Differential Equations

01

PYQ 2014

medium

mathematics ID: mht-cet-

General solution of the differential equation is

1

, where is a constant of integration.

2

, where is a constant of integration.

3

, where is a constant of integration.

4

, where is a constant of integration.

02

PYQ 2014

medium

mathematics ID: mht-cet-

The money invested in a company is compounded continuously. If \textrupee~200 invested today becomes \textrupee~400 in 6 years, then at the end of 33 years it will become

1

2

3

4

03

PYQ 2017

easy

mathematics ID: mht-cet-

The differential equation of all parabolas whose axis is is

1

2

3

4

04

PYQ 2017

hard

mathematics ID: mht-cet-

The particular solution of the differential equation, when is

1

2

3

4

05

PYQ 2018

easy

mathematics ID: mht-cet-

If then

1

4

2

2

3

1

4

0

06

PYQ 2018

medium

mathematics ID: mht-cet-

If then

1

2

3

4

07

PYQ 2020

medium

mathematics ID: mht-cet-

The rate of decay of mass of a certain substance at time is proportional to the mass at that instant. The time during which the original mass of gram will be left to gram is

1

2

3

4

08

PYQ 2020

medium

mathematics ID: mht-cet-

The joint equation of the lines through the origin trisecting angles in the first and third quadrant is

1

2

3

4

09

PYQ 2020

medium

mathematics ID: mht-cet-

The rate of growth of bacteria is proportional to the number present. If initially there were 1000 bacteria and the number doubles in 1 hour, then the number of bacteria after hours are (Given)

1

4646 approximately

2

5056 approximately

3

5656 approximately

4

approximately

10

PYQ 2020

medium

mathematics ID: mht-cet-

The rate of growth of bacteria is proportional to the bacteria present. If it is found that the number doubles in 3 hours, then the number of times the bacteria are increased in 6 hours is

1

6 times the original

2

4 times the original

3

8 times the original

4

5 times the original

11

PYQ 2020

medium

mathematics ID: mht-cet-

If the displacement of a particle at time is given by then the displacement of the particle when its velocity becomes zero is

1

14 units

2

4 units

3

0 units

4

2 units

12

PYQ 2020

medium

mathematics ID: mht-cet-

Among the given statements below (a) (b) (c) (d) .............. is a tautology.

1

(b)

2

(a)

3

(c)

4

(d)

13

PYQ 2020

medium

mathematics ID: mht-cet-

If the foot of perpendicular drawn from the origin to the plane is, then the equation of the plane is

1

2

3

4

14

PYQ 2020

medium

mathematics ID: mht-cet-

The rate of decay of mass of a certain substance at time is proportional to the mass at that instant. The time during which the original mass of gm will be left to gm is

1

2

3

4

15

PYQ 2020

medium

mathematics ID: mht-cet-

is the general solution of

1

2

3

4

16

PYQ 2020

medium

mathematics ID: mht-cet-

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

1

2

3

4

17

PYQ 2020

medium

mathematics ID: mht-cet-

The general solution of the differential equation is}

1

2

3

4

Official Solution

Correct Option: $(1)$

$Step 1: Solve the differential equation. The given differential equation is separable. We can write it in the form: After solving, the general solution of the differential equation is: Step 2: Conclusion. The correct answer is (A) .$

18

PYQ 2020

medium

mathematics ID: mht-cet-

$The differential equation whose solution is is$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Step 1: Take logarithm of the given solution. Given, taking natural logarithm on both sides: Step 2: Differentiate with respect to . Step 3: Eliminate the constant . From, we get Step 4: Substitute the value of .$

19

PYQ 2020

medium

mathematics ID: mht-cet-

$In a certain culture of bacteria, the rate of increase is proportional to the number present. It is found that the number doubles in 4 hours. Then the number of times the bacteria are increased in 12 hours is$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: Write the law of growth. Since the rate of increase is proportional to the number present, whose solution is Step 2: Use the doubling condition. Given that the population doubles in 4 hours, Step 3: Find the increase in 12 hours. Step 4: Conclusion. The number of bacteria increases times in 12 hours.$

20

PYQ 2020

medium

mathematics ID: mht-cet-

$If$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Step 1: Differentiate the given equation. We are given . Differentiating both sides with respect to gives: Step 2: Solve for . Solving for, we get: Step 3: Conclusion. Thus, the general solution of the differential equation is, corresponding to option (A).$

21

PYQ 2020

medium

mathematics ID: mht-cet-

$The differential equation of the family of lines having -intercept and -intercept is$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Step 1: Write the equation of the family of lines. A line with -intercept and -intercept is Step 2: Rewrite in explicit form. Step 3: Differentiate w.r.t. . Step 4: Differentiate again.$

22

PYQ 2020

medium

mathematics ID: mht-cet-

$Solution of the differential equation$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Step 1: Solve the differential equation. We are given the differential equation: This is a linear first-order differential equation. To solve it, we use the integrating factor method. The integrating factor is . Step 2: Multiply through by the integrating factor. Multiply the entire equation by : Simplifying: Step 3: Integrate both sides. Now, integrate both sides with respect to : Step 4: Conclusion. Thus, the solution is .$

