Question

If \( \alpha \) and \( \beta \) are two zeroes of a polynomial \( f(x) = px^2 - 2x + 3p \) and \( \alpha + \beta = \alpha\beta \), then value of p is :

• A. \( -\frac{2}{3} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{1}{3} \)
• D. \( -\frac{1}{3} \)

Hint:

When solving for \( p \), always ensure \( p \neq 0 \) since it is the leading coefficient of a quadratic equation.

Answer 1

The Correct Option is B

To determine the value of \( p \) given that \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( f(x) = px^2 - 2x + 3p \) and the condition \( \alpha + \beta = \alpha \beta \), we can make use of the properties of polynomials and their roots.

The sum and product of the roots (zeroes) of a quadratic polynomial \( ax^2 + bx + c = 0 \) are given by:

• \(\alpha + \beta = -\frac{b}{a}\)
• \(\alpha \beta = \frac{c}{a}\)

For the given polynomial \( f(x) = px^2 - 2x + 3p \), we identify the coefficients: \( a = p \), \( b = -2 \), \( c = 3p \).

Thus, we find:

• \(\alpha + \beta = -\frac{-2}{p} = \frac{2}{p}\)
• \(\alpha \beta = \frac{3p}{p} = 3\)

According to the problem, \( \alpha + \beta = \alpha \beta \). Therefore, we set:

\(\frac{2}{p} = 3\)

Solving for \( p \):

\(\frac{2}{p} = 3\)
\(2 = 3p\)
\(p = \frac{2}{3}\)

Thus, the value of \( p \) is \(\frac{2}{3}\).

The correct answer is:

• \(\frac{2}{3}\)