Question:
In the expansion of \(\left(\sqrt{x} + \frac{1}{3x^2}\right)^{10}\), the value of the constant term (independent of \(x\)) is:
\(\left(\sqrt{x} + \frac{1}{3x^2}\right)^{10}\) के विस्तार में स्थिर पद (\(x\) से स्वतंत्र पद) का मान है:
Options:
- 5
- 8
- 45
- 90
- 5
- 8
- 45
- 90
Correct Answer: a)
Explanation:
General term \(T_{r+1} = {}^{10}C_r (x^{1/2})^{10-r} (3x^2)^{-r} = {}^{10}C_r 3^{-r} x^{(10-5r)/2}\).
For independence of \(x\):
\[\frac{10-5r}{2} = 0 \implies r = 2\]
Constant term:
\[T_3 = {}^{10}C_2 \cdot \frac{1}{3^2} = \frac{45}{9} = 5\]
Hence, the correct option is (a).
द्विपद प्रसार में व्यापक पद \(T_{r+1} = {}^{10}C_r \cdot 3^{-r} \cdot x^{(10-5r)/2}\)।
\(x\) से स्वतंत्र पद हेतु:
\[10 – 5r = 0 \implies r = 2\]
अचर पद:
\[T_3 = {}^{10}C_2 \cdot \frac{1}{9} = \frac{45}{9} = 5\]
अतः सही विकल्प (a) है।