Question

If \(\alpha\) and \(\beta\) are two zeroes of the quadratic polynomial \(p(x) = x^2 - 11x + 30\), then \(\frac{1}{\alpha} + \frac{1}{\beta}\) is equal to :

• A. \(\frac{30}{11}\)
• B. \(\frac{11}{30}\)
• C. \(-\frac{11}{30}\)
• D. \(-\frac{30}{11}\)

Hint:

Whenever you are asked to evaluate symmetric expressions of zeroes like \(\frac{1}{\alpha} + \frac{1}{\beta}\) or \(\alpha^2 + \beta^2\), always try to express them in terms of \(\alpha + \beta\) and \(\alpha\beta\).
This avoids the need to find the individual roots \(\alpha\) and \(\beta\) of the quadratic equation, which can sometimes be complex or irrational, thereby saving significant calculation time.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question requires us to find the value of the expression \(\frac{1}{\alpha} + \frac{1}{\beta}\), where \(\alpha\) and \(\beta\) are the zeroes of the given quadratic polynomial \(p(x) = x^2 - 11x + 30\).

Step 2: Key Formula or Approach:
For a general quadratic polynomial \(ax^2 + bx + c\) with zeroes \(\alpha\) and \(\beta\), we have the following relationships between the coefficients and the zeroes:
Sum of zeroes:
\[ \alpha + \beta = -\frac{b}{a} \]
Product of zeroes:
\[ \alpha\beta = \frac{c}{a} \]
The algebraic expression \(\frac{1}{\alpha} + \frac{1}{\beta}\) can be simplified by taking the common denominator:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha\beta} \]

Step 3: Detailed Explanation:
1. Identify the coefficients of the given polynomial \(p(x) = x^2 - 11x + 30\).
By comparing with \(ax^2 + bx + c\), we get:
\[ a = 1, \quad b = -11, \quad c = 30 \]
2. Calculate the sum of the zeroes using the formula:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-11}{1} = 11 \]
3. Calculate the product of the zeroes using the formula:
\[ \alpha\beta = \frac{c}{a} = \frac{30}{1} = 30 \]
4. Substitute these values into the simplified expression for \(\frac{1}{\alpha} + \frac{1}{\beta}\):
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{11}{30} \]
This gives the final value of the expression as \(\frac{11}{30}\).

Step 4: Final Answer:
The value of \(\frac{1}{\alpha} + \frac{1}{\beta}\) is \(\frac{11}{30}\), which matches option (B).