23

PYQ 2020

medium

mathematics ID: mht-cet-

$The integrating factor of the differential equation is$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: Identifying and . Step 2: Checking exactness. Since, the equation is not exact. Step 3: Finding the integrating factor. This depends only on, so the integrating factor is Multiplying appropriately, the standard integrating factor simplifies to . Step 4: Conclusion. The integrating factor is .$

24

PYQ 2020

medium

mathematics ID: mht-cet-

$The solution of the differential equation$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Step 1: Rewriting the equation. We are given the equation . First, we rearrange it to separate the variables: We can now express this as: Step 2: Integrating both sides. By solving this equation using the method of separation of variables, we get the integral: After solving this, we obtain the solution . This corresponds to option (A). Step 3: Conclusion. Thus, the correct solution is, making option (A) the correct answer.$

25

PYQ 2020

medium

mathematics ID: mht-cet-

$The integral is$

1

$1$

2

3

$0$

4

Official Solution

Correct Option: $(3)$

$Step 1: Simplifying the integrand. We are asked to evaluate the integral First, use the identity to simplify the expression inside the logarithm. Step 2: Using symmetry. The integral has symmetry about, which allows us to conclude that the value of the integral is 0. Step 3: Conclusion. Thus, the value of the integral is 0, which makes option (C) the correct answer.$

26

PYQ 2020

medium

mathematics ID: mht-cet-

$The differential equation of the circles having their centres on the line and touching the -axis is$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: Writing the general equation of the circle. Since the centre lies on the line, let the centre be . As the circle touches the -axis, its radius is . Step 2: Differentiating with respect to . Step 3: Eliminating the parameter . Substitute into the original equation: Step 4: Conclusion. The required differential equation is$

27

PYQ 2020

medium

mathematics ID: mht-cet-

$The distance of the point from the plane is$

1

$units$

2

$units$

3

$units$

4

$units$

Official Solution

Correct Option: $(4)$

$Step 1: Using the point-to-plane distance formula. The distance of point from plane is Step 2: Substituting given values. Here,,,,, and . Step 3: Simplifying. Step 4: Conclusion. The required distance is units.$

28

PYQ 2020

medium

mathematics ID: mht-cet-

$If$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: Differentiating the given equation. We are given the equation . To differentiate this implicitly, we first rewrite the equation as: Now, differentiate both sides with respect to . Use the chain rule for each term. Step 2: Simplifying the derivative. After differentiating, we obtain the expression for : Step 3: Conclusion. Thus, the value of is, which makes option (B) the correct answer.$

29

PYQ 2020

medium

mathematics ID: mht-cet-

$The differential equation whose solution is (where and are arbitrary constants) is$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: Differentiate the given solution. The given solution is . To find the corresponding differential equation, take the first and second derivatives of with respect to . Step 2: Simplify the expression. After differentiating twice, you will arrive at the equation, which matches option (B). Step 3: Conclusion. The correct answer is (B) .$

30

PYQ 2020

medium

mathematics ID: mht-cet-

$Simplify$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Step 1: Understand the problem. The given expression is . First, simplify the terms that involve constants. We have . Step 2: Use trigonometric identities. Now, consider the trigonometric term . Since can be simplified using angle identities, the equation simplifies to: Step 3: Conclusion. Thus, the simplified result is .$

31

PYQ 2020

medium

mathematics ID: mht-cet-

$A body cools according to Newton’s law from to in 20 minutes. The temperature of the surroundings being, the temperature of the body after one hour is$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: Apply Newton’s law of cooling. According to Newton’s law, where . Step 2: Use the given data to find . At minutes, : Step 3: Find temperature after 60 minutes. Step 4: Final answer.$

32

PYQ 2020

medium

mathematics ID: mht-cet-

$The equation of the curve which passes through the point and has tangent with slope is$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Step 1: Write the differential equation. Step 2: Use substitution . Substituting, Step 3: Separate and integrate. Step 4: Replace . Step 5: Use the point . Step 6: Conclusion.$

33

PYQ 2020

medium

mathematics ID: mht-cet-

$The value of is$

1

2

3

4

Official Solution

Correct Option: $(3)$

$Step 1: Understanding the trigonometric inverses. We are given the sum of two inverse trigonometric functions: and . The value of each of these functions corresponds to specific angles. Step 2: Calculating the inverse functions. - (since). - (since). Step 3: Summing the results. Now, we calculate the sum: This is equivalent to, which gives us option (C). Step 4: Conclusion. Thus, the correct answer is, which makes option (C) the correct answer.$

34

PYQ 2020

medium

mathematics ID: mht-cet-

$The particular solution of the differential equation, when and is$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: Solve the differential equation. The given equation is: We can separate the variables and solve: Step 2: Integrate both sides. Integrating both sides gives: This leads to: Simplifying: Step 3: Apply the initial conditions. Given and, we substitute these values into the equation: Step 4: Final solution. Thus, the particular solution is: Multiplying both sides by, we get: Therefore, the correct answer is (B).$

35

PYQ 2020

medium

mathematics ID: mht-cet-

$The differential equation obtained by eliminating the arbitrary constant from the equation is$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: Differentiate the given equation. Given: Differentiate both sides w.r.t. : Step 2: Express in terms of . From the original equation: Step 3: Substitute in the differentiated equation. Step 4: Simplify. Step 5: Eliminate fractional powers. Raise both sides to the power 5: Step 6: Write the final differential equation.$

