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

To determine the correct answer, we evaluate both the assertion (A) and the reason (R) given in the question:

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

Let's analyze both statements.

Step-by-step Analysis:

1. Understanding the Polynomial:
   • The polynomial given is \(p(y) = y^2 + 4y + 3\), which is a quadratic polynomial.
   • A quadratic polynomial is of the form \(ax^2 + bx + c\), where \(a \neq 0\).
2. Number of Zeroes of a Quadratic Polynomial:
   • The fundamental theorem of algebra states that a polynomial of degree \(n\) has exactly \(n\) zeroes (considering multiplicity and within the complex number system).
   • For a quadratic polynomial (degree 2), it can have at most two zeroes.
3. Verifying Assertion (A):
   • We can find the zeroes of the polynomial using factoring: \(p(y) = y^2 + 4y + 3\).
   • Factoring gives \[p(y) = (y + 3)(y + 1)\].
   • Setting each factor to zero gives the zeroes: \(y + 3 = 0\) or \(y + 1 = 0\), hence, \(y = -3\) and \(y = -1\).
   • Thus, the assertion that the polynomial has two zeroes is correct.
4. Verifying Reason (R):
   • The statement that a quadratic polynomial can have at most two zeroes is indeed correct and aligns with the properties of quadratic polynomials.

Conclusion:

Both (A) and (R) are true. Additionally, (R) provides the correct explanation for why the polynomial \(p(y) = y^2 + 4y + 3\) has two zeroes, because quadratic polynomials always have at most two zeroes. Thus, the correct answer is:

Both (A) and (R) are true and (R) is the correct explanation of (A).