Question:
If the term free from $\mathrm{x}$ in the expansion of $\sqrt{x}-\frac{k}{x^{2}}$ is $\mathbf{4 0 5}$, find the value of $k$.
Solution:
Given $\sqrt{x}-\frac{k}{x^{2}}$
From the standard formula of Tro1 we can write given expression as
$T_{r+1}={ }^{10} C_{r}(\sqrt{x})^{10-r}\left(\frac{-k}{x^{2}}\right)^{r}={ }^{10} C_{r}(x)^{\frac{1}{2}(10-r)}(-k)^{r} x^{-2 r}$
$={ }^{10} C_{r}(x)^{5-\frac{r}{2}-2 r}(-k)^{r}={ }^{10} C_{r} x^{\frac{10-5 r}{2}}(-k)^{r}$
For the term free from $x$ we have $(10-5 r) / 2=0$
Which implies $r=2$
So, the term free from $x$ is
$T_{2+1}={ }^{10} C_{2}(-k)^{2}$
${ }^{10} C_{2}(-k)^{2}=405$
$\frac{10 \times 9 \times 8 !}{2 ! \times 8 !}(-k)^{2}=405$
$45 k^{2}=405 \quad \Rightarrow \quad k^{2}=9 \quad \therefore k=\pm 3$
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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
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- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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