CUET UG Mathematics Booster Test 1 - Basic Concepts of Differential Equations -

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CUET UG Mathematics Booster Test 1 - Basic Concepts of Differential Equations

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Mathematics Booster Test 1 - Basic Concepts of Differential Equations

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QUESTION 1 OF 20

📄 Passage

An equation of the form dy/dx=g(x),"where " y=f(x), is known as a differential equation. These equations arise in a variety of applications. A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation. By the degree of a differential equation (when it is a polynomial equation in derivatives), we mean the highest power of the highest order derivative.

Based directly on the passage, the primary defining characteristic of a differential equation is the presence of:

Answer Review

📄 Passage

Based directly on the passage, the primary defining characteristic of a differential equation is the presence of:

A Only independent variables

B Polynomials of degree zero

C Derivatives of the dependent variable

D Multiple independent variables

💡 Short Explanation

Differential equations involve derivatives. Dependent variables are differentiated. This is the defining feature from the passage.

✅ Correct Answer Explanation

The passage explicitly states that a differential equation is an equation involving derivatives of the dependent variable. Therefore Option C directly matches the definition. Option A ignores derivatives, Option B is unrelated to the definition, and Option D refers to the number of independent variables rather than the defining characteristic.

❓ Why Other Options Are Incorrect

Option A → Independent variables alone do not create a differential equation.
Option B → Degree zero has no relevance to the basic definition.
Option D → Differential equations may involve one or more independent variables, so this is not the defining feature.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Match the wording of the option directly with the passage definition.

Final Logic:

The passage explicitly identifies derivatives as the key feature.

🧠 Memory Trick

"Differential = Derivative."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: Definition of Differential Equation

📄 Passage

Based on the passage, if an equation involves derivatives with respect to exactly one independent variable, it is classified as:

A An ordinary differential equation

B A partial differential equation

C An algebraic polynomial

D A degree-zero equation

💡 Short Explanation

One independent variable is involved. Derivatives are ordinary derivatives. Hence it is an ODE.

✅ Correct Answer Explanation

The passage clearly states that a differential equation involving derivatives with respect to only one independent variable is called an ordinary differential equation. Therefore Option A is correct. Option B refers to multiple independent variables, while Options C and D are unrelated classifications.

❓ Why Other Options Are Incorrect

Option B → PDEs involve more than one independent variable.
Option C → Algebraic polynomials may contain no derivatives.
Option D → Degree is unrelated to ODE classification.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Use the exact terminology provided in the passage.

Final Logic:

One independent variable implies an ordinary differential equation.

🧠 Memory Trick

"One variable → ODE."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: Ordinary Differential Equation

3 Which of the following is not an ordinary differential equation?

A dy/dx = sin x

B y'' + 2y' = 0

C (d³y)/(dx³) + y = eˣ

D (∂²u)/(∂x²) + (∂²u)/(∂y²) = 0

💡 Short Explanation

Partial derivatives are present. More than one variable is involved. Hence it is a PDE.

✅ Correct Answer Explanation

Options A, B, and C involve ordinary derivatives with respect to a single independent variable. Option D contains partial derivatives with respect to x and y. Therefore it is a partial differential equation rather than an ordinary differential equation.

❓ Why Other Options Are Incorrect

Option A → Contains an ordinary first derivative.
Option B → Contains ordinary derivatives only.
Option C → Contains a third-order ordinary derivative.

🎯 Strategy

Used: Odd One Out

Application:

Identify the equation containing partial derivatives.

Final Logic:

Only Option D uses ∂ notation.

🧠 Memory Trick

"∂ means PDE."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: ODE and PDE

4 Match the concepts in List I with their definitions in List II.

List I	List II
1. Ordinary Differential Equation	a. Involves derivatives with respect to more than one independent variable
2. Partial Differential Equation	b. Highest power of the highest order derivative in a polynomial differential equation
3. Order	c. Involves derivatives with respect to only one independent variable
4. Degree	d. Order of the highest derivative present

A 1–a, 2–c, 3–d, 4–b

B 1–c, 2–a, 3–d, 4–b

C 1–c, 2–a, 3–b, 4–d

D 1–b, 2–a, 3–d, 4–c

💡 Short Explanation

ODE uses one variable. PDE uses multiple variables. Order and degree follow standard definitions.

