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

Hint:

To check if Reason (R) explains Assertion (A), ask: "Does the truth of R guarantee the truth of A?"
Since a quadratic polynomial can have at most 2 zeroes (which includes 0 or 1 zero), it doesn't automatically mean our given polynomial MUST have 2 zeroes.
Therefore, R does not fully explain A, leading to option (B).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This is an Assertion-Reason type of question.
We need to evaluate the truth value of:
- Assertion (A): The quadratic polynomial $p(y) = y^2 + 4y + 3$ has two zeroes.
- Reason (R): A quadratic polynomial can have at most two zeroes.
Then, we determine if Reason (R) is the correct explanation for Assertion (A).

Step 2: Key Formula or Approach:
1. A quadratic polynomial of the form $ay^2 + by + c$ can have at most two real zeroes.
2. The number of real zeroes of a quadratic polynomial depends on its discriminant $D = b^2 - 4ac$.
- If $D \gt 0$, it has two distinct real zeroes.
- If $D = 0$, it has two equal real zeroes (one distinct real zero).
- If $D \lt 0$, it has no real zeroes.

Step 3: Detailed Explanation:

• Evaluate Assertion (A):
Let us find the zeroes of the given polynomial $p(y) = y^2 + 4y + 3$:
Set $p(y) = 0$:
\[ y^2 + 4y + 3 = 0 \]
Factorizing the quadratic expression by splitting the middle term:
\[ y^2 + 3y + y + 3 = 0 \]
\[ y(y + 3) + 1(y + 3) = 0 \]
\[ (y + 1)(y + 3) = 0 \]
This gives the zeroes as:
\[ y = -1 \quad \text{and} \quad y = -3 \]
Since we found two distinct real zeroes, Assertion (A) is true.

• Evaluate Reason (R):
The statement "A quadratic polynomial can have at most two zeroes" is a standard mathematical theorem.
A polynomial of degree $n$ has at most $n$ zeroes. Since a quadratic polynomial has degree 2, it can have at most 2 zeroes.
Thus, Reason (R) is a true statement.

• Determine if Reason (R) is the correct explanation of Assertion (A):
While it is true that a quadratic polynomial can have *at most* two zeroes, this general boundary does not guarantee that *every* quadratic polynomial has exactly two zeroes.
For instance, $p(y) = y^2 + 1$ has zero real zeroes, and $p(y) = y^2 + 2y + 1$ has only one distinct real zero.
The reason why this specific polynomial $p(y) = y^2 + 4y + 3$ has exactly two zeroes is because its discriminant is positive ($D = 4^2 - 4(1)(3) = 4 \gt 0$).
Therefore, Reason (R) is not the complete or correct explanation of Assertion (A).


Step 4: Final Answer:
Both A and R are true, but R is not the correct explanation of A.
Hence, option (B) is correct.