Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Differentiation

NDA 2022 · Mathematics · Differentiation

If (d)/(d x)((1+x^4+x^8)/(1-x^2+x^4))=a x+b x^3 , then which one of...

If \frac{d}{d x}\left(\frac{1+x^4+x^8}{1-x^2+x^4}\right)=a x+b x^3$, then which one of the following is correct?

Options

  1. Aa=ba=b
  2. Ba=2ba=2 b
  3. Ca+b=0a+b=0
  4. D2a=b2 a=b

Correct answer

D. 2a=b2 a=b

Step-by-step solution

 Let f(x)=x8+x4+1x4−x2+1=x8+2x4+1−x4x4−x2+1=(x4+1)2−(x2)2(x4−x2+1)=(x4−x2+1)(x4+x2+1)x4−x2+1f(x)=x4+x2+1f′(x)=4x3+2x=ax+bx3⇒a=2,b=4⇒b=2a\begin{aligned} & \text{ Let } f(x)=\frac{x^8+x^4+1}{x^4-x^2+1} \\ & =\frac{x^8+2 x^4+1-x^4}{x^4-x^2+1}=\frac{\left(x^4+1\right)^2-\left(x^2\right)^2}{\left(x^4-x^2+1\right)} \\ & =\frac{\left(x^4-x^2+1\right)\left(x^4+x^2+1\right)}{x^4-x^2+1} \\ & f(x)=x^4+x^2+1 \\ & f^{\prime}(x)=4 x^3+2 x=a x+b x^3 \\ & \Rightarrow a=2, b=4 \Rightarrow b=2 \mathrm{a}\end{aligned}
Practice Differentiation on Quantrex Academy →

More from Differentiation

Passage: Let (e^y)^x-y=0 , where y is a function of x whose domain is (0,10] . Question… NDA 2026If [4] y =x+ x^2+4 , then what is x^2+4 \,(dy)/(dx) equal to ? NDA 2026Passage: Let (e^y)^x-y=0 , where y is a function of x whose domain is (0,10] . Question… NDA 2026For the following two (02) items: Consider the function f(x) = cases 4(5^x), & x What is… NDA 2025Consider the following for the two (02) items that follow: Let x = θ - θ and y = ^4θ… NDA 2025Consider the following for the two (02) items that follow: Let (x+y)^ p+q = x^p y^q… NDA 2025Consider the following for the two (02) items that follow: Let y = ^ -1 (x… NDA 2025For the following three (03) items: Let y = f(x) = x ^ -1 x 1 - x^2 + 1 - x^2 . If x = θ… NDA 2025Consider the following for the two (02) items that follow: Let (x+y)^ p+q = x^p y^q… NDA 2025For the following three (03) items: Let y = f(x) = x ^ -1 x 1 - x^2 + 1 - x^2 . What is… NDA 2025Consider the following for the two (02) items that follow: Let x = θ - θ and y = ^4θ… NDA 2025Paragraph: Let f:(-1,1) arrow R be a differentiable function with f(0)=-1 and f^ (0)=1 .… NDA 2024Let f(x) and g(x) be two functions such that g(x)=x-(1)/(x) and f o g(x)=x^3-(1)/(x^3) .… NDA 2024Paragraph: Let f:(-1,1) arrow R be a differentiable function with f(0)=-1 and f^ (0)=1 .… NDA 2024Let a function f be defined on ℝ-\ 0\ and 2 f(x)+f((1)/(x))=x+3 If f is differentiable… NDA 2023If f(x)=| | x| | where 0 < x < 1 , then what is f^ (0.5) equal to? NDA 2023 Full Differentiation list Differentiation JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs