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Differential Equations and Modeling Applied Maths - Differential Equations and Modeling

CBSE - STD 12 Science - Applied Maths (CBSE - English Medium). 66 questions in this chapter.

Question types

Differential Equations and Modeling question types

66 questions across 5 question groups — pick any mix to generate a Applied Maths paper with step-by-step answer keys.

66
Questions
5
Question groups
5
Question types
Sample Questions

Differential Equations and Modeling questions

One sample from each question group in this chapter. Select any group above to see the full set with answer keys.

Q 1MCQ1 Mark
Systems that exhibit exponential growth have a constant doubling time, which is given by:
  • ✓
    $\frac{\log 2}{k}$
  • B
    $\frac{k}{\log 2}$
  • C
    k. $\log 2$
  • D
    None of these

Answer: A.

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Q 2MCQ1 Mark
Systems that exhibit exponential decay have a constant half-life, which is given by:
  • ✓
    $\frac{\log 2}{k}$
  • B
    $\frac{k}{\log 2}$
  • C
    $k \cdot \log 2$
  • D
    None of these

Answer: A.

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Q 3MCQ1 Mark
Systems that exhibit exponential decay follow a model of the form:
  • A
    $y=y_0 e^{k t}$
  • ✓
    $y=y_0 e^{-k t}$
  • C
    $y=y_0 e^{k / t}$
  • D
    None of these

Answer: B.

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Q 4MCQ1 Mark
Systems that exhibit exponential growth increase according to the mathematical model $y=y_0 e^{k t}$ where $y_0$ represents the initial state of the system and k > 0 is a constant, is called the.........constant
  • ✓
    Growth
  • B
    Decay
  • C
    Simple
  • D
    Normal

Answer: A.

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Q 5MCQ1 Mark
The order and degree (if defined) of differential equation $y^{\prime \prime}+y^2+e^{y^{\prime}}=0$ are:
  • A
    3,1
  • ✓
    3, not defined
  • C
    not defined, 3
  • D
    1,3

Answer: B.

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Q 62 Marks Questions2 Marks
A cell culture in a biology lab currently holds 1 million cells. The cells have a constant continuous birth rate of 1.5% and death rate of 0.5% per hour. Cells are extracted from the culture for an experiment at the rate of 5000 per hour. How many cells will be in the culture 10 hours from now?
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Q 113 Marks Question3 Marks
Doctors have shown certain drugs leave a person's bloodstream at a rate that is proportional to the amount present. In an experiment a patient is injected with 450 mg of a substance. Seven hours later it is found that 50 mg of the substance remains. Assuming the proportional model is correct for the particular substance.
(a) Express the differential equation that models this scenario.
(b) Find the time constant, k.(log 9 = 2.197225).
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Q 123 Marks Question3 Marks
The surface area of a balloon being inflated changes at a constant rate. If initially, its radius is 3 units and after 2 seconds, it is 5 units, find the radius after t seconds.
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Q 133 Marks Question3 Marks
The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
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Q 165 Marks Questions5 Marks
The population of a village increases at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20,000 in 1999 and 25,000 in the year 2004, what was its population in 2009 ?
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Q 175 Marks Questions5 Marks
It is given that radium decomposes at a rate proportional to the amount present. If P% of the original amount of radium disappears in I years, what percentage of it will remain after 2l years?
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Q 185 Marks Questions5 Marks
Suppose the growth of a population is proportional to the number present. If the population of a colony doubles in 25 days, in how many days will the population become triple ?
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Q 195 Marks Questions5 Marks
A radio active substance has a half life of h days. Find a formula for its mass m in terms of t the time, if the initial mass is $m_0$. What is the initial decay rate ?
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Q 205 Marks Questions5 Marks
Assume that a spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally in 3 mm and 1 hour later has been reduced to 2 mm, find an expression for the radius of the rain drop at any time.
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Q 21Case study (4 Marks)4 Marks
Mr. Lal was 75 year old businessman. He had two sons and three daughters. On the silver jubilee of his company, he gifted an equal amount of money among his children. One of his daughters buys jewellery from the gifted money whereas the other daughter invested his money in bank.
It is known that if the interest is compounded continuously, the principal changes at the rate equal to the product of the rate of bank interest per annum and principal.

Image

Based on the above given information answer the following questions:
Q.1. If P denotes the principal at time t and the rate of interest be r% per annum compounded continuously, then the differential equation that models this scenario is:
(A) $\frac{d P}{d t}=\frac{P r}{100}$
(B) $\frac{d P}{d t}=\frac{100}{P_r}$
(C) $\frac{d P}{d t}=P r$
(D) $\frac{d t}{d P}=\frac{P r}{100}$
Q.2. The particular solution of the associated differential equation is:
(A) $\log \left(\frac{P}{P_0}\right)=\frac{100}{r t}$
(B) $\log \left(\frac{P}{P_0}\right)=\frac{r t}{100}$
(C) $\log \left(\frac{P_0}{P}\right)=\frac{100}{r t}$
(D) $\log \left(\frac{P_0}{P}\right)=\frac{r t}{100}$

Q.3. If the interest is compounded continuously at 5% per annum, in how many years will ₹100 double itself? $\left(\log _e 2=0.6931\right)$
(A) 13.86 (B) 13 (C) 14 (D) 14.82

Q.4. At what rate interest rate will₹ 100 double itself in 10 years? $\left(\log _e 2=0.6931\right)$
(A) 6% (B) 7% (C) 6.93% (D) 7.36%
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Q 22Case study (4 Marks)4 Marks
Read the following text and answer the following questions on the basis of the same:
Riya a biologist runs an experiment in the laboratory involving a culture of bacteria. She is doing this experiment first time. She notices that the mass of the bacteria in culture increases exponentially. The bacteria count in the culture is 1,00,000. The rate of growth of bacteria is proportional to the number of bacteria present in the culture. Also, the number of bacteria is increased by 10% in 2 hours.
Image

Q.1. If y be the number of bacteria at time t, then the differential equation that models this scenario is
(A) $\frac{d y}{d x}=k y$
(B) $\frac{d y}{d t}=k y$
(C) $\frac{d y}{d t}=k t$
(D) $\frac{d y}{d x}=k t$

Q.2. The general solution of the associated differential equation is:
(A) $\log y=k t+c$
(B) $\log x=k t+c$
(C) $\log y=k x+c$
(D) $\log y=k t$

Q.3. The value of integration constant 'c' is:
(A) 1,00,000
(B) log 1000
(C) log 1 ,00,000
(D) 1,10,000

Q.4. Count of bacteria after 2 hours:
(A) 1,00,000
(B) log 1000
(C) log 1,00,000
(D) 1,10,000
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