Question

Assertion (A): Zeroes of a polynomial \(p(x) = x^2 − 2x − 3\) are -1 and 3.
Reason (R): The graph of polynomial \(p(x) = x^2 − 2x − 3\) intersects the x-axis at (-1, 0) and (3, 0).

• A. Both, Assertion (A) and Reason (R) are true. Reason (R) explains Assertion (A) completely.
• B. Both, Assertion (A) and Reason (R) are true. Reason (R) does not explain Assertion (A).
• C. Assertion (A) is true but Reason (R) is false.
• D. Assertion (A) is false but Reason (R) is true.

Answer 1

The Correct Option is A

Step 1: Understanding the Assertion (A):

We are given the polynomial \( p(x) = x^2 - 2x - 3 \). To find the zeroes of the polynomial, we solve the equation \( x^2 - 2x - 3 = 0 \) by factoring it.
To factor the quadratic expression, we look for two numbers whose product is \( -3 \) (the constant term) and whose sum is \( -2 \) (the coefficient of \( x \)). These numbers are \( -3 \) and \( 1 \), because:
\[ (-3) \times 1 = -3 \quad \text{and} \quad (-3) + 1 = -2 \] So, we can factor the quadratic as:
\[ x^2 - 2x - 3 = (x - 3)(x + 1) = 0 \] To find the roots (zeroes) of the polynomial, we set each factor equal to zero:
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] \[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \] Thus, the zeroes of the polynomial are \( x = 3 \) and \( x = -1 \). This matches the assertion that the zeroes of the polynomial are \( -1 \) and \( 3 \).

Step 2: Understanding the Reason (R):

The graph of a quadratic polynomial intersects the x-axis at the points where the value of the polynomial is zero. The x-intercepts of the graph correspond to the zeroes of the polynomial. Since we have already established that the zeroes of the polynomial \( p(x) = x^2 - 2x - 3 \) are \( x = -1 \) and \( x = 3 \), it follows that the graph of this polynomial intersects the x-axis at the points \( (-1, 0) \) and \( (3, 0) \). This confirms the reason that the graph of the polynomial intersects the x-axis at these points.

Step 3: Conclusion:

Both the assertion and the reason are true:
- The assertion is correct because we have verified that the zeroes of \( p(x) = x^2 - 2x - 3 \) are indeed \( -1 \) and \( 3 \).
- The reason is also true, as the graph of the polynomial intersects the x-axis at these points, corresponding to the zeroes of the polynomial.
Therefore, the correct answer is:
Both, Assertion (A) and Reason (R) are true. Reason (R) explains Assertion (A) completely.