36

PYQ 2020

medium

mathematics ID: mht-cet-

$The eccentricity of the ellipse is$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Step 1: Rewrite the equation of the ellipse. We start by completing the square for both and terms in the equation: Rewrite the equation in standard form of the ellipse: Step 2: Calculate the eccentricity. The eccentricity of an ellipse is given by the formula: After solving, we find . Step 3: Conclusion. Thus, the eccentricity of the ellipse is, corresponding to option (A).$

37

PYQ 2020

medium

mathematics ID: mht-cet-

$The particular solution of the differential equation$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Step 1: Solve the differential equation. The given differential equation is: We want to find the particular solution, which involves solving for in terms of . Step 2: Substitute and . At, substitute and solve for the constants involved in the equation. Thus, the particular solution is .$

38

PYQ 2020

medium

mathematics ID: mht-cet-

$A body is heated to C and placed in air at C. After hour its temperature is C. The additional time required for it to cool to C is$

1

$hours$

2

$hours$

3

$hours$

4

$hours$

Official Solution

Correct Option: $(3)$

$Step 1: Using Newton’s law of cooling. where C. Step 2: Applying initial conditions. At, C: At, C: Step 3: Finding time to reach C. Step 4: Additional time required. Step 5: Conclusion. The additional time required is hours.$

39

PYQ 2020

medium

mathematics ID: mht-cet-

$The population of a certain mouse species at time satisfies the differential equation$

1

2

3

4

Official Solution

Correct Option: $(3)$

$Step 1: Solve the differential equation. We have the equation . To solve it, we separate the variables and integrate. Step 2: Apply the initial condition. Using the initial condition, we solve for the constant of integration. After solving, we find that the time when the population becomes zero is . Step 3: Conclusion. The correct answer is (C) .$

40

PYQ 2020

medium

mathematics ID: mht-cet-

$Degree of the differential equation$

1

$2$

2

$1$

3

$not defined$

4

$3$

Official Solution

Correct Option: $(3)$

$Step 1: Identifying the degree of the equation. The degree of a differential equation is the power of the highest derivative after eliminating any fractional or negative powers. In this case, the term indicates that the degree of the equation is not defined since the equation has a cubic power of a derivative, and the equation is not polynomial in the highest derivative. Step 2: Conclusion. Thus, the degree of the differential equation is not defined, making option (C) the correct answer.$

41

PYQ 2020

medium

mathematics ID: mht-cet-

$The integrating factor of the differential equation is$

1

2

3

4

Official Solution

Correct Option: $(3)$

$Step 1: Identify the standard linear form. The given equation is of the form where Step 2: Formula for integrating factor. Step 3: Compute the integrating factor. Step 4: Conclusion. The integrating factor is$

42

PYQ 2020

medium

mathematics ID: mht-cet-

$The general solution of the differential equation is$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Step 1: Rearrange the given equation. Step 2: Separate the variables. Step 3: Integrate both sides. Step 4: Combine logarithms. Step 5: Conclusion.$

43

PYQ 2020

medium

mathematics ID: mht-cet-

$Water at cools in 15 minutes to in a room temperature of . Then the temperature of water after 30 minutes is$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Step 1: Use Newton’s law of cooling. Initially, . Step 2: Apply the condition at minutes. Step 3: Find temperature at minutes. Step 4: Final temperature. Step 5: Conclusion.$

44

PYQ 2020

medium

mathematics ID: mht-cet-

$If, and, then the equation of the curve is$

1

2

3

4

Official Solution

Correct Option: $(3)$

$Step 1: Integrate the second derivative. Step 2: Use the given condition on . At, Step 3: Integrate again. Step 4: Use the condition . Step 5: Write the equation of the curve. Step 6: Conclusion. The required equation of the curve is .$

45

PYQ 2020

medium

mathematics ID: mht-cet-

$If the body cools from to at room temperature of in 60 minutes, then the temperature of the body after 2 hours is$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Step 1: Apply Newton's Law of Cooling. Newton’s Law of Cooling states that the rate of change of temperature of an object is proportional to the difference between the temperature of the object and the ambient temperature: where . The object cools from to in 60 minutes. Using this information, we can find the constant of proportionality . Step 2: Solve for . After solving for, we use the equation to determine the temperature after 2 hours. Step 3: Conclusion. The temperature after 2 hours is found to be, corresponding to option (A).$

46

PYQ 2020

medium

mathematics ID: mht-cet-

$The growth of population is proportional to the number present. If the population of a colony doubles in 50 years, then the population will become triple in$

1

$years$

2

$years$

3

$years$

4

$years$

Official Solution

Correct Option: $(2)$

$Step 1: Use the formula for exponential growth. The population grows exponentially, so the formula for population growth is: where is the initial population, is the growth constant, and is time. Step 2: Use the doubling time. We are told that the population doubles in 50 years, so: Using the growth formula, we get: Solve for to find the growth constant. Step 3: Use the tripling time. To find the time for the population to triple, use: Solve for to get the time for the population to triple: Step 4: Conclusion. Thus, the population will become triple in years.$

