Question:
If the fourth term in the binomial expansion of
$\left(\frac{2}{x}+x^{\log _{8} x}\right)^{6}(x>0)$ is $20 \times 8^{7}$, then a value of
$x$ is :
8
$8^{2}$
$8^{-2}$
$8^{3}$
Correct Option: , 2
Solution:
$\mathrm{T}_{4}=\mathrm{T}_{3+1}=\left(\begin{array}{l}6 \\ 3\end{array}\right)\left(\frac{2}{\mathrm{x}}\right)^{3} \cdot\left(\mathrm{x}^{\log _{\mathrm{s}} \mathrm{x}}\right)^{3}$
$20 \times 8^{7}=\frac{160}{x^{3}} \cdot x^{3 \log _{8} x}$
$8^{6}=x^{\log _{2} x}-3$
$2^{18}=x^{\log _{2} x-3}$
$\Rightarrow 18=\left(\log _{2} x-3\right)\left(\log _{2} x\right)$
Let $\log _{2} x=t$
$\Rightarrow t^{2}-3 t-18=0$
$\Rightarrow(t-6)(t+3)=0$
$\Rightarrow t=6,-3$
$\log _{2} x=6 \Rightarrow x=2^{6}=8^{2}$
$\log _{2} x=-3 \Rightarrow x=2^{-3}=8^{-1}$
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- 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,
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