Question

If \( \alpha \) and \( \beta \) are the roots of \( 2x^2 - 4x + 1 = 0 \). Then \( \frac{1}{\alpha^2\beta} + \frac{1}{\alpha\beta^2} = \)

• A. -4
• B. 8
• C. 1
• D. 0

Hint:

1. For \(2x^2-4x+1=0\): Sum of roots \(\alpha+\beta = -(-4)/2 = 2\). Product of roots \(\alpha\beta = 1/2\). 2. Simplify the target expression: \( \frac{1}{\alpha^2\beta} + \frac{1}{\alpha\beta^2} \) Common denominator is \(\alpha^2\beta^2 = (\alpha\beta)^2\). So, \( \frac{\beta}{\alpha^2\beta^2} + \frac{\alpha}{\alpha^2\beta^2} = \frac{\alpha+\beta}{(\alpha\beta)^2} \). 3. Substitute values from step 1: \( \frac{2}{(1/2)^2} = \frac{2}{1/4} = 2 \times 4 = 8 \).

Answer 1

The Correct Option is B

Concept: For a quadratic equation \(ax^2+bx+c=0\), if \(\alpha\) and \(\beta\) are its roots, then:
Sum of roots: \(\alpha + \beta = -b/a\)
Product of roots: \(\alpha\beta = c/a\) We need to simplify the given expression in terms of \(\alpha+\beta\) and \(\alpha\beta\). Step 1: Find the sum and product of the roots for the given equation The equation is \( 2x^2 - 4x + 1 = 0 \). Here, \(a=2\), \(b=-4\), \(c=1\). Sum of roots: \(\alpha + \beta = -(-4)/2 = 4/2 = 2\). Product of roots: \(\alpha\beta = 1/2\). Step 2: Simplify the expression to be evaluated The expression is \( \frac{1}{\alpha^2\beta} + \frac{1}{\alpha\beta^2} \). To add these fractions, find a common denominator. The common denominator is \(\alpha^2\beta^2\). \[ \frac{1}{\alpha^2\beta} + \frac{1}{\alpha\beta^2} = \frac{\beta}{\alpha^2\beta^2} + \frac{\alpha}{\alpha^2\beta^2} \] \[ = \frac{\beta + \alpha}{\alpha^2\beta^2} \] This can be written as: \[ = \frac{\alpha + \beta}{(\alpha\beta)^2} \] Step 3: Substitute the values of \(\alpha+\beta\) and \(\alpha\beta\) We found: \(\alpha + \beta = 2\) \(\alpha\beta = 1/2\) Substitute these into the simplified expression: \[ \frac{2}{\left(\frac{1}{2}\right)^2} \] First, calculate \((\frac{1}{2})^2\): \[ \left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2} = \frac{1}{4} \] Now the expression becomes: \[ \frac{2}{\frac{1}{4}} \] To divide by a fraction, multiply by its reciprocal: \[ 2 \times \frac{4}{1} = 2 \times 4 = 8 \] The value of the expression is 8.