Question

Assertion (A) : The polynomial \(p(y) = y^2 + 4y + 3\) has two zeroes.
Reason (R) : A quadratic polynomial can have at most two zeroes.

• A. Both (A) and (R) are true and (R) is the correct explanation of (A).
• B. Both A and R are true but R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true.

Hint:

While a quadratic can have "at most" two zeroes, they may be distinct, equal (one repeated zero), or non-real (no real zeroes).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A quadratic polynomial is of the form \(ay^2 + by + c\). The number of zeroes of a polynomial is determined by its degree.
Step 2: Key Formula or Approach:
1. Degree of \(p(y) = y^2 + 4y + 3\) is 2. 2. Check the discriminant \(D = b^2 - 4ac\) to see if zeroes are real.
Step 3: Detailed Explanation:
1. Evaluating Assertion (A): For \(y^2 + 4y + 3\), the zeroes are found by factoring: \((y+3)(y+1) = 0\), which gives \(y = -3\) and \(y = -1\). It has exactly two real zeroes. Assertion (A) is true. 2. Evaluating Reason (R): It is a fundamental theorem of algebra that a polynomial of degree \(n\) has at most \(n\) zeroes. For a quadratic (\(n=2\)), it has at most 2 zeroes. Reason (R) is true. 3. Connection: Since \(p(y)\) is a quadratic polynomial, the reason correctly explains why it is limited to (and in this case, has) two zeroes.
Step 4: Final Answer:
Both Assertion and Reason are true, and Reason is the correct explanation.