✅ Correct Answer Explanation

An ODE involves one independent variable (c), a PDE involves more than one (a), order is the order of the highest derivative (d), and degree is the highest power of the highest-order derivative in polynomial form (b). Thus the correct matching is Option B.

❓ Why Other Options Are Incorrect

Option A → Reverses ODE and PDE definitions.
Option C → Exchanges order and degree definitions.
Option D → Incorrectly matches ODE and degree.

🎯 Strategy

Used: Option Grouping

Application:

Match standard definitions one by one.

Final Logic:

All four standard definitions align with Option B.

🧠 Memory Trick

"ODE-One, PDE-Plural."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: Definitions and Classification

5 Which of the following correctly represent the second derivative of y with respect to x?
I. y''
II. y₂
III. (d²y)/(dx²)
IV. (dy/dx)²

A I and III only

B I, II, and III only

C I, III, and IV only

D All of the above

💡 Short Explanation

y'' and d²y/dx² denote second derivative. y₂ is not standard NCERT notation. (dy/dx)² is square of first derivative.

✅ Correct Answer Explanation

The standard notations for the second derivative are y'' and d²y/dx². The symbol y₂ is not used in NCERT as a second-derivative notation. Further, (dy/dx)² means the square of the first derivative, not the second derivative. Therefore the correct answer is A, not B.

❓ Why Other Options Are Incorrect

Option B → Incorrect because y₂ is not a standard second-derivative notation in NCERT.
Option C → Includes (dy/dx)², which is not a second derivative.
Option D → Includes both incorrect representations.

🎯 Strategy

Used: Elimination

Application:

Separate derivative notation from powers of derivatives.

Final Logic:

Only I and III are universally accepted second-derivative notations.

🧠 Memory Trick

"Double dash = second derivative."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: Derivative Notations

6 Let yₙ denote the n-th order derivative.
Given:
(d⁴y)/(dx⁴) + y₃ = 0
Find the sum of the orders of the derivatives present.

A 3

B 4

C 7

D 1

💡 Short Explanation

Fourth derivative is present. Third derivative is present. Sum = 4 + 3 = 7.

✅ Correct Answer Explanation

The equation contains a fourth-order derivative d⁴y/dx⁴ and a third-order derivative y₃. Adding their orders gives 4 + 3 = 7. Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Includes only the third-order derivative.
Option B → Includes only the fourth-order derivative.
Option D → Does not correspond to any derivative order present.

🎯 Strategy

Used: Substitution

Application:

Replace y₃ by "third derivative" and count orders.

Final Logic:

4 + 3 = 7.

🧠 Memory Trick

"Add derivative orders directly."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: Higher Order Derivatives

7 Differentiating the relation
∫(d²y)/(dx²) dx = cos x
with respect to x, what is the order of the resulting differential equation?

A 2

B 1

C 3

D Not defined

💡 Short Explanation

Integral of y'' gives y'. Differentiating yields y'' = −sin x. Highest derivative is second order.

✅ Correct Answer Explanation

From ∫y''dx = cos x, we obtain y' = cos x. Differentiating both sides gives y'' = −sin x. The resulting differential equation contains the second derivative as the highest derivative. Therefore the order is 2. The provided answer A is correct.

❓ Why Other Options Are Incorrect

Option B → Ignores the resulting second derivative.
Option C → No third derivative appears.
Option D → Order is clearly defined.

🎯 Strategy

Used: Substitution

Application:

Simplify the integral first and then differentiate.

Final Logic:

Resulting equation contains y'', so order = 2.

🧠 Memory Trick

"y'' survives after differentiation."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: Order of Differential Equation

8 Assertion (A):
The order of the differential equation
(dy/dx)³ + y = 0
is 3.
Reason (R):
Order is defined as the order of the highest order derivative present in the equation.

A Both A and R are false

B A is true, R is false

C Both A and R are true, and R explains A

D A is false, R is true

💡 Short Explanation

Highest derivative is dy/dx. Its order is one. Cubing does not change order.

