Question

Assertion (A): A monic cubic polynomial having 3, 2, –1 as its zeros is $$p(x)=x^{3}+4x^{2}-x+6$$.

Reason (R): The monic cubic polynomial having $$\alpha,\beta,\gamma$$ as its zeros is given by $$x^{3}-(\alpha+\beta+\gamma)x^{2}+(\alpha\beta+\beta\gamma+\gamma\alpha)x-(\alpha\beta\gamma)$$.

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Assertion (A) is false and Reason (R) is true.

Explanation:

Step 1 – Assertion: A is false. Using the given formula for zeros $$3,2,-1$$:

$$\alpha+\beta+\gamma=3+2-1=4$$

$$\alpha\beta+\beta\gamma+\gamma\alpha=6-2-3=1$$

$$\alpha\beta\gamma=3(2)(-1)=-6$$

Therefore, the polynomial is $$x^{3}-4x^{2}+x+6$$, not the polynomial stated in the Assertion.

Step 2 – Reason: R is true. It is the standard formula for the monic cubic polynomial with zeros $$\alpha,\beta,\gamma$$.