Question:
If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2x^2 – 7x + 4 = 0\), find the value of \(\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha}\).
यदि \(\alpha\) और \(\beta\) द्विघात समीकरण \(2x^2 – 7x + 4 = 0\) के मूल हैं, तो \(\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha}\) का मान ज्ञात कीजिए।
Options:
- \(\frac{175}{16}\)
- \(\frac{147}{16}\)
- \(\frac{189}{16}\)
- \(\frac{161}{8}\)
- \(\frac{175}{16}\)
- \(\frac{147}{16}\)
- \(\frac{189}{16}\)
- \(\frac{161}{8}\)
Correct Answer: a)
Explanation:
For \(2x^2 – 7x + 4 = 0\):
Sum of roots: \(\alpha + \beta = \frac{7}{2}\)
Product of roots: \(\alpha \beta = \frac{4}{2} = 2\)
We need to evaluate:
\(\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} = \frac{\alpha^3 + \beta^3}{\alpha \beta}\).
Using identity \(\alpha^3 + \beta^3 = (\alpha + \beta)^3 – 3\alpha \beta (\alpha + \beta)\):
\(\alpha^3 + \beta^3 = \left(\frac{7}{2}\right)^3 – 3(2)\left(\frac{7}{2}\right) = \frac{343}{8} – 21 = \frac{343 – 168}{8} = \frac{175}{8}\).
Therefore:
\(\frac{\alpha^3 + \beta^3}{\alpha \beta} = \frac{175/8}{2} = \frac{175}{16}\).
Sum of roots: \(\alpha + \beta = \frac{7}{2}\)
Product of roots: \(\alpha \beta = \frac{4}{2} = 2\)
We need to evaluate:
\(\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} = \frac{\alpha^3 + \beta^3}{\alpha \beta}\).
Using identity \(\alpha^3 + \beta^3 = (\alpha + \beta)^3 – 3\alpha \beta (\alpha + \beta)\):
\(\alpha^3 + \beta^3 = \left(\frac{7}{2}\right)^3 – 3(2)\left(\frac{7}{2}\right) = \frac{343}{8} – 21 = \frac{343 – 168}{8} = \frac{175}{8}\).
Therefore:
\(\frac{\alpha^3 + \beta^3}{\alpha \beta} = \frac{175/8}{2} = \frac{175}{16}\).
मूलों का योग \(\alpha + \beta = \frac{7}{2}\), गुणनफल \(\alpha \beta = 2\)
\(\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} = \frac{\alpha^3 + \beta^3}{\alpha \beta}\)
\(\alpha^3 + \beta^3 = (\alpha + \beta)^3 – 3\alpha\beta(\alpha + \beta) = \frac{343}{8} – 21 = \frac{175}{8}\)
अतः मान = \(\frac{175/8}{2} = \frac{175}{16}\)।
\(\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} = \frac{\alpha^3 + \beta^3}{\alpha \beta}\)
\(\alpha^3 + \beta^3 = (\alpha + \beta)^3 – 3\alpha\beta(\alpha + \beta) = \frac{343}{8} – 21 = \frac{175}{8}\)
अतः मान = \(\frac{175/8}{2} = \frac{175}{16}\)।