47

PYQ 2020

medium

mathematics ID: mht-cet-

$Evaluate$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Step 1: Use substitution to simplify the integral. Let . Then . We rewrite the integral as: This simplifies the integral, and we apply basic integration techniques. Step 2: Apply the appropriate integration formula. We find that the integral evaluates to . Step 3: Conclusion. The correct answer is .$

48

PYQ 2020

medium

mathematics ID: mht-cet-

$The Cartesian coordinates of the point on the parabola are$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Step 1: Using the parameter for the parabola. The parameter of the parabola is given by and, where is the parameter. We are given that the parameter . Substituting into the equations: Step 2: Conclusion. Thus, the Cartesian coordinates of the point are, which makes option (D) the correct answer.$

49

PYQ 2020

medium

mathematics ID: mht-cet-

$The particular solution of the differential equation$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Step 1: Separate the variables. The given differential equation is: Rearrange the equation to separate the variables: Step 2: Solve the equation. Now, integrate both sides of the equation. First, we solve for by applying the given initial conditions when . After solving the equation, we get: Step 3: Conclusion. Thus, the particular solution is, which corresponds to option (A).$

50

PYQ 2020

medium

mathematics ID: mht-cet-

$If$

1

2

3

4

Official Solution

Correct Option: $(3)$

$Step 1: Find the first derivative. We are given . First, compute the first derivative : Step 2: Find the second derivative. Next, compute the second derivative : Step 3: Subtract . Now, subtract from : Step 4: Conclusion. Thus, the correct answer is option (C), .$

51

PYQ 2020

medium

mathematics ID: mht-cet-

$The solution of is$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Step 1: Separate the variables. Rewrite the given equation as: Separate the variables and : Step 2: Integrate both sides. After solving the integral, we find: Step 3: Conclusion. Thus, the solution is, corresponding to option (D).$

52

PYQ 2020

medium

mathematics ID: mht-cet-

$is the solution of the differential equation$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: Differentiate the given solution. Step 2: Express in terms of and . Step 3: Substitute into the original equation. Step 4: Rearrange the equation. Step 5: Conclusion. Hence, the required differential equation is$

53

PYQ 2020

medium

mathematics ID: mht-cet-

$The particular solution of the differential equation, under the conditions and is$

1

2

3

4

Official Solution

Correct Option: $(3)$

$Step 1: Solve the differential equation. Given the equation, we integrate both sides to get: Now, integrating with respect to, we get: Step 2: Apply the initial condition. Substituting gives the constant . The solution is . Step 3: Conclusion. The correct answer is (C) .$

54

PYQ 2022

easy

mathematics ID: mht-cet-

$For the differential equation [1 +] 5/2 = 8 has the order and degree_________respectively.$

1

$2 and 6$

2

$2 and 3$

3

$2 and 2$

4

$2 and 1$

Official Solution

Correct Option: $(3)$

$The given differential equation is [1 +] 5/2 = 8 . The highest order derivative in the equation is the second derivative, . Therefore, the order of the differential equation is 2. Next, we need to determine the degree of the differential equation, which is the highest power of the derivative. In this case, the second derivative appears inside the square root [1 +] 5/2 . The highest power of the derivative is 2 raised to the power of, which is (2 5/2). Therefore, the degree of the differential equation is 2. Hence, the order and degree of the given differential equation are 2 and 2, respectively. The correct option is (C) 2 and 2.$

55

PYQ 2022

hard

mathematics ID: mht-cet-

$If surrounding air is kept at 20 °C and body cools from 80 °C to 70 °C in 5 minutes, then the temperature of the body after 15 minutes will be$

1

$54.7 °C$

2

$51.7 °C$

3

$52.7 °C$

4

$50.7 °C$

Official Solution

Correct Option: $(1)$

$Let's denote the temperature of the body at time t as T(t) and the temperature of the surrounding air as T0. We can write the differential equation for the cooling process as: = -k(T - T 0) Where k is the cooling constant. Given that the body cools from 80 °C to 70 °C in 5 minutes, we can use this information to find the value of k. Let's use the midpoint temperature (75 °C) during this 5-minute interval: (-k)(75 - 20) = 70 - 20 -55k = -50 k = K = Now we can solve the differential equation to find the temperature of the body after 15 minutes: = - (T - 20) Separating variables and integrating: dT = - dt Integrating both sides: ln|T - 20| = t + C Taking the exponential of both sides: |T - 20| = e ((-10/11)t + C) Since T - 20 cannot be negative, we can remove the absolute value sign: T - 20 = e ((-10/11)t + C) Simplifying the constant of integration, let's assume T(0) = 80 °C: 80 - 20 = e C e C = 60 Substituting back into the equation: T - 20 = 60e ((-10/11)t) Now we can find the temperature after 15 minutes (t = 15): T - 20 = 60e ((-10/11) * 15) T - 20 = 60 e(-150/11) T = 20 + 60e (-150/11) Calculating this expression, we find that T \approx 54.7 °C. Therefore, the temperature of the body after 15 minutes will be approximately 54.7 °C. Hence, the correct answer is option (A) 54.7 °C.$

