UG Applied Mathematics Booster Test 2 - Introduction to Differential Equations -

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UG Applied Mathematics Booster Test 2 - Introduction to Differential Equations

⏱️ 30 Minutes

❓ 20 Questions

📌 Answers are locked once submitted — results and explanations appear at the end.

UG Applied Mathematics Booster Test 2 - Introduction to Differential Equations

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QUESTION 1 OF 20

If a mathematical model describes the rate of change of population P with respect to time t using the equation dP/dt = kP, this equation is rigorously classified as:

Answer Review

1 If a mathematical model describes the rate of change of population P with respect to time t using the equation dP/dt = kP, this equation is rigorously classified as:

A A partial differential equation

B An ordinary differential equation of order 1

C An ordinary differential equation of order 2

D A linear algebraic equation

💡 Short Explanation

One independent variable (t) First derivative present Hence first-order ODE

✅ Correct Answer Explanation

dP/dt = kP involves derivative w.r.t. single variable t → Only first derivative appears → Therefore, it is a first-order ordinary differential equation

❓ Why Other Options Are Incorrect

Option A → PDE requires multiple independent variables
Option C → no second derivative present
Option D → derivative exists, so not algebraic

🎯 Strategy

Used

Definition Recall

Application: Identify type via variables and derivative order.

Final Logic: Single variable + first derivative → first-order ODE

🧠 Memory Trick

"One variable, one derivative → ODE 1st order."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Basic Definition

2 Match the physical problem to its corresponding mathematical differential equation model:

List I	List II
1. Population Growth	a. dT/dt = -k(T-A)
2. Newton's Law of Cooling	b. dP/dt = αP
3. Compound Interest	c. dA/dt = rA
4. Exponential Decay (Carbon Dating)	d. C(t) = C₀ e^(-kt)

A (1)-(b), (2)-(a), (3)-(c), (4)-(d)

B (1)-(a), (2)-(b), (3)-(c), (4)-(d)

C (1)-(c), (2)-(d), (3)-(a), (4)-(b)

D (1)-(b), (2)-(c), (3)-(a), (4)-(a)

💡 Short Explanation

Population growth → proportional model Cooling → temperature difference model Compound interest → growth rate model Decay → exponential decay

✅ Correct Answer Explanation

(A) Population Growth → dP/dt = αP → (II) → (B) Newton's Law of Cooling → dT/dt = −k(T−A) → (I) → (C) Compound Interest → dA/dt = rA → (III) → (D) Carbon Dating → C(t) = C₀e^(−kt) → (IV)

❓ Why Other Options Are Incorrect

Option B → swaps cooling and growth models
Option C → mismatches exponential decay and cooling
Option D → repetition and incorrect mapping

🎯 Strategy

Used

Concept Mapping

Application: Match real-world processes to standard DE models.

Final Logic: Each physical law has fixed DE form

🧠 Memory Trick

"Growth, Cooling, Interest, Decay = standard four models."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Applications

3 In the context of the differential equation for drug absorption dy/dt = -ky, which reasoning statements are valid?
1. y represents the dependent variable (amount of drug).
2. t represents the independent variable (time).
3. The negative sign indicates that the dependent variable is decreasing over time.
4. k is the independent variable instead of t.

A 1, 2, 4

B 1, 2

C 1, 2, 3

D 1, 2, 3, 4

💡 Short Explanation

y depends on time Negative sign implies decay k is constant, not independent variable

✅ Correct Answer Explanation

1: y is dependent variable → 2: t is independent variable → 3: negative sign indicates decay → 4: false because k is constant, not variable

❓ Why Other Options Are Incorrect

Option A → includes incorrect 4
Option B → incomplete
Option D → includes false statement

🎯 Strategy

Used

Elimination

Application: Remove incorrect role of k.

Final Logic: 1, 2, 3 are correct

🧠 Memory Trick

"k is constant, not variable."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Variables

4 Identify the incorrect statement regarding Carbon Dating formulation:

A Time t is independent variable

B Mass C(t) is dependent variable

C Rate of change is taken w.r.t dependent variable

D dC/dt proportional to C

💡 Short Explanation

Rate must be w.r.t independent variable Time is independent Statement C contradicts DE definition

✅ Correct Answer Explanation

Differential equations define rate of change w.r.t independent variable → Here t is independent variable → So C is incorrect

❓ Why Other Options Are Incorrect

Option A → correct
Option B → correct
Option D → correct decay model

🎯 Strategy

Used

Definition Check

Application: Verify correct variable roles.

