Question

Assertion :If $y=(1+x)(1+x^2)(1+x^4)...(1+x^{2^n})$, then $\frac{dy}{dx}$ at $x=0$ is $1$ Reason: $y=\frac{1-x^{2^{n+1}}}{1-x}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

We have,

$y=(1+x)(1+x^2)(1+x^4)...(1+2^{2^n})$

$=\frac{(1-x)(1+x)(1+x^2)(1+x^4)...(1+x^{2^n})}{1-x}=\frac{1-x^{2^{n+1}}}{1-x}$

$\Rightarrow \frac{dy}{dx}=\frac{(1-x).-2^{n+1}.x^{2^{n+1}}+(1-x^{2^{n+1}})}{{(1-x)}^2}$

$\therefore {\left[\frac{dy}{dx}\right]}_{x=0}=1$