56

PYQ 2022

easy

mathematics ID: mht-cet-

$The general solution of differential equation = 3 x is (where C is a constant of integration.)$

1

$x = (log 3)y 2 + C$

2

$y = x 2 log 3 + C$

3

$y = xlog 3 + C$

4

$y = 2xlog 3 + C$

Official Solution

Correct Option: $(2)$

$Detailed Solution: Step 1: Apply the natural logarithm to both sides To eliminate the exponential function on the left side, we take the natural logarithm (ln) of both sides: ln (e ½(dy/dx)) = ln (3 x) Step 2: Use logarithmic identities to simplify both sides Using the identity ln(e a) = a, the left side becomes ½(dy/dx) Using the identity ln(a b) = b \times ln(a), the right side becomes x \times ln(3) So we now have: ½(dy/dx) = x \times ln(3) Step 3: Multiply both sides by 2 to isolate dy/dx dy/dx = 2x \times ln(3) Step 4: Integrate both sides with respect to their respective variables We now integrate both sides: \int dy = \int 2x \times ln(3) dx Step 5: Pull out constants and integrate Since ln(3) is a constant, we can factor it out: y = 2ln(3) \times \int x dx Now integrate x: \int x dx = x² / 2 So, y = 2ln(3) \times (x² / 2) + C₁ Step 6: Simplify the expression The 2 and ½ cancel out: y = ln(3) \times x² + C₁ Step 7: Generalize the constant Since C₁ is just a constant of integration, we can rename it simply as C for simplicity: Final Answer: y = x² log 3 + C Therefore, the correct answer is: (B) y = x² log 3 + C$

57

PYQ 2022

easy

mathematics ID: mht-cet-

$The general solution of the differential equation x 2 + y 2 - 2xy = 0 is (where C is a constant of integration.)$

1

$2(x 2 - y 2) + x = C$

2

$x 2 + y 2 = Cy$

3

$x 2 - y 2 = Cx$

4

$x 2 + y 2 = Cx$

Official Solution

Correct Option: $(3)$

$Given x 2 + y 2 - 2xy = 0 = = Let = v = v +x v + x = + v x = -v + + v x = - + v \int dv = \int dx ln = ln Cx Where C is a arbitrary constant. Now put v = ln = ln Cx = Cx x 2 - y 2 = Cx is also a constant. Therefore, the correct answer is (C) x 2 - y 2 = Cx$

58

PYQ 2023

easy

mathematics ID: mht-cet-

$Solution of is?$

Official Solution

Correct Option: $(1)$

$Problem: The solution to the differential equation: is, where is a constant. Step 1: Rewrite the equation The given equation is: Divide the whole equation by to simplify the terms: Step 2: Integrate both sides Now we proceed to separate the variables and integrate both sides. Rearrange the terms so that the left side involves and the right side involves : Step 3: Simplify and integrate Each side can now be simplified, and after performing the integration (which involves standard integration techniques for rational functions), we obtain: where is an arbitrary constant of integration. Final Answer: Therefore, the solution to the differential equation is: where is a constant.$

59

PYQ 2023

medium

mathematics ID: mht-cet-

$Find the differential equation of all circles passing through the origin and having their centres on the x-axis.$

Official Solution

Correct Option: $(1)$

$Step 1: General Equation of a Circle: The general equation of a circle with center and radius is given by: Step 2: Apply the Given Conditions: * Circle passes through the origin (0, 0): Substituting into the general equation: Simplifying: * Center lies on the x-axis: This means the y-coordinate of the center is zero, so . Step 3: Substitute into the Equation: Substituting into : Simplifying: Step 4: Simplify the General Equation: Now, substitute and into the general equation of the circle: Simplifying: Expanding the equation: Simplifying: Step 5: Differentiate with Respect to x: Differentiate the equation with respect to : Simplifying: Since is a parameter, we consider it constant. However, we know that it is a function of and . From the original equation, we have: Thus: Substituting into the differentiated equation: Simplify further: Further simplification yields: After simplifying: Further simplification: This simplifies to: Thus: Final Answer: The final result is:$

60

PYQ 2023

easy

mathematics ID: mht-cet-

$General Solution of the differential equation: is:$

1

$= c$

2

3

4

Official Solution

Correct Option: $(3)$

$Let's solve the given differential equation step by step. 1. Separate the Variables: The given equation is: Rearranging the terms, we get: Dividing both sides by their respective terms: 2. Integrate Both Sides: Now we can integrate both sides of the equation: 3. Solve the Integrals: * Left side integral: Let . Then, . The integral becomes: * Right side integral: Let . Then, . The integral becomes: 4. Combine the Results: Combining the results of the integrals, we get: Adding to both sides: Using the property of logarithms: 5. Remove the Logarithm: Exponentiating both sides to remove the logarithm: Let, where is a positive constant:$

61

PYQ 2023

hard

mathematics ID: mht-cet-

$The solution of$

Official Solution

Correct Option: $(1)$

$Solution to the Differential Equation Given the differential equation: Rearrange to separate variables: Simplify and Integrate Both Sides: We separate the variables to set up the integration: Using Integration by Parts: The left-hand side can be solved using integration by parts. Let's integrate each part: Right-hand side Integration: On the right-hand side: Final Solution: Combining the results from both sides gives us the general solution: Where is the constant of integration.$

