Question

Assertion: If (x − 1) is a factor of p(x) = x² + kx + 1, then k = −2.
Reason: If (x − a) is a factor of p(x), then p(a) = 0.

(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true and but Reason is not a correct explanation of assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

Answer 1

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

Assertion:
Let:
$p\left(x\right)=x^2+kx+1$
Now,
$x-1=0\Rightarrow x=1$
$$\begin{gathered} \left(x-1\right)\text{is a factor of}p\left(x\right). \\ \Rightarrow p\left(1\right)=0 \\ \Rightarrow 1^2+k\times 1+1=0 \\ \Rightarrow 1+k+1=0 \\ \Rightarrow k+2=0 \\ \Rightarrow k=-2 \end{gathered}$$
Hence, Assertion is true.
Reason: If (x − a) is a factor of p(x), then p(a) = 0.
The given statement is true.
Therefore, both Assertion and Reason are true and Reason is a correct explanation of Assertion.