Final Logic: Rate is not w.r.t dependent variable

🧠 Memory Trick

"Derivative always w.r.t independent variable."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Modeling

5 A mixture of derivatives is present in the equation xy(d²y/dx²) + x(dy/dx)² - y(dy/dx) = 0. What is the highest order derivative present?

A dy/dx

B (dy/dx)²

C d²y/dx²

D xy(d²y/dx²)

💡 Short Explanation

Highest derivative determines order Second derivative present Powers do not matter

✅ Correct Answer Explanation

Highest derivative is d²y/dx² → Therefore order = 2

❓ Why Other Options Are Incorrect

Option A → first order only
Option B → power irrelevant
Option D → coefficient not derivative

🎯 Strategy

Used

Highest Derivative Rule

Application: Ignore powers and coefficients.

Final Logic: Second derivative dominates

🧠 Memory Trick

"Look only at highest derivative."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Order

6 Subject to the constraint y ≠ 3, the differential equation dy/dx = (2-x)/(y-3) has what order:

A 0

B 1

C 2

D 3

💡 Short Explanation

First derivative only No higher derivatives Domain restriction does not affect order

✅ Correct Answer Explanation

Only dy/dx appears → Hence first order differential equation

❓ Why Other Options Are Incorrect

Option A → no derivative is false
Option C → no second derivative
Option D → equation is defined

🎯 Strategy

Used

Direct Identification

Application: Identify highest derivative.

Final Logic: Only first derivative present

🧠 Memory Trick

"One derivative = order one."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Order

7 Calculating the degree is systematic like calculating an EMI step-by-step. For the equation d²y/dx² + y² + e^(dy/dx) = 0, why is the degree NOT defined?

A Because it contains a second order derivative.

B Because the highest order derivative has no power.

C Because the equation is not a polynomial equation in its derivatives.

D Because the dependent variable y is squared.

💡 Short Explanation

Must be polynomial in derivatives Exponential violates rule Degree undefined

✅ Correct Answer Explanation

Degree exists only if polynomial in derivatives → e^(dy/dx) is non-polynomial → Hence degree not defined

❓ Why Other Options Are Incorrect

Option A → irrelevant
Option B → order unrelated
Option D → y² not issue

🎯 Strategy

Used

Rule Violation Check

Application: Detect exponential term.

Final Logic: Non-polynomial → undefined degree

🧠 Memory Trick

"Exponential breaks degree."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Degree

8 Averaging the powers is incorrect; we must find the highest power of the highest order derivative. What is the degree of (ds/dt)⁴ + 2s(d²s/dt²)=0

A 4

B 2

C 1

D 0

💡 Short Explanation

Highest order derivative is first derivative Its power is 4 Degree = 4

✅ Correct Answer Explanation

Highest order derivative: ds/dt → Power = 4 → Degree = 4

❓ Why Other Options Are Incorrect

Option B → confusion with second derivative
Option C → order mistaken
Option D → incorrect assumption

🎯 Strategy

Used

Order-Power Identification

Application: Identify highest derivative and exponent.

Final Logic: First derivative raised to 4

🧠 Memory Trick

"Power of highest derivative = degree."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Degree – Page 193

9 What is the probability that the equation (1 + dy/dx)³ = (d²y/dx²)² has a degree of 2?

A 0

B 0.5

C 0.75

D 1

💡 Short Explanation

Equation is polynomial in derivatives Degree is always defined Highest power is 2

✅ Correct Answer Explanation

Polynomial structure ensures degree exists → Highest order derivative (second derivative) has power 2 → So probability = 1

❓ Why Other Options Are Incorrect

Option A → impossible
Option B → arbitrary
Option C → incorrect probabilistic value

🎯 Strategy

Used

Certainty Principle

Application: Polynomial guarantees defined degree.

Final Logic: Always degree exists → probability 1

🧠 Memory Trick

"Polynomial → sure degree."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Degree

10 If a differential equation contains a vector-like exponential translation element such as e^(y'), what happens to its degree?