✅ Correct Answer Explanation

The highest derivative present is dy/dx, which is first order. Raising it to the third power affects degree, not order. Therefore Assertion is false. The Reason correctly states the definition of order. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Reason is true.
Option B → Assertion is false.
Option C → Assertion is not true.

🎯 Strategy

Used: Elimination

Application:

Separate derivative order from derivative power.

Final Logic:

Order depends on derivative level, not exponent.

🧠 Memory Trick

"Power changes degree, not order."

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

9 Identify the incorrect statement regarding the degree of a differential equation.

A The degree is defined for all differential equations

B The equation must be polynomial in derivatives

C Equations like sin(y') = 0 have no defined degree

D Degree is the highest power of the highest order derivative

💡 Short Explanation

Degree requires polynomial form. Many equations lack a defined degree. Therefore Option A is incorrect.

✅ Correct Answer Explanation

Degree is defined only when the differential equation is polynomial in its derivatives. Therefore Option A is false. Options B, C, and D correctly describe the NCERT definition and conditions required for degree determination.

❓ Why Other Options Are Incorrect

Option B → Correct statement.
Option C → Correct because trigonometric derivatives prevent degree definition.
Option D → Correct NCERT definition of degree.

🎯 Strategy

Used: Elimination

Application:

Recall the polynomial condition for degree.

Final Logic:

Degree is not defined for all differential equations.

🧠 Memory Trick

"No polynomial, no degree."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: Degree of Differential Equation

10 Arrange the following equations in increasing order of their degree.
1. y''' + (y'')³ = 0
2. (y')⁴ + y = 0
3. y'' + y = 0

A 1, 2, 3

B 2, 1, 3

C 3, 1, 2

D 3, 2, 1

💡 Short Explanation

Equation 3 has degree 1. Equation 1 has degree 3. Equation 2 has degree 4.

✅ Correct Answer Explanation

For Equation 3, degree = 1. For Equation 1, the highest-order derivative is y''' with power 1, so degree = 1, not 3. For Equation 2, degree = 4. Therefore increasing order is actually 1 = 3 < 2, making the provided answer questionable. Based on NCERT definition, the correct arrangement should be 1, 3, 2 (not listed exactly).

❓ Why Other Options Are Incorrect

Option A → Does not follow actual degree values.
Option B → Places highest degree first.
Option D → Reverses the increasing order.

🎯 Strategy

Used: Option Grouping

Application:

Determine the degree of each equation separately.

Final Logic:

Degree depends on the highest-order derivative, not any derivative.

🧠 Memory Trick

"Degree follows highest-order derivative only."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: Degree of Differential Equation

11 The velocity vector v⃗ gives the system:
dx/dt = −y, dy/dt = x
Eliminating y yields:
(d²x)/(dt²) + x = 0
Determine the order and degree respectively.

A Order 1, Degree 2

B Order 2, Degree 2

C Order 1, Degree 1

D Order 2, Degree 1

💡 Short Explanation

Highest derivative is d²x/dt². Order equals 2. Highest-order derivative has power 1.

✅ Correct Answer Explanation

The equation d²x/dt² + x = 0 contains the second derivative as its highest derivative, so the order is 2. The equation is polynomial in derivatives and the highest-order derivative appears to the first power. Hence degree = 1. Therefore the correct answer is Option D.

❓ Why Other Options Are Incorrect

Option A → Order is incorrectly taken as 1.
Option B → Degree is not 2 because the second derivative has power 1.
Option C → Both order and degree are incorrectly identified.

🎯 Strategy

Used: Elimination

Application:

Find the highest derivative and then determine its power.

Final Logic:

Second derivative present with power 1 ⇒ Order 2, Degree 1.

🧠 Memory Trick

"Highest derivative → Order, its power → Degree."

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

12 The differential equation governing a bounded area A is:
e^(dA/dx) + x = 0
What is the degree of this differential equation?

A 0

B 1

C 2

D Not defined

💡 Short Explanation

Degree requires polynomial derivatives. Derivative appears in exponential form. Therefore degree is undefined.

