JEE Mains · Maths · STD 12 - 9. differential equations
The temperature \(\mathrm{T}(\mathrm{t})\) of a body at time \(\mathrm{t}=0\) is \(160^{\circ}\) \(\mathrm{F}\) and it decreases continuously as per the differential equation \(\frac{\mathrm{dT}}{\mathrm{dt}}=-\mathrm{K}(\mathrm{T}-80)\), where \(\mathrm{K}\) is positive constant. If \(\mathrm{T}(15)=120^{\circ} \mathrm{F}\), then \(\mathrm{T}(45)\) is equal to ...........
- A \(85^{\circ} \mathrm{F}\)
- B \(95^{\circ} \mathrm{F}\)
- C \(90^{\circ} \mathrm{F}\)
- D \(80^{\circ} \mathrm{F}\)
Answer & Solution
Correct Answer
(C) \(90^{\circ} \mathrm{F}\)
Step-by-step Solution
Detailed explanation
\( \frac{\mathrm{dT}}{\mathrm{dt}}=-\mathrm{k}(\mathrm{T}-80) \) \( \int_{160}^{\mathrm{T}} \frac{\mathrm{dT}}{(\mathrm{T}-80)}=\int_0^{\mathrm{t}}-\mathrm{Kdt} \) \( {[\ln |\mathrm{T}-80|]_{160}^{\mathrm{T}}=-\mathrm{kt}} \) \( \ln |\mathrm{T}-80|-\ln 80=-\mathrm{kt}\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(\vec{a}=\hat{i}+\hat{j}+2 \hat{k}, \vec{b}=2 \hat{i}-3 \hat{j}+\hat{k}\) and \(\overrightarrow{ c }=\hat{ i }-\hat{ j }+\hat{ k }\) be three given vectors. Let \(\vec{v}\) be a vector in the plane of \(\vec{a}\) and \(\overrightarrow{ b }\) whose projection on \(\overrightarrow{ c }\) is \(\frac{2}{\sqrt{3}}\). If \(\overrightarrow{ v } . \hat{ j }=7\), then \(\overrightarrow{ v } \cdot(\hat{ i }+\hat{ k })\) is equal toJEE Mains 2022 Medium
- Let \(\overrightarrow{ a }=2 \hat{ i }+\hat{ j }+\hat{ k }\), and \(\overrightarrow{ b }\) and \(\overrightarrow{ c }\) be two nonzero vectors such that \(|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}| \quad\) and \(\vec{b} \cdot \vec{c}=0\). Consider the following two statement: \((A)\) \(|\overrightarrow{ a }+\lambda \overrightarrow{ c }| \geq|\overrightarrow{ a }|\) for all \(\lambda \in R\). \((B)\) \(\overrightarrow{ a }\) and \(\overrightarrow{ c }\) are always parallelJEE Mains 2023 Hard
- Let \(3,7,11,15, \ldots, 403\) and \(2,5,8,11, \ldots, 404\) be two arithmetic progressions. Then the sum, of the common terms in them, is equal to ...........JEE Mains 2024 Hard
- Let \(\mathrm{A}\) and \(\mathrm{B}\) be two square matrices of order \(3\) such that \(|A|=3\) and \(|B|=2\). Then \(\left|\mathrm{A}^{\mathrm{T}} \mathrm{A}(\operatorname{adj}(2 \mathrm{~A}))^{-1}(\operatorname{adj}(4 \mathrm{~B}))(\operatorname{adj}(\mathrm{AB}))^{-1} \mathrm{AA}^{\mathrm{T}}\right|\) is equal to :JEE Mains 2024 Hard
- If the equation \(\cos ^{4} \theta+\sin ^{4} \theta+\lambda=0\) has real solutions for \(\theta,\) then \(\lambda\) lies in the intervalJEE Mains 2020 Hard
- Let \(z\) and \(w\) be two complex numbers such that \(w=z \bar{z}-2 z+2,\left|\frac{z+i}{z-3 i}\right|=1\) and \(\operatorname{Re}(w)\) has minimum value. Then, the minimum value of \(n \in N\) for which \(w ^{ n }\) is real, is equal to..........JEE Mains 2021 Hard
More PYQs from JEE Mains
- If the system of equations \( 11 x+y+\lambda z=-5 \) \( 2 x+3 y+5 z=3 \) \( 8 x-19 y-39 z=\mu\) has infinitely many solutions, then \(\lambda^4-\mu\) is equal to :JEE Mains 2024 Hard
- Let \(P\) be an arbitrary point having sum of the squares of the distance from the planes \(x + y + z =0, l x - nz =0\) and \(x -2 y + z =0\) equal to \(9 .\) If the locus of the point \(P\) is \(x ^{2}+ y ^{2}+ z ^{2}=9,\) then the value of \(l- n\) is equal to ...... .JEE Mains 2021 Hard
- If the function \(\mathrm{f}\) defined on \(\left(-\frac{1}{3}, \frac{1}{3}\right)\) by \(f(x)=\left\{\begin{array}{ll}{\frac{1}{x} \log _{e}\left(\frac{1+3 x}{1-2 x}\right)} & {, \text { when } x \neq 0} \\ {k} & {, \text { when } x=0}\end{array}\right.\) is continuous, then \(\mathrm{k}\) is equal toJEE Mains 2020 Hard
- If the shortest distance between the lines \(\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}\) and \(\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}\) is \(\frac{38}{3 \sqrt{5}} \mathrm{k}\) and \(\int_0^{\mathrm{k}}\left[\mathrm{x}^2\right] \mathrm{dx}=\alpha-\sqrt{\alpha}\), where \([\mathrm{x}]\) denotes the greatest integer function, then \(6 \alpha^3\) is equal to ...........JEE Mains 2024 Hard
- If the area of the region bounded by the curves \(y^2-2 y=-x, x+y=0\) is \(A\), then \(8 A\) is equal toJEE Mains 2023 Hard
- \(\mathop \smallint \limits_2^4 \frac{{\log {x^2}}}{{\log {x^2} + {\rm{log}}\left( {36 - 12x + {x^2}} \right)}}\;dx = \)JEE Mains 2015 Medium