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Choose the correct alternative. Bacteria increases at the rate proportional to the number present. If the original number M doubles in 3 hours, then the number of bacteria will be 4M in

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प्रश्न

Choose the correct alternative.

Bacteria increases at the rate proportional to the number present. If the original number M doubles in 3 hours, then the number of bacteria will be 4M in

पर्याय

  • 4 hours

  • 6 hours

  • 8 hours

  • 10 hours

MCQ
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उत्तर

Bacteria increases at the rate proportional to the number present. If the original number M doubles in 3 hours, then the number of bacteria will be 4M in 6 hours

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पाठ 8: Differential Equation and Applications - Miscellaneous Exercise 8 [पृष्ठ १७२]

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बालभारती Mathematics and Statistics 1 (Commerce) [English] Standard 12 Maharashtra State Board
पाठ 8 Differential Equation and Applications
Miscellaneous Exercise 8 | Q 1.08 | पृष्ठ १७२

संबंधित प्रश्‍न

For the following differential equation verify that the accompanying function is a solution:

Differential equation Function
\[y = \left( \frac{dy}{dx} \right)^2\]
\[y = \frac{1}{4} \left( x \pm a \right)^2\]

\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]

\[\cos x\frac{dy}{dx} - \cos 2x = \cos 3x\]

C' (x) = 2 + 0.15 x ; C(0) = 100


\[\frac{dy}{dx} = \frac{1 - \cos 2y}{1 + \cos 2y}\]

x cos2 y  dx = y cos2 x dy


Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]


\[\frac{dr}{dt} = - rt, r\left( 0 \right) = r_0\]

\[\frac{dy}{dx} = \frac{\left( x - y \right) + 3}{2\left( x - y \right) + 5}\]

\[\frac{dy}{dx} = \frac{x + y}{x - y}\]

(x + 2y) dx − (2x − y) dy = 0


A population grows at the rate of 5% per year. How long does it take for the population to double?


The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.


In a simple circuit of resistance R, self inductance L and voltage E, the current `i` at any time `t` is given by L \[\frac{di}{dt}\]+ R i = E. If E is constant and initially no current passes through the circuit, prove that \[i = \frac{E}{R}\left\{ 1 - e^{- \left( R/L \right)t} \right\}.\]


Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of  radium to decompose?


Show that all curves for which the slope at any point (x, y) on it is \[\frac{x^2 + y^2}{2xy}\]  are rectangular hyperbola.


Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y\] sin x = 1, is


The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution


Choose the correct option from the given alternatives:

The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is


Determine the order and degree of the following differential equations.

Solution D.E.
y = 1 − logx `x^2(d^2y)/dx^2 = 1`

y2 dx + (xy + x2)dy = 0


`xy dy/dx  = x^2 + 2y^2`


Select and write the correct alternative from the given option for the question 

Differential equation of the function c + 4yx = 0 is


Verify y = `a + b/x` is solution of `x(d^2y)/(dx^2) + 2 (dy)/(dx)` = 0

y = `a + b/x`

`(dy)/(dx) = square`

`(d^2y)/(dx^2) = square`

Consider `x(d^2y)/(dx^2) + 2(dy)/(dx)`

= `x square + 2 square`

= `square`

Hence y = `a + b/x` is solution of `square`


Solve the following differential equation `("d"y)/("d"x)` = cos(x + y)

Solution: `("d"y)/("d"x)` = cos(x + y)    ......(1)

Put `square`

∴ `1 + ("d"y)/("d"x) = "dv"/("d"x)`

∴ `("d"y)/("d"x) = "dv"/("d"x) - 1`

∴ (1) becomes `"dv"/("d"x) - 1` = cos v

∴ `"dv"/("d"x)` = 1 + cos v

∴ `square` dv = dx

Integrating, we get

`int 1/(1 + cos "v")  "d"v = int  "d"x`

∴ `int 1/(2cos^2 ("v"/2))  "dv" = int  "d"x`

∴ `1/2 int square  "dv" = int  "d"x`

∴ `1/2* (tan("v"/2))/(1/2)` = x + c

∴ `square` = x + c


`d/(dx)(tan^-1  (sqrt(1 + x^2) - 1)/x)` is equal to:


Solve the differential equation

`y (dy)/(dx) + x` = 0


The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.


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