A The degree becomes 1.

B The degree becomes the order.

C The degree is not defined.

D The degree is infinite.

💡 Short Explanation

Non-polynomial term Exponential derivative Degree undefined

✅ Correct Answer Explanation

Degree defined only for polynomial DE → e^(y') violates condition → Hence undefined

❓ Why Other Options Are Incorrect

Option A → incorrect
Option B → unrelated
Option D → meaningless concept

🎯 Strategy

Used

Structural Check

Application: Identify exponential expression.

Final Logic: Non-polynomial → no degree

🧠 Memory Trick

"No polynomial → no degree."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Degree Undefined

11 To find the area, we might integrate a first order equation. The equation y dx + x log(y/x) dy − 2x dy = 0 can be rewritten to find its order and degree. What are they?

A Order 2, Degree 1

B Order 1, Degree 2

C Order 1, Degree 1

D Order 1, Degree Not Defined

💡 Short Explanation

Highest derivative is dy/dx Appears to first power No higher-order derivative

✅ Correct Answer Explanation

The equation contains only dy/dx implicitly after rearrangement → Highest derivative is first order → Derivative appears in power 1 → degree 1

❓ Why Other Options Are Incorrect

Option A → second order not present
Option B → degree is not 2
Option D → equation is polynomial in derivative

🎯 Strategy

Used

Structural Identification

Application: Convert and identify highest derivative.

Final Logic: First-order linear form → degree 1

🧠 Memory Trick

"Only dy/dx → order 1."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Order and Degree

12 Integrating the second-order equation of a family of parabolas having vertex at origin and axis along positive y-axis requires understanding its form. What is the order of the original differential equation for x² = 4ay?

A 1

B 2

C 3

D 4

💡 Short Explanation

Parabola equation One differentiation gives DE Hence first order

✅ Correct Answer Explanation

x² = 4ay represents a parabola family → Differentiation once gives dy/dx relation → Hence original DE is first order

❓ Why Other Options Are Incorrect

Option B → second order not required
Option C → irrelevant
Option D → irrelevant

🎯 Strategy

Used

Reverse Differentiation

Application: Identify number of differentiations needed.

Final Logic: One differentiation → order 1

🧠 Memory Trick

"One curve family → first order."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Formation of DE

13 How many arbitrary constants must be eliminated from a general solution to form a third-order differential equation?

A 1

B 2

C 3

D 4

💡 Short Explanation

Order equals constants eliminated Third order → 3 constants Standard rule

✅ Correct Answer Explanation

n-th order DE is obtained by eliminating n constants → For third order → 3 constants removed

❓ Why Other Options Are Incorrect

Option A → first order case
Option B → second order case
Option D → over-elimination

🎯 Strategy

Used

Pattern Rule

Application: Use order = number of constants.

Final Logic: 3rd order → 3 constants

🧠 Memory Trick

"Order = constants removed."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Formation Rule

14 The differential equation representing y = a sin(x+b), where a, b are constants is:

A y'' + y' = 0

B y'' + y = 0

C y'' − y = 0

D y' + y = 0

💡 Short Explanation

Standard sine family Eliminates constants a, b Leads to harmonic equation

✅ Correct Answer Explanation

y = a sin(x+b) is a harmonic function → Second differentiation removes constants → Gives y'' + y = 0

❓ Why Other Options Are Incorrect

Option A → incorrect structure
Option C → exponential type equation
Option D → first order only

🎯 Strategy

Used

Standard Form Recognition

Application: Recall harmonic differential equation.

Final Logic: Sin function → y'' + y = 0

🧠 Memory Trick

"Sin → y'' + y."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Standard Solutions

15 The general solution of dy/dx = (x+1)/(2-y) will contain how many arbitrary constants?

A 0

B 1

C 2

D Infinite

💡 Short Explanation

First order DE One integration → one constant Standard rule

✅ Correct Answer Explanation

First-order differential equation → Integration introduces one constant c → Hence one arbitrary constant

❓ Why Other Options Are Incorrect

Option A → integration always adds constant
Option C → second order required
Option D → not possible

🎯 Strategy

Used

Order-Constant Relation

Application: First order → one constant.