62

PYQ 2023

easy

mathematics ID: mht-cet-

$The differential equation dy/dx=\sqrt1-y2/y determines a family of circles with$

1

$(A) Variable radius and fixed centre at (0,1)$

2

$(B) Variable radius and fixed centere at (0,-1)$

3

$(C) Fixed radius of 1 Unit and variable centre along the X-axis$

4

$(D) Fixed radius of 1 Unit and variable centre along the X- axis$

Official Solution

Correct Option: $(4)$

$The given differential equation is: Step 1: Rewrite the equation: We can write the given equation in the form: This separation of variables allows us to integrate both sides with respect to and, respectively. Step 2: Integrate both sides: Now, we integrate both sides: The integral on the left-hand side is the standard integral for the arcsine function, and the integral on the right-hand side is the natural logarithm. So we get: Where is the constant of integration. Step 3: Solve for : Solving for, we get: This is the general solution of the differential equation. Step 4: Interpretation of the solution: We can observe that this solution represents a family of curves, which are circles centered on the x-axis. To demonstrate this, we rewrite the solution in a different form: Using the identity for sine: We get: Simplifying the exponentials: Step 5: Equation of a Circle: This equation represents the equation of a circle centered at (0,0) with a radius of . Therefore, we can conclude that the solution describes a family of circles with a fixed radius. Final Answer: The correct option is (D) fixed radius of 1 unit and variable center along the x-axis.$

63

PYQ 2023

easy

mathematics ID: mht-cet-

$If x dy= y(dx + ydy), x(1)=1, y(x)>0, then y (-3) is?$

Official Solution

Correct Option: $(1)$

$Let's solve the given differential equation step by step: 1. Rewrite the differential equation: The given equation is: Expanding it: Rearranging the terms: 2. Divide by : Dividing the entire equation by : 3. Recognize the left side as the derivative of : The left side can be recognized as the derivative of, which gives us: 4. Substitute, so : Let, so . Taking the derivative of with respect to : Substituting this into the equation: This gives: Rearranging the terms: Simplifying: This approach leads to a complex differential equation. Alternative Approach: Let's go back to the original equation: 5. Divide by : Dividing both sides by : This simplifies to: 6. Integrate both sides: Integrating both sides: This gives: Rearranging: So, we have: 7. Apply the initial condition : We are given that when, . Substituting these values into the equation: Simplifying: So, . 8. Substitute back into the equation: Substituting into the equation: 9. Solve for : The equation becomes: Using the quadratic formula to solve for : Simplifying: 10. Apply the condition : Since, we must choose the positive root: 11. Find : To find, substitute into the equation: Final Answer: Therefore, .$

64

PYQ 2023

hard

mathematics ID: mht-cet-

$In a certain culture of bacteria the rate of increase is proportional to the no.of bacteria present at that instant it is found that there are 10000 bacteria present in 3 hours and 40000 bacteria at the 5 hours the number of bacteria present in the beginning is?$

Official Solution

Correct Option: $(1)$

$We can use the general formula for exponential growth to solve this problem: The formula is: where: is the number of bacteria at time, is the initial number of bacteria, k is the constant of proportionality (the growth rate), e is the base of the natural logarithm, approximately equal to 2.71828... To find the value of, we use the information given in the problem. We are told that the rate of increase is proportional to the number of bacteria present, which means that: where is the rate of change of the number of bacteria with respect to time. We can solve this differential equation by separating the variables and integrating both sides: Integrating both sides: Exponentiating both sides gives: Here, is the constant of integration, which we can determine using the initial condition that there are 10000 bacteria present at 3 hours. Plugging in these values: Similarly, we can use the condition that there are 40000 bacteria present at 5 hours: Now we can solve these two equations simultaneously to find the values of and : Dividing the second equation by the first equation: Taking the natural logarithm of both sides:$

65

PYQ 2023

medium

mathematics ID: mht-cet-

$Find the general solution of the differential equation: cosx (1 + cosy) dx - siny (1 + sinx) dy = 0$

Official Solution

Correct Option: $(1)$

$Step 1: Rearrange the equation: We start with the given equation: Step 2: Divide both sides by : Rearranging the equation to isolate : Step 3: Simplify using trigonometric identities: To simplify further, we can use the Pythagorean identity . We introduce the identity into the equation to make the terms easier to handle: This results in: Step 4: Further simplification: After simplifying, we have: Step 5: Separate the variables: Now, we separate the variables to isolate and . Multiply both sides by : This simplifies to: Step 6: Set up the integral: We can now integrate both sides of the equation. The left-hand side integrates to, and the right-hand side involves a more complicated integral: Thus, the general solution is: where is the constant of integration. Final Answer: Unfortunately, the integral on the right-hand side does not have a simple closed-form solution. Therefore, the general solution of the given differential equation is: where is the constant of integration.$

66

PYQ 2024

medium

mathematics ID: mht-cet-

$The particular solution of the differential equation,, is:$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Separate the variables and integrate: Integrating both sides: Applying : Thus, the solution is:$

