<45^{\circ}\right)$, and $\alpha<\beta$. Then $\sum_{n=0}^{\infty}\left(\alpha^{n}+\frac{(-1)^{n}}{\beta^{n}}\right)$
is equal to :-
$\frac{1}{1-\cos \theta}+\frac{1}{1+\sin \theta}$
$\frac{1}{1+\cos \theta}+\frac{1}{1-\sin \theta}$
$\frac{1}{1-\cos \theta}-\frac{1}{1+\sin \theta}$
$\frac{1}{1+\cos \theta}-\frac{1}{1-\sin \theta}$
Correct Option: 1
$\mathrm{D}=(1+\sin \theta \cos \theta)^{2}-4 \sin \theta \cos \theta=(1-\sin \theta \cos \theta)^{2}$
$\Rightarrow$ roots are $\beta=\operatorname{cosec} \theta$ and $\alpha=\cos \theta$
$\Rightarrow \sum_{n=0}^{\infty}\left(\alpha^{n}+\left(-\frac{1}{\beta}\right)^{n}\right)=\sum_{n=0}^{\infty}(\cos \theta)^{n}+\sum_{n=0}^{n}(-\sin \theta)^{n}$
$=\frac{1}{1-\cos \theta}+\frac{1}{1+\sin \theta}$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,