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Bacterial increases at the rate proportional to the number present. If original number M doubles in 3 hours, then number of bacteria will be 4M in - Mathematics and Statistics

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प्रश्न

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

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

विकल्प

  • 4 hours

  • 6 hours

  • 8 hours

  • 10 hours

MCQ
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उत्तर

6 hours

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अध्याय 2.6: Differential Equations - MCQ

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संबंधित प्रश्न

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.


Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]


Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]


\[\frac{dy}{dx} = x^5 + x^2 - \frac{2}{x}, x \neq 0\]

\[\frac{dy}{dx} + 2x = e^{3x}\]

\[\frac{1}{x}\frac{dy}{dx} = \tan^{- 1} x, x \neq 0\]

\[\frac{dy}{dx} = x^5 \tan^{- 1} \left( x^3 \right)\]

C' (x) = 2 + 0.15 x ; C(0) = 100


\[\frac{dy}{dx} = \sin^2 y\]

(1 + x2) dy = xy dx


\[\frac{dy}{dx} = \left( e^x + 1 \right) y\]

(ey + 1) cos x dx + ey sin x dy = 0


(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0


(y2 + 1) dx − (x2 + 1) dy = 0


In a culture the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present.


Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.


\[\frac{dy}{dx}\cos\left( x - y \right) = 1\]

(x + y) (dx − dy) = dx + dy


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

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

\[2xy\frac{dy}{dx} = x^2 + y^2\]

\[x^2 \frac{dy}{dx} = x^2 - 2 y^2 + xy\]

(x + 2y) dx − (2x − y) dy = 0


\[x\frac{dy}{dx} = y - x \cos^2 \left( \frac{y}{x} \right)\]

Solve the following initial value problem:-

\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]


Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]


Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]


Solve the following initial value problem:-

\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]


The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.


If the interest is compounded continuously at 6% per annum, how much worth Rs 1000 will be after 10 years? How long will it take to double Rs 1000?


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.


Find the equation of the curve passing through the point \[\left( 1, \frac{\pi}{4} \right)\]  and tangent at any point of which makes an angle tan−1  \[\left( \frac{y}{x} - \cos^2 \frac{y}{x} \right)\] with x-axis.


Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.


The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.


The slope of the tangent at each point of a curve is equal to the sum of the coordinates of the point. Find the curve that passes through the origin.


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


The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is


The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when


The differential equation satisfied by ax2 + by2 = 1 is


The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]


Solve the following differential equation : \[y^2 dx + \left( x^2 - xy + y^2 \right)dy = 0\] .


Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.


Find the differential equation whose general solution is

x3 + y3 = 35ax.


Solve the following differential equation.

y2 dx + (xy + x2 ) dy = 0


State whether the following is True or False:

The degree of a differential equation is the power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any.


x2y dx – (x3 + y3) dy = 0


Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0


Solve the following differential equation

`y log y ("d"x)/("d"y) + x` = log y


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`


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