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

NDA 2022 · Mathematics · Differential Equations

A radioactive substance decays at a rate proportional to the amount...

A radioactive substance decays at a rate proportional to the amount of substance present. If half of the substance decays in 100 years, then what is the decay constant (proportionality constant)?

Options

  1. Aℓn2100\frac{\ell \mathrm{n} 2}{100}
  2. Bℓn5100\frac{\ell \mathrm{n} 5}{100}
  3. Cℓn10100\frac{\ell \mathrm{n} 10}{100}
  4. D2ℓn2100\frac{2 \ell \mathrm{n} 2}{100}

Correct answer

A. ℓn2100\frac{\ell \mathrm{n} 2}{100}

Step-by-step solution

dPdt=−kP∫1pdp=−k∫dt\begin{aligned} & \frac{\mathrm{dP}}{\mathrm{dt}}=-\mathrm{kP} \\ & \int \frac{1}{\mathrm{p}} \mathrm{dp}=-\mathrm{k} \int \mathrm{dt}\end{aligned}⇒ln⁡p=−kt+CWhen ℓ=0,P=8Pln⁡=C=put t=100 and C=ℓnpℓn(p/2)=−100k+lnp⁡⇒ln⁡n−ln⁡2=−100k=ℓnp⇒k=ℓn2100\begin{aligned} & \Rightarrow \ln \mathrm{p}=-\mathrm{kt}+\mathrm{C} \\ & \text{When } \ell=0, \mathrm{P}=8 \mathrm{P} \\ & \ln =\mathrm{C}= \\ & \text{put } \mathrm{t}=100 \text{ and } \mathrm{C}=\ell \mathrm{np} \\ & \ell \mathrm{n}(\mathrm{p} / 2)=-100 \mathrm{k}+\operatorname{lnp} \\ & \Rightarrow \ln \mathrm{n}-\ln 2=-100 \mathrm{k}=\ell \mathrm{np} \\ & \Rightarrow \mathrm{k}=\frac{\ell \mathrm{n} 2}{100}\end{aligned}$
Practice Differential Equations on Quantrex Academy →

More from Differential Equations

Passage: Consider the differential equation e^ x+y (dy)/(dx)=e^ x-y . Question: What is… NDA 2026What is the solution of the differential equation ((dy)/(dx))=p when y(0)=q ? NDA 2026Passage: Consider the differential equation e^ x+y (dy)/(dx)=e^ x-y . Question: What is… NDA 2026What is the degree of the differential equation ((d^2y)/(dx^2))^ (3)/(2) = ((dy)/(dx))^… NDA 2025If k is an arbitrary constant, then what is the general solution of the equation (x +… NDA 2025Consider the following statements:I. y = xe^ 2x is the solution of (dy)/(dx) = y (2 +… NDA 2025What are the order and degree respectively of the differential equation \ 2-((d y)/(d… NDA 2024Paragraph: Let y d x+(x-y^3) d y=0 be a differential equation. Question: What is the… NDA 2024The differential equation, representing the curve y=e^x(a x+b x) where a and b are… NDA 2024Paragraph: Let y d x+(x-y^3) d y=0 be a differential equation. Question: What are the… NDA 2024If (d y)/(d x)=2 e^x y^3, y(0)=(1)/(2) then what is 4 y^2 (2-e^x) equal to? NDA 2024Let y_1(x) and y_2(x) be two solutions of the differential equation (d y)/(d x)=x . If… NDA 2024A parabola pases through (1,2) and satisfies the differential equation (d y)/(d x)=(2… NDA 2023A parabola pases through (1,2) and satisfies the differential equation (d y)/(d x)=(2… NDA 2023What are the order and the degree respectively of the differential equation x^2((d^3… NDA 2023What is the differential equation of all parabolas of the type y^2=4 a(x-b) ? NDA 2023 Full Differential Equations list Differential Equations JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs