If the coefficient of $x^{15}$ in the expansion of $\left(a x^3+\frac{1}{b x^{1 / 3}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $\left(a x^{1 / 3}-\frac{1}{b x^3}\right)^{15}$, where a and $b$ are positive real numbers, then for each such ordered pair $(a, b)$ :
- $a=3 b$
- $a=b$
- $ab =1$
- $a b=3$
The Correct Option is C
Approach Solution - 1
Step 1: Coefficient of \(x^{15}\) in \((ax^3 + \frac{1}{bx^3})^{15}\)
The general term in the expansion of \((ax^3 + \frac{1}{bx^3})^{15}\) is given by:
\[ T_{r+1} = \binom{15}{r} \cdot a^{15-r} \cdot \left(\frac{1}{b}\right)^r \cdot x^{3(15-r) - 3r}. \]
Simplify the powers of \(x\):
\[ T_{r+1} = \binom{15}{r} \cdot a^{15-r} \cdot b^{-r} \cdot x^{45 - 6r}. \]
For the coefficient of \(x^{15}\), set \(45 - 6r = 15\):
\[ 45 - 15 = 6r \implies r = 9. \]
Thus, the coefficient of \(x^{15}\) is:
\[ \binom{15}{9} \cdot a^{6} \cdot b^{-9}. \]
Step 2: Coefficient of \(x^{-15}\) in \(\left(\frac{a}{x^3} - \frac{1}{bx^3}\right)^{15}\)
The general term in the expansion of \(\left(\frac{a}{x^3} - \frac{1}{bx^3}\right)^{15}\) is given by:
\[ T_{r+1} = \binom{15}{r} \cdot \left(\frac{a}{x^3}\right)^{15-r} \cdot \left(-\frac{1}{bx^3}\right)^r. \]
Simplify the powers of \(x\):
\[ T_{r+1} = \binom{15}{r} \cdot a^{15-r} \cdot b^{-r} \cdot (-1)^r \cdot x^{-3(15-r) - 3r}. \]
The exponent of \(x\) becomes:
\[ -45 + 6r. \]
For the coefficient of \(x^{-15}\), set \(-45 + 6r = -15\):
\[ 6r = 30 \implies r = 6. \]
Thus, the coefficient of \(x^{-15}\) is:
\[ \binom{15}{6} \cdot a^{9} \cdot b^{-6}. \]
Step 3: Equating the Coefficients
Equate the coefficients of \(x^{15}\) and \(x^{-15}\):
\[ \binom{15}{9} \cdot a^{6} \cdot b^{-9} = \binom{15}{6} \cdot a^{9} \cdot b^{-6}. \]
Since \(\binom{15}{9} = \binom{15}{6}\), cancel these terms:
\[ a^{6} \cdot b^{-9} = a^{9} \cdot b^{-6}. \]
Rearranging gives:
\[ \frac{a^6}{b^6} = \frac{b^9}{a^9}. \]
Cross-multiply:
\[ a^{15} \cdot b^9 = b^{15} \cdot a^9. \]
Divide both sides by \(a^9 b^9\):
\[ a^{6} = b^{6} \implies \frac{a}{b} = 1 \implies ab = 1. \]
Conclusion
The correct ordered pair satisfies \(ab = 1\).
Approach Solution -2
Coefficient of
Coefficient of
Coefficient
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.