Final Logic: Order 1 → constant 1

🧠 Memory Trick

"1st order → 1 constant."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Solution Structure

16 Eliminating m and c from y = mx + c yields the differential equation y'' = 0. This implies:

A y = mx+c is a particular solution

B y = mx+c is the general solution representing a family of curves for y'' = 0

C The family of curves are parabolas

D The degree of the equation is 0

💡 Short Explanation

y = mx + c contains constants Represents straight lines General solution of y'' = 0

✅ Correct Answer Explanation

y = mx + c is family of straight lines → Eliminating constants gives y'' = 0 → Hence it is general solution

❓ Why Other Options Are Incorrect

Option A → too specific
Option C → incomplete interpretation
Option D → irrelevant

🎯 Strategy

Used

Concept Mapping

Application: Link equation form to DE.

Final Logic: Linear family → general solution

🧠 Memory Trick

"Lines → y'' = 0."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Formation – Page 203

17 To find a particular solution for dy/dx = y+1 given y(0)=3, what step is applied after integration?

A Differentiate the result again

B Substitute x=0 and y=3 to fix the value of the arbitrary constant c

C Set the arbitrary constant to zero automatically

D Assume c=3 without solving

💡 Short Explanation

Use initial condition Determine constant Fix specific solution

✅ Correct Answer Explanation

After integration, solution contains constant c → Use condition y(0)=3 → Substitute to find c

❓ Why Other Options Are Incorrect

Option A → wrong step
Option C → unjustified
Option D → arbitrary assumption

🎯 Strategy

Used

Substitution Method

Application: Apply initial condition.

Final Logic: Use given values to find constant

🧠 Memory Trick

"Plug condition → find constant."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Particular Solution

18 A specific solution curve is obtained when:

A All parameters arbitrary

B Particular values are given to arbitrary constants to satisfy a specific condition

C The differential equation is fundamentally not integrable

D The degree is determined to be undefined

💡 Short Explanation

Fix arbitrary constants One unique curve Derived from general solution

✅ Correct Answer Explanation

General solution contains constants → Assigning values gives specific curve → Hence particular solution

❓ Why Other Options Are Incorrect

Option A → general solution
Option C → irrelevant
Option D → unrelated

🎯 Strategy

Used

Concept Elimination

Application: Identify effect of constants.

Final Logic: Fixed constants → specific solution

🧠 Memory Trick

"Fix constants → fix curve."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Particular Solution

📄 Passage

The formulation of a real-world problem in mathematical terms is the construction of a mathematical model. In many natural phenomenon, a quantity changes at a rate proportional to the amount present. The mathematical model for exponential growth or decay is given by dy/dt = ky or f(t) = Ae^(kt). If k > 0, the formula represents exponential growth. If k < 0, it represents exponential decay.

Question:
According to the passage, substituting k < 0 into the mathematical model verifies which physical process?

A Constant population

B Exponential growth

C Exponential decay

D Zero rate of change

💡 Short Explanation

Negative exponent Exponential decreases Models decay

✅ Correct Answer Explanation

k < 0 makes e^(kt) decrease with time → Function decreases exponentially → Represents decay

❓ Why Other Options Are Incorrect

Option A → constant requires k=0
Option B → k>0 gives growth
Option D → not implied

🎯 Strategy

Used

Sign Analysis

Application: Check exponent sign.

Final Logic: Negative k → decay

🧠 Memory Trick

"Negative k → decay."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Exponential Model

📄 Passage

Question:
Differentiating the model f(t) = Ae^(kt) with respect to t directly yields:

A f'(t) = Ake^(kt)

B f'(t) = Ae^(kt)

C f'(t) = k

D f'(t) = At

💡 Short Explanation

Chain rule Exponential derivative Multiply by k

✅ Correct Answer Explanation

d/dt of e^(kt) = k e^(kt) → Multiply by A → f'(t) = Ake^(kt)

❓ Why Other Options Are Incorrect

Option B → missing k
Option C → incorrect form
Option D → unrelated

🎯 Strategy

Used

Differentiation Rule

Application: Apply chain rule.

Final Logic: Exponential derivative rule

🧠 Memory Trick

"e^(kt) → multiply by k."

📚 Class XI – NCERT Mathematics – Chapter: Differential Equations – Topic: Exponential Function

UG Applied Mathematics Booster Test 2 - Introduction to Differential Equations

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