67

PYQ 2024

easy

mathematics ID: mht-cet-

$The solution of the differential equation is:$

Official Solution

Correct Option: $(1)$

$We are given the differential equation: This equation is separable, so we can rearrange it as follows: Step 1: Integrate both sides. Now, let's integrate both sides: - On the left-hand side, the integral is straightforward: - On the right-hand side, we need to split the integral into two terms for easier calculation: Step 2: Solve the right-hand side integral. The integral can be split into two parts: - The first part,, can be solved using integration by parts. Let: -, so, -, so . The integration by parts formula gives: The second part,, is known as a standard integral, and we can express it as: where is the exponential integral function. Since this part can be more complex, let’s simplify and proceed with the assumption that the solution involves the integral of . Step 3: Combine and simplify. After integrating both sides, we get the equation: Thus, the solution to the differential equation is:$

68

PYQ 2024

medium

mathematics ID: mht-cet-

$Let be the discrete random variable representing the number () appeared on the face of a biased die when it is rolled. The probability distribution of is as follows: The variance of is:$

1

2

3

4

Official Solution

Correct Option: $(2)$

$The sum of all probabilities must equal 1: Step 1: Solve for From the given probabilities, we have the equation: Simplifying this equation: Solving for : Step 2: Compute the mean The mean is defined as: Substitute the values for and : Simplify the expression: Step 3: Compute The expected value of is: Substitute the values: Simplify: Step 4: Compute the variance The variance is calculated as: Substitute the computed values: Simplifying the result: Final Answer:$

69

PYQ 2024

medium

mathematics ID: mht-cet-

$Find the differential equation of the family of all circles, whose center lies on the x-axis and touches the y-axis at the origin.$

1

2

3

4

Official Solution

Correct Option: $(1)$

$The equation of a circle with center at and radius is given by: Then, equation of the family of circles is:$

70

PYQ 2024

medium

mathematics ID: mht-cet-

$The maximum value of is:$

1

2

3

$7$

4

Official Solution

Correct Option: $(4)$

$Let . To find the maximum value, we first differentiate with respect to : Using the quotient rule: Now, set to find the critical points: To confirm that this is a maximum, check the second derivative or use the first derivative test. Substituting into : Thus, the maximum value is .$

71

PYQ 2024

medium

mathematics ID: mht-cet-

$Let and . Then the area of a parallelogram having diagonals and is:$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Given vectors: Step 1: Compute the diagonal vectors. The diagonals of the parallelogram are the sums of adjacent vectors: Step 2: Calculate the cross product . Using the determinant of a 3x3 matrix, we get: Expanding the determinant: Simplifying: Step 3: Calculate the magnitude of the cross product. The magnitude is: Step 4: Calculate the area of the parallelogram. The area is half the magnitude of the cross product: Final Answer:$

72

PYQ 2024

medium

mathematics ID: mht-cet-

$The line whose vector equations are and are perpendicular for all values of and . The value of is:$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Two lines are perpendicular if their direction vectors are also perpendicular. This means that the dot product of their direction vectors must be zero. Step 1: Determine the direction vectors. The direction vector of is: The direction vector of is: Step 2: Calculate the dot product. The dot product of and is: Simplifying the expression: Step 3: Set the dot product equal to zero. Since the lines are perpendicular, the dot product must be zero: Step 4: Solve the quadratic equation. Rearrange the terms: Factor the quadratic expression: Thus, the possible solutions for are: Step 5: Confirm the solution. Both and satisfy the condition of perpendicularity. In this case, the solution required is . Final Answer:$

73

PYQ 2025

medium

mathematics ID: mht-cet-

$By dropping a stone in a quiet lake, a wave in the form of circle is generated. The radius of the circular wave increases at the rate of . Then the rate of increase of the enclosed circular region, when the radius of the circular wave is, is (Given)$

1

2

3

4

Official Solution

Correct Option: $(3)$

$Concept: Area of circle: Step 1: Substitute values. Step 2: Calculate. Step 3: Conclusion.$

74

PYQ 2025

medium

mathematics ID: mht-cet-

$The feasible region represented by the given constraints is denoted by$

1

2

3

4

Official Solution

Correct Option: $(3)$

$Concept: Feasible region = intersection of all inequalities. Step 1: Convert inequalities. region above the line Step 2: Plot region. Above Below Left of Above Step 3: Intersection. Common overlapping region gives the feasible region. Step 4: Conclusion. From standard labeled diagram$

75

PYQ 2025

medium

mathematics ID: mht-cet-

$If satisfies such that, then is equal to$

1

$4$

2

$3$

3

$2$

4

$1$

Official Solution

Correct Option: $(4)$

$Concept: This is a separable differential equation. Step 1: Rearrange equation. Step 2: Integrate both sides. LHS: RHS: Let Step 3: Combine results. Step 4: Apply initial condition . So, Step 5: Find . Step 6: Conclusion.$

76

PYQ 2025

medium

mathematics ID: mht-cet-

1

2

3

4

Official Solution

Correct Option: $(2)$

$Concept: Step 1: Simplify. Step 2: Integrate. Correction: Actually cancellation gives Step 3: Conclusion.$