✅ Correct Answer Explanation

In e^(dA/dx) + x = 0, the derivative dA/dx appears inside an exponential function. Since the equation is not polynomial in its derivatives, the degree cannot be defined. NCERT specifies that degree exists only for polynomial differential equations.

❓ Why Other Options Are Incorrect

Option A → Degree is not zero; it is undefined.
Option B → Polynomial condition is not satisfied.
Option C → No valid degree calculation exists.

🎯 Strategy

Used: Elimination

Application:

Check whether derivatives appear inside transcendental functions.

Final Logic:

Exponential derivative ⇒ Degree not defined.

🧠 Memory Trick

"Exp on derivative = No degree."

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

13 For any differential equation, when defined, the order and degree must be:

A Any real number

B Positive integers

C Whole numbers including zero

D Rational numbers

💡 Short Explanation

Order counts derivative level. Degree counts derivative power. Both are positive integers.

✅ Correct Answer Explanation

Order is the order of the highest derivative present, and degree is the power of the highest-order derivative when defined. Both are counting quantities and therefore must be positive integers according to NCERT definitions.

❓ Why Other Options Are Incorrect

Option A → Order and degree cannot be arbitrary real numbers.
Option C → Zero is not accepted as order or degree.
Option D → Fractional values are not allowed.

🎯 Strategy

Used: Contextual/Tonal Matching

Application:

Recall the formal definitions of order and degree.

Final Logic:

Both are positive integer values.

🧠 Memory Trick

"Order and Degree always count positively."

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

14 Consider the equation:
y'' + (y')² + cos(y') = 0
Determine its order and degree respectively.

A 2 and 2

B 1 and Not defined

C 2 and Not defined

D Not defined and 2

💡 Short Explanation

Highest derivative is y''. Order equals 2. cos(y') makes degree undefined.

✅ Correct Answer Explanation

The highest-order derivative present is y'', so the order is 2. However, the derivative y' appears inside cos(y'), making the equation non-polynomial in derivatives. Therefore degree cannot be defined. Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Degree cannot be assigned due to cos(y').
Option B → Order is not 1 because y'' is present.
Option D → Order is defined and equals 2.

🎯 Strategy

Used: Elimination

Application:

Determine order first, then check polynomial condition for degree.

Final Logic:

y'' gives order 2; cosine term destroys degree.

🧠 Memory Trick

"Cos derivative ⇒ Degree undefined."

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

15 The relation x² + y² = r² represents a circle. Differentiating with respect to x, which of the following equations is obtained?

A x + y dy/dx = 0

B 2x + y = 0

C dy/dx = x/y

D x² + y dy/dx = 0

💡 Short Explanation

Differentiate both sides. 2x + 2y(dy/dx) = 0. Dividing by 2 gives Option A.

✅ Correct Answer Explanation

Differentiating x² + y² = r² gives: 2x + 2y(dy/dx) = 0 Dividing throughout by 2: x + y(dy/dx) = 0 Therefore Option A is correct. This is a standard example of forming a differential equation from a relation.

❓ Why Other Options Are Incorrect

Option B → Omits the derivative term.
Option C → Solving Option A gives dy/dx = −x/y, not x/y.
Option D → Differentiation of x² is incorrect.

🎯 Strategy

Used: Substitution

Application:

Apply implicit differentiation to the circle equation.

Final Logic:

Derivative of y² gives 2y(dy/dx).

🧠 Memory Trick

"Differentiate y² ⇒ 2y y'."

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

16 Which of the following is a general solution of the differential equation:
(d²y)/(dx²) + y = 0?

A y = x²

B y = a cos x + b sin x

C y = e²ˣ

D y = ln x

💡 Short Explanation

Characteristic equation: m² + 1 = 0. Roots are ±i. Solution involves sine and cosine.

✅ Correct Answer Explanation

For y'' + y = 0, the auxiliary equation is m² + 1 = 0, giving roots ±i. Hence the general solution is y = a cos x + b sin x. The remaining options do not satisfy the differential equation for arbitrary constants.

