Question

If the zeroes of the polynomial \(p(x) = 2x^2 - 7x + 6\) are \(\alpha\) and \(\beta\), then the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\) is

• A. \(\frac{7}{2}\)
• B. \(\frac{6}{7}\)
• C. \(\frac{-7}{6}\)
• D. \(\frac{7}{6}\)

Hint:

Directly memorize the simplified relation for the sum of reciprocals of roots:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = -\frac{b}{c} \]
For our polynomial, this yields \(-\frac{-7}{6} = \frac{7}{6}\) instantly, eliminating the need to compute sum and product separately.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a quadratic polynomial \(p(x) = 2x^2 - 7x + 6\) with roots or zeroes \(\alpha\) and \(\beta\).
We need to determine the numerical value of the expression \(\frac{1}{\alpha} + \frac{1}{\beta}\).

Step 2: Key Formula or Approach:
For any standard quadratic polynomial of the form \(ax^2 + bx + c\) having zeroes \(\alpha\) and \(\beta\), the relationship between the zeroes and coefficients is given by:
1. Sum of zeroes:
\[ \alpha + \beta = -\frac{b}{a} \]
2. Product of zeroes:
\[ \alpha\beta = \frac{c}{a} \]
We can simplify the given expression algebraically before substituting these values:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha\beta} = \frac{\alpha + \beta}{\alpha\beta} \]

Step 3: Detailed Explanation:

• Compare the given polynomial \(p(x) = 2x^2 - 7x + 6\) with the general standard form \(ax^2 + bx + c\).

• From the comparison, we identify the coefficients as:
\(a = 2\)
\(b = -7\)
\(c = 6\)

• Now, let us calculate the sum of the zeroes using the formula \(\alpha + \beta = -\frac{b}{a}\):
\[ \alpha + \beta = -\frac{-7}{2} = \frac{7}{2} \]

• Next, let us calculate the product of the zeroes using the formula \(\alpha\beta = \frac{c}{a}\):
\[ \alpha\beta = \frac{6}{2} = 3 \]

• The expression we need to evaluate is \(\frac{1}{\alpha} + \frac{1}{\beta}\).

• As established, simplifying this expression gives:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} \]

• Now, substitute the calculated values of the sum and product of the zeroes into this expression:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\frac{7}{2}}{3} \]

• Simplifying the fraction division:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{7}{2 \times 3} = \frac{7}{6} \]


Step 4: Final Answer:
The value of \(\frac{1}{\alpha} + \frac{1}{\beta}\) is \(\frac{7}{6}\). Hence, the correct option is (D).