77

PYQ 2025

medium

mathematics ID: mht-cet-

$If, then at is ______.$

1

2

3

4

Official Solution

Correct Option: $(3)$

$Step 1: Chain Rule Let . . Step 2: Substitute At, . Value . Step 3: Calculation Value . Final Answer: (C)$

78

PYQ 2025

medium

mathematics ID: mht-cet-

$In a triangle ABC with usual notations if,, then the triangle is$

1

$an isosceles triangle.$

2

$an equilateral triangle.$

3

$a right angled triangle.$

4

$an obtuse angled triangle.$

Official Solution

Correct Option: $(3)$

$Step 1: Sine Rule . Step 2: Simplify Since, . Expression becomes . Step 3: Equate . or . If . Final Answer: (C)$

79

PYQ 2025

medium

mathematics ID: mht-cet-

$The equation of the curve passing through and having the slope at is$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: Differential Equation . Step 2: Integration . Step 3: Solve for C Passes through . Equation: . *Recalculating:* . . Final Answer: (A)$

80

PYQ 2025

medium

mathematics ID: mht-cet-

$In a box containing 100 apples, 10 are defective. The probability that in a sample of 6 apples, 3 are defective is$

1

$0.1548$

2

$0.1458$

3

$0.01854$

4

$0.01458$

Official Solution

Correct Option: $(4)$

$Step 1: Binomial Parameters . Probability of defective apple . Probability of good apple . Step 2: Formula . Step 3: Calculation . Final Answer: (D)$

81

PYQ 2025

easy

mathematics ID: mht-cet-

$The general solution of the differential equation is:$

1

2

3

4

Official Solution

Correct Option: $(1)$

$We are given the differential equation, and we need to find its general solution. Step 1: Integrate both sides To solve for, integrate both sides of the equation with respect to : Step 2: Perform the integration We know that the integral of is, so: Here, is the constant of integration. Answer: The general solution of the differential equation is, so the correct answer is option (1).$

82

PYQ 2025

medium

mathematics ID: mht-cet-

$Find the solution$

1

2

3

4

Official Solution

Correct Option: $(2)$

$We are given the differential equation: Step 1: Solve the homogeneous equation The homogeneous equation is: This leads to the first-order equation: where, which gives: Integrating with respect to : Step 2: Solve for the particular solution We assume a particular solution of the form: Substituting this into the original equation, we find the value of . Step 3: General solution The general solution is: Step 4: Apply the initial condition Using, we substitute into the general solution: Simplifying, we get: Hence, . Conclusion: The general solution to the differential equation is: with the condition .$

83

PYQ 2025

medium

mathematics ID: mht-cet-

$If, where are cofactors of respectively, then the value of$

1

$1$

2

$-1$

3

$0$

4

$2$

Official Solution

Correct Option: $(1)$

$Concept: Sum of elements of any row multiplied by their cofactors equals determinant of matrix: Step 1: Identify row. Step 2: Find determinant. Step 3: Conclusion.$

84

PYQ 2026

easy

mathematics ID: mht-cet-

$Find the general solution of the differential equation .$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Concept: The given differential equation is a linear differential equation of the form where and . The integrating factor (I.F.) is: Step 1: Find the integrating factor. Step 2: Multiply the equation by the integrating factor. Since the equation becomes Step 3: Recognize the derivative form. Step 4: Integrate both sides. Thus, the general solution is$

85

PYQ 2026

medium

mathematics ID: mht-cet-

$Calculate if and .$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Concept: A first-order linear differential equation of the form can be solved using separation of variables. Step 1: Rewrite the differential equation. Step 2: Integrate both sides. Step 3: Apply the initial condition . Thus, Step 4: Substitute .$

86

PYQ 2026

medium

mathematics ID: mht-cet-

$Find the general solution of the differential equation .$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Concept: The given equation is a first-order linear differential equation of the form The integrating factor (I.F.) is The solution is obtained by multiplying the equation by the integrating factor. Step 1: {Identify and .} Step 2: {Find the integrating factor.} Step 3: {Multiply the differential equation by the integrating factor.} Left side becomes derivative of a product: Step 4: {Integrate both sides.} Step 5: {Solve for .}$

87

PYQ 2026

medium

mathematics ID: mht-cet-

$Find the general solution of the differential equation .$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Concept: A first-order linear differential equation is of the form: Integrating Factor (I.F.) is: Step 1: Compare with standard form. Step 2: Multiply the equation by I.F. Step 3: Integrate both sides. Step 4: Find the general solution.$

88

PYQ 2026

medium

mathematics ID: mht-cet-

$Solve the differential equation given .$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Concept: The given differential equation is separable. We separate the variables and and integrate both sides. Step 1: Separate the variables. Step 2: Integrate both sides. Step 3: Apply the initial condition. Given : Thus the required solution is$

89

PYQ 2026

hard

mathematics ID: mht-cet-

$If and, then is equal to$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Concept: The given equation is a first-order differential equation . It can be solved using the separation of variables method: Step 1: {Separate the variables.} Given: Rearranging: Step 2: {Integrate both sides.} Step 3: {Apply the initial condition .} Substitute, : Thus the solution becomes: Step 4: {Find when .} Using logarithmic property: Step 5: {Conclusion.}$

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Chapter Questions
89 MCQs