❓ Why Other Options Are Incorrect

Option A → y'' + y = 2 + x² ≠ 0.
Option C → y'' + y = 5e²ˣ ≠ 0.
Option D → Does not satisfy the equation.

🎯 Strategy

Used: Substitution

Application:

Check which function satisfies y'' + y = 0.

Final Logic:

Trigonometric combination is the standard general solution.

🧠 Memory Trick

"y'' + y = 0 ⇒ Sin-Cos pair."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: General Solution

17 The equation
dP/dt = P/20
models continuous growth. What is its order?

A 1

B 2

C 0

D Not defined

💡 Short Explanation

Only first derivative appears. Highest derivative is dP/dt. Hence order is 1.

✅ Correct Answer Explanation

The equation contains only the first derivative dP/dt. Since order is determined by the highest derivative present, the order equals 1. This is a first-order differential equation commonly used for growth and decay models.

❓ Why Other Options Are Incorrect

Option B → No second derivative appears.
Option C → Presence of a derivative rules out order zero.
Option D → Order is clearly defined.

🎯 Strategy

Used: Elimination

Application:

Identify the highest derivative appearing.

Final Logic:

dP/dt is first order.

🧠 Memory Trick

"One derivative ⇒ Order one."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: Order of Differential Equation

18 The equation
dB/dt = kB
represents bacterial growth. What is its classification?

A Second-order ordinary differential equation

B Partial differential equation

C Not a differential equation

D First-order, first-degree ordinary differential equation

💡 Short Explanation

Highest derivative is first order. Derivative power is one. Hence first-order, first-degree ODE.

✅ Correct Answer Explanation

The equation contains only dB/dt, making it first order. The derivative appears to power one, so the degree is one. Since differentiation is with respect to a single variable t, it is an ordinary differential equation. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → No second derivative exists.
Option B → No partial derivatives are present.
Option C → It clearly contains a derivative.

🎯 Strategy

Used: Elimination

Application:

Determine order, degree, and variable count.

Final Logic:

First derivative, power one, one variable ⇒ first-order first-degree ODE.

🧠 Memory Trick

"dB/dt alone ⇒ First-first ODE."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: Classification of Differential Equations

19 Compare x + y = 7 and x(dy/dx) + y = 0. What distinguishes the second equation?

A It contains only independent variables

B It involves the derivative of a dependent variable

C The first is a partial differential equation

D The second has no defined order

💡 Short Explanation

Second equation contains dy/dx. First equation contains no derivative. This creates the distinction.

✅ Correct Answer Explanation

The equation x(dy/dx) + y = 0 contains the derivative dy/dx and is therefore a differential equation. The equation x + y = 7 is purely algebraic. The presence of the derivative of a dependent variable is the distinguishing feature.

❓ Why Other Options Are Incorrect

Option A → dy/dx appears, so not only independent variables.
Option C → The first equation is algebraic, not PDE.
Option D → The second equation has order 1.

🎯 Strategy

Used: Odd One Out

Application:

Identify the equation containing a derivative.

Final Logic:

Derivative presence distinguishes the second equation.

🧠 Memory Trick

"Derivative present ⇒ Differential equation."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: Definition of Differential Equation

20 Determine the order and degree of:
(d⁴y)/(dx⁴) + sin(y''') = 0

A Order 3, Degree 1

B Order 4, Degree 1

C Order 4, Degree not defined

D Order 3, Degree not defined

💡 Short Explanation

Highest derivative is y⁽⁴⁾. Order equals 4. Trigonometric derivative prevents degree definition.

✅ Correct Answer Explanation

The highest derivative present is d⁴y/dx⁴, so the order is 4. However, the derivative y''' appears inside sin(y'''), making the equation non-polynomial in derivatives. Therefore the degree is not defined. Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Ignores the fourth derivative.
Option B → Degree cannot be defined due to sine.
Option D → Order is 4, not 3.

🎯 Strategy

Used: Elimination

Application:

Determine order first, then test the polynomial condition.

Final Logic:

Fourth derivative exists; sine term destroys degree.

🧠 Memory Trick

"Sin on derivative ⇒ No degree."

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

CUET UG Mathematics Booster Test 1 - Basic Concepts of Differential Equations

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