Find the ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of $\left(x^{2}+\frac{2}{x}\right)^{15}$
To Find: the ratio of the coefficient of $x^{15}$ to the term independent of $x$
Formula : $\mathrm{t}_{\mathrm{r}+1}=\left(\begin{array}{l}\mathrm{n} \\ \mathrm{r}\end{array}\right) \mathrm{a}^{\mathrm{n}-\mathrm{r}} \mathrm{b}^{\mathrm{r}}$
Here, $a=x^{2}, \quad b=\frac{2}{x}$ and $n=15$
We have a formula,
$\mathrm{t}_{\mathrm{r}+1}=\left(\begin{array}{l}\mathrm{n} \\ \mathrm{r}\end{array}\right) \mathrm{a}^{\mathrm{n}-\mathrm{r}} \mathrm{b}^{\mathrm{r}}$
$=\left(\begin{array}{c}15 \\ r\end{array}\right)\left(x^{2}\right)^{15-r}\left(\frac{2}{x}\right)^{r}$
$=\left(\begin{array}{c}15 \\ r\end{array}\right)(x)^{30-2 r}(2)^{r}(x)^{-r}$
$=\left(\begin{array}{c}15 \\ r\end{array}\right)(x)^{30-2 r-r}(2)^{r}$
$=\left(\begin{array}{c}15 \\ r\end{array}\right)(2)^{r}(x)^{30-3 r}$
To get coefficient of $x^{15}$ we must have,
$(x)^{30-3 r}=x^{15}$
- $30-3 r=15$
- $3 r=15$
$\cdot r=5$
Therefore, coefficient of $x^{15}=\left(\begin{array}{c}15 \\ 5\end{array}\right)(2)^{5}$
Now, to get coefficient of term independent of $x$ that is coefficient of $x^{0}$ we must have,
$(x)^{30-3 r}=x^{0}$
$-30-3 r=0$
$\cdot 3 r=30$
$\cdot r=10$
Therefore, coefficient of $x^{0}=\left(\begin{array}{l}15 \\ 10\end{array}\right)(2)^{10}$
But $\left.\left(\begin{array}{l}15 \\ 10\end{array}\right)=\left(\begin{array}{c}15 \\ 5\end{array}\right) \ldots \ldots \ldots\left(\begin{array}{l}\mathrm{n} \\ \mathrm{r}\end{array}\right)=\left(\begin{array}{c}\mathrm{n} \\ \mathrm{n}-\mathrm{r}\end{array}\right)\right]$
Therefore, the coefficient of x0 $=\left(\begin{array}{c}15 \\ 5\end{array}\right)(2)^{10}$
Therefore,
$\frac{\text { coefficient of } x^{15}}{\text { coefficient of } x^{0}}=\frac{\left(\begin{array}{c}15 \\ 5\end{array}\right)(2)^{5}}{\left(\begin{array}{c}15 \\ 5\end{array}\right)(2)^{10}}$
$=\frac{1}{(2)^{5}}$
$=\frac{1}{32}$
Hence, coefficient of $x^{15}:$ coefficient of $x^{0}=1: 32$
$\underline{\text { Conclusion }}$ : The ratio of coefficient of $x^{15}$ to coefficient of $x^{0}=1: 32$
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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,