Question

Assertion (A): If on dividing the polynomial $$p(x)=x^{2}-3ax+3a-7$$ by $$(x+1)$$, we get 6 as remainder then $$a=4$$.

Reason (R): When a polynomial $$p(x)$$ is divided by $$(x-\alpha)$$ then the remainder is $$p(\alpha)$$.

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. By the remainder theorem, division by $$x+1=x-(-1)$$ gives remainder $$p(-1)$$. Setting this equal to 6:

$$p(-1)=(-1)^{2}-3a(-1)+3a-7=6$$

$$1+3a+3a-7=6\Rightarrow 6a=12\Rightarrow a=2$$

Thus, the assertion's value $$a=4$$ is incorrect.

Step 2 – Reason: R is true. It is the remainder theorem, applied here with $$\alpha=-1$$.