Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Application of Derivatives

NDA 2018 · Mathematics · Application of Derivatives

What is the maximum value of 16 θ-12 ^2 θ ?

What is the maximum value of 16sin⁡θ−12sin⁡2θ16 \sin \theta-12 \sin ^2 \theta ?

Options

  1. A34\frac{3}{4}
  2. B43\frac{4}{3}
  3. C163\frac{16}{3}
  4. D4

Correct answer

C. 163\frac{16}{3}

Step-by-step solution

f(θ)=16sin⁡θ−12sin⁡2θf′(θ)=16cos⁡θ−24sin⁡θcos⁡θf′(θ)=0⇒cos⁡θ(16−24sin⁡θ)=0⇒cos⁡θ=0 or  16−24sin⁡θ=0⇒θ=π2 or sin⁡θ=23∴f(π2)=16sin⁡π2−12sin⁡2π2=16−12=4and f(sin⁡θ=23)=16(23)−12(23)2=323−489=323−163=163\begin{aligned} & f(\theta)=16 \sin \theta-12 \sin ^2 \theta \\ & f^{\prime}(\theta)=16 \cos \theta-24 \sin \theta \cos \theta \\ & f^{\prime}(\theta)=0 \Rightarrow \cos \theta(16-24 \sin \theta)=0 \\ & \Rightarrow \cos \theta=0 \text{\text{ or } } 16-24 \sin \theta=0 \\ & \Rightarrow \theta=\frac{\pi}{2} \text{ or } \sin \theta=\frac{2}{3} \\ & \therefore \mathrm{f}\left(\frac{\pi}{2}\right)=16 \sin \frac{\pi}{2}-12 \sin ^2 \frac{\pi}{2}=16-12=4 \\ & \text{and } \mathrm{f}\left(\sin \theta=\frac{2}{3}\right)=16\left(\frac{2}{3}\right)-12\left(\frac{2}{3}\right)^2 \\ & =\frac{32}{3}-\frac{48}{9}=\frac{32}{3}-\frac{16}{3}=\frac{16}{3}\end{aligned}
Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

Passage: Let S and T be the sets where f(x)=(x^3)/(3)-(5x^2)/(2)+6x+7 decreases and… NDA 2026What is the length of the longest interval in which the function f(x)=2 ^2 x-1 is… NDA 2026Passage: Let S and T be the sets where f(x)=(x^3)/(3)-(5x^2)/(2)+6x+7 decreases and… NDA 2026For the following three (03) items: Let y = f(x) = x ^ -1 x 1 - x^2 + 1 - x^2 . What is… NDA 2025Consider the following statements regarding the function f(x) = (1)/(x - 5)… NDA 2025Consider the following for the two (02) items that follow: Let ABC be a triangle… NDA 2025Consider the following for the two (02) items that follow: Let ABC be a triangle… NDA 2025A wire of length 20 cm is to be bent into a rectangle. Which of the following statements… NDA 2025Consider the following for the two (02) items that follow: Let the function f(x) = x^2 +… NDA 2025For the following two (02) items: Consider the function f(x) = 1 - [3] (x-1)^2 . The… NDA 2025Consider the following for the two (02) items that follow: The slope of the tangent to… NDA 2025For the following three (03) items: Consider the function f(x) = x|x| . Consider the… NDA 2025Consider the following statements: Statement-I: The function f(x) = (x^3 + 128)/(x) has a… NDA 2025Paragraph: Let f(x)= 2 x+x on [(-π)/(2), (π)/(2)] Question: What is the greatest value of… NDA 2024Let f(x)=(x)/( x) ;(x > 1) Consider the following statements: 1. f(x) is increasing in… NDA 2024Paragraph: Let f(x)= 2 x+x on [(-π)/(2), (π)/(2)] Question: What is the least value of… NDA 2024 Full Application of Derivatives list Application of Derivatives JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs