CUET UG Mathematics Booster Test 1 - Introduction & Basics -

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CUET UG Mathematics Booster Test 1 - Introduction & Basics

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Mathematics Booster Test 1 - Introduction & Basics

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QUESTION 1 OF 20

In a graphical context, while differential calculus aims to find the tangent slope to a curve, integral calculus primarily aims to find:

Answer Review

1 In a graphical context, while differential calculus aims to find the tangent slope to a curve, integral calculus primarily aims to find:

A The concavity of the curve.

B The area bounded by the graph of the curve.

C The exact midpoint of the curve.

D The absolute maximum point.

💡 Short Explanation

Differential calculus studies slopes and rates of change. Integral calculus studies accumulation and area. Area under a curve is a primary application of integration.

✅ Correct Answer Explanation

Differential calculus focuses on derivatives, tangent slopes, and rates of change. Integral calculus developed from problems involving area under curves and accumulation of quantities. Therefore Option B is correct. Options A, C, and D concern other geometric properties and are not the primary objective of integration.

❓ Why Other Options Are Incorrect

Option A → Concavity is determined using second derivatives.
Option C → Midpoint determination is not a central goal of integration.
Option D → Absolute maxima involve optimization methods.

🎯 Strategy

Used: Odd One Out

Application: Distinguish area-related concepts from derivative-based concepts.

Final Logic: Integration is fundamentally associated with area calculation.

🧠 Memory Trick

"Derivative → Slope, Integral → Area."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Introduction to Integration

2 Using the area interpretation of integrals, what is the value of
∫₀ᵃ √(a²−x²) dx,
representing a quarter circle of radius a?

A πa²

B 1/2 πa²

C 1/4 πa²

D 2πa

💡 Short Explanation

Equation represents the upper semicircle. Limits 0 to a select one-quarter circle. Area equals one-fourth of πa².

✅ Correct Answer Explanation

The curve y=√(a²−x²) represents the upper half of a circle of radius a. Integrating from x=0 to x=a covers exactly one-quarter of the circle. Therefore the area equals (1/4)πa². Options A and B represent larger portions of the circle, while D is circumference-related.

❓ Why Other Options Are Incorrect

Option A → Represents the entire circle area.
Option B → Represents a semicircle area.
Option D → Represents circumference formula dimensions.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Recognize the graph as a quarter circle.

Final Logic: Quarter circle area = πa²/4.

🧠 Memory Trick

"Quarter Circle = πr²/4."

📚 Class XII – NCERT Mathematics – Chapter: Application of Integrals – Topic: Area Under Curves

3 Arrange the sequence to find the anti-derivative of x³:
1. Apply the power rule for integration xⁿ⁺¹/(n+1).
2. Add the constant of integration C.
3. Substitute n=3 to get x⁴/4.

A 1, 2, 3

B 2, 3, 1

C 3, 1, 2

D 1, 3, 2

💡 Short Explanation

Start with the integration rule. Substitute the exponent value. Add the integration constant last.

✅ Correct Answer Explanation

The correct procedure begins with applying the power rule. Next substitute n=3 to obtain x⁴/4. Finally add the arbitrary constant C to represent the complete family of anti-derivatives. Thus the proper order is 1 → 3 → 2.

❓ Why Other Options Are Incorrect

Option A → Adds C before completing integration.
Option B → Starts with the final step.
Option C → Substitutes before applying the rule.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Follow the logical sequence of integration.

Final Logic: Rule → Substitute → Add C.

🧠 Memory Trick

"Rule, Result, Constant."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Power Rule

4 Match the families of curves to their general representations:

List I	List II
1. Family of anti-derivatives of 2x	a. eˣ+C
2. Family of anti-derivatives of cos x	b. sin x+C
3. Family of anti-derivatives of 1	c. x+C
4. Family of anti-derivatives of eˣ	d. x²+C

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

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

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

D 1-c, 2-d, 3-a, 4-b

💡 Short Explanation

Integrate each function separately. Match with corresponding anti-derivative family. Include arbitrary constant C.

✅ Correct Answer Explanation

∫2x dx = x²+C, ∫cos x dx = sin x+C, ∫1 dx = x+C, and ∫eˣ dx = eˣ+C. Therefore the matching is 1-d, 2-b, 3-c, 4-a. This corresponds exactly to Option A.

❓ Why Other Options Are Incorrect

Option B → Incorrectly matches 2x and cos x.
Option C → Swaps anti-derivatives of cos x and eˣ.
Option D → Multiple mismatches occur.

🎯 Strategy

Used: Option Grouping

Application: Compute each anti-derivative and compare pairings.

Final Logic: Match each derivative with its anti-derivative family.

🧠 Memory Trick

"2x→x², cos→sin, 1→x, eˣ→eˣ."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Standard Anti-derivatives

5 The indefinite integral
∫(f(x)±g(x)) dx
is equivalent to which of the following properties?
1. ∫f(x) dx ± ∫g(x) dx
2. Represents a family of curves.
3. Evaluates to a discrete integer.

A 1 and 3 only

B 1 and 2 only

C 2 and 3 only

D 1, 2, and 3

💡 Short Explanation

Integration obeys linearity. Indefinite integrals form families of functions. They do not yield fixed integers.

✅ Correct Answer Explanation

The sum and difference rule gives ∫(f±g)dx = ∫f dx ± ∫g dx. An indefinite integral represents a family of curves because of the arbitrary constant C. Statement 3 is false since indefinite integrals are functions, not specific numerical values.

❓ Why Other Options Are Incorrect

Option A → Statement 3 is incorrect.
Option C → Statement 1 is also true.
Option D → Statement 3 remains false.

🎯 Strategy

Used: Option Grouping

Application: Verify each statement independently.

Final Logic: Only statements 1 and 2 describe indefinite integrals.

🧠 Memory Trick

"Linearity + Family."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Properties of Indefinite Integrals

6 Identify the INCORRECT statement concerning integral notation:

A ∫ represents the entire class of anti-derivatives.

B dx indicates the variable of integration is x.

C The interval of integration must always be explicitly stated in an indefinite integral.

D f(x) is the integrand.

💡 Short Explanation

Indefinite integrals have no limits. Limits are required only for definite integrals. Therefore statement C is incorrect.

✅ Correct Answer Explanation

An indefinite integral does not require upper or lower limits. It represents a family of anti-derivatives. Option B correctly identifies the variable of integration, and Option D correctly identifies the integrand. Therefore Option C is the incorrect statement.

❓ Why Other Options Are Incorrect

Option A → Correctly relates integration to anti-derivative families.
Option B → dx specifies the integration variable.
Option D → f(x) is indeed the integrand.

🎯 Strategy

Used: Elimination

Application: Separate properties of definite and indefinite integrals.

Final Logic: Only definite integrals require intervals.

🧠 Memory Trick

"No Limits = Indefinite."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Integral Notation

7 A particle has velocity vector
v⃗(t)=sin t î+cos t ĵ.
If the initial position is (0,1) at t=0, integrating v⃗(t) gives its position. What is the position vector at t=π/2?

A î+ĵ

B -î+2ĵ

C 0î+0ĵ

D î+2ĵ

💡 Short Explanation

Integrate each velocity component. Apply initial position conditions. Evaluate at t=π/2.

✅ Correct Answer Explanation

Integrating gives x(t)=−cos t+C₁ and y(t)=sin t+C₂. Using position (0,1) at t=0 gives C₁=1 and C₂=1. Hence x(t)=1−cos t and y(t)=sin t+1. At t=π/2, position=(1,2). Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect y-coordinate.
Option B → Incorrect x-coordinate sign.
Option C → Ignores displacement from initial conditions.

🎯 Strategy

Used: Substitution

Application: Integrate velocity and apply initial conditions.

Final Logic: Position equals integrated velocity plus constants.

🧠 Memory Trick

"Velocity Integral = Position."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Applications of Integration

8 The probability of a random event occurring between x=0 and x=π/2 is modeled by
∫₀^(π/2) cos x dx.
Evaluate this exact probability.

A 1

B 0

C π/2

D -1

💡 Short Explanation

Integral of cos x is sin x. Apply limits 0 and π/2. Result equals 1.

✅ Correct Answer Explanation

∫₀^(π/2) cos x dx = [sin x]₀^(π/2) = sin(π/2)−sin(0)=1−0=1. Therefore Option A is correct. The remaining options do not follow from evaluating the definite integral.

❓ Why Other Options Are Incorrect

Option B → Ignores positive accumulated area.
Option C → Confuses result with interval length.
Option D → Integral is positive, not negative.

🎯 Strategy

Used: Substitution

Application: Evaluate the anti-derivative at limits.

Final Logic: sin(π/2)−sin(0)=1.

🧠 Memory Trick

"Cos → Sin → One."

📚 Class XII – NCERT Mathematics – Chapter: Definite Integrals – Topic: Evaluation Using Anti-derivatives

9 Assertion (A): Evaluating ∫eˣ dx gives exactly one specific curve, y=eˣ.
Reason (R): The indefinite integral produces a family of anti-derivatives containing an arbitrary constant C.

A Both false

B A false, R true

C Both true, R explains A

D A true, R false

💡 Short Explanation

Indefinite integrals include constant C. Infinite anti-derivatives exist. Assertion ignores the constant.

✅ Correct Answer Explanation

The integral ∫eˣdx equals eˣ+C, not a single unique curve. Therefore the assertion is false. The reason is true because indefinite integrals represent a family of anti-derivatives differing by constants. Hence Option B is correct.

❓ Why Other Options Are Incorrect

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

🎯 Strategy

Used: Elimination

Application: Check assertion and reason independently.

Final Logic: Missing +C makes the assertion false.

🧠 Memory Trick

"Never Forget +C."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Family of Anti-derivatives

10 Evaluate the definite integral:
∫₀^(π/4) sec²x dx.

A 0

B π/4

C 1

D -1

💡 Short Explanation

Integral of sec²x is tan x. Evaluate between limits. tan(π/4)=1 and tan(0)=0.

✅ Correct Answer Explanation

Using the standard integral, ∫sec²x dx = tan x. Applying limits gives tan(π/4)−tan(0)=1−0=1. Therefore Option C is correct. Other options arise from incorrect use of trigonometric values or anti-derivatives.

❓ Why Other Options Are Incorrect

Option A → Integral over a positive interval cannot be zero.
Option B → Confuses answer with upper limit.
Option D → Result is positive, not negative.

🎯 Strategy

Used: Substitution

Application: Use the standard anti-derivative and evaluate limits.

Final Logic: tan(π/4)−tan(0)=1.

🧠 Memory Trick

"sec² → tan."

📚 Class XII – NCERT Mathematics – Chapter: Definite Integrals – Topic: Standard Integrals

11 Data of a machine's production rate is continuously modeled by
R(t)=eᵗ.
Calculate the total integrated production data from t=0 to t=ln 5.

A 5

B e⁵

C 3

D 4

💡 Short Explanation

Total production equals the definite integral of the rate. Anti-derivative of eᵗ is eᵗ. Evaluate between 0 and ln 5.

✅ Correct Answer Explanation

Total production is ∫₀ˡⁿ⁵ eᵗ dt = [eᵗ]₀ˡⁿ⁵ = eˡⁿ⁵−e⁰ = 5−1 = 4. Therefore Option D is correct. Option A ignores subtraction of the lower limit, while Options B and C arise from incorrect evaluation.

❓ Why Other Options Are Incorrect

Option A → Uses only the upper-limit value.
Option B → Incorrectly evaluates e^(ln 5).
Option C → Does not satisfy definite integral computation.

🎯 Strategy

Used: Substitution

Application: Find the anti-derivative and apply upper-minus-lower evaluation.

Final Logic: e^(ln5)−1 = 5−1 = 4.

🧠 Memory Trick

"e and ln cancel."

📚 Class XII – NCERT Mathematics – Chapter: Definite Integrals – Topic: Evaluation Using Fundamental Theorem

12 The connection between definite and indefinite integrals states
∫ₐᵇf(x)dx = F(b)−F(a).
Use this to find the value of
∫₁² (1/x) dx.

A ln 2

B ln 3

C e²

D 1/2

💡 Short Explanation

Anti-derivative of 1/x is ln|x|. Apply limits 1 and 2. Result simplifies to ln 2.

✅ Correct Answer Explanation

Using F(x)=ln|x|, we obtain ∫₁²(1/x)dx = ln2−ln1. Since ln1=0, the value is ln2. Therefore Option A is correct. The other options do not follow from applying the Fundamental Theorem of Calculus correctly.

❓ Why Other Options Are Incorrect

Option B → Would require different limits.
Option C → Not obtained from logarithmic integration.
Option D → Incorrect numerical interpretation.

🎯 Strategy

Used: Substitution

Application: Apply the Fundamental Theorem directly.

Final Logic: ln2−0 = ln2.

🧠 Memory Trick

"1/x → Log."

📚 Class XII – NCERT Mathematics – Chapter: Definite Integrals – Topic: Fundamental Theorem of Calculus

13 In the equation
∫xeˣ dx = eˣ(x−1)+C,
the integrand is:

A eˣ(x−1)

B xeˣ

C dx

D C

💡 Short Explanation

Integrand is the function being integrated. It appears between ∫ and dx. Here it is xeˣ.

✅ Correct Answer Explanation

The integrand is the expression placed immediately after the integral sign and before dx. In ∫xeˣdx, the integrand is xeˣ. Option A is part of the anti-derivative, Option C is the differential, and Option D is the integration constant.

❓ Why Other Options Are Incorrect

Option A → Represents the anti-derivative result.
Option C → Indicates the variable of integration.
Option D → Constant added after integration.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Identify standard parts of integral notation.

Final Logic: Function inside the integral sign is the integrand.

🧠 Memory Trick

"Inside ∫ = Integrand."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Terminology of Integration

14 Match the integral notation to its variable of integration:

List I	List II
1. ∫sin θ dθ	a. y
2. ∫2y dy	b. u
3. ∫eᵘ du	c. θ
4. ∫x² dx	d. x

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

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

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

D 1-d, 2-c, 3-b, 4-a

💡 Short Explanation

Variable of integration is indicated by the differential. dθ corresponds to θ, dy to y. Match each notation accordingly.

✅ Correct Answer Explanation

The variable of integration is identified by the symbol following d. Thus dθ→θ, dy→y, du→u, and dx→x. Hence the correct matching is 1-c, 2-a, 3-b, 4-d, which corresponds to Option C.

❓ Why Other Options Are Incorrect

Option A → Incorrectly swaps θ and u.
Option B → Multiple variables mismatched.
Option D → Assigns incorrect variables throughout.

🎯 Strategy

Used: Option Grouping

Application: Match each differential with its variable.

Final Logic: Symbol after d determines the integration variable.

🧠 Memory Trick

"dx→x, dy→y, du→u."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Integral Notation

15 Assertion (A): Two indefinite integrals with the same derivative lead to different families of curves.
Reason (R): If
d/dx ∫f(x)dx = d/dx ∫g(x)dx,
then
∫f(x)dx = ∫g(x)dx + C,
showing equivalence.

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Same derivative implies functions differ only by a constant. They belong to the same family. Reason correctly states this property.

✅ Correct Answer Explanation

If two functions have the same derivative, they differ only by a constant. Therefore they belong to the same family of anti-derivatives, making the assertion false. The reason correctly expresses this relationship through the constant C. 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: Test the assertion and reason independently.

Final Logic: Same derivative ⇒ same family up to a constant.

🧠 Memory Trick

"Same Derivative, Same Family."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Family of Anti-derivatives

16 Which of the following is INCORRECT regarding reverse differentiation?

A ∫sin x dx = cos x + C

B ∫cos x dx = sin x + C

C ∫sec²x dx = tan x + C

D ∫eˣ dx = eˣ + C

💡 Short Explanation

Integral reverses differentiation. Derivative of cos x is −sin x. Correct integral is −cos x + C.

✅ Correct Answer Explanation

Since d/dx(−cos x)=sin x, we have ∫sin x dx=−cos x+C. Therefore Option A is incorrect. Options B, C, and D are standard integration formulas and are correctly stated.

❓ Why Other Options Are Incorrect

Option B → Correct because d/dx(sin x)=cos x.
Option C → Correct because d/dx(tan x)=sec²x.
Option D → Correct because eˣ is its own derivative.

🎯 Strategy

Used: Substitution

Application: Differentiate each proposed anti-derivative.

Final Logic: Only Option A differentiates to −sin x.

🧠 Memory Trick

"Sin integrates to −Cos."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Standard Integrals

17 Why is the arbitrary constant C referred to as a parameter?
1. It determines a specific unique function if an initial condition is given.
2. Varying C yields different anti-derivatives.
3. It changes the derivative of the function entirely.

A 1 and 3 only

B 1 and 2 only

C 2 and 3 only

D 1, 2, and 3

💡 Short Explanation

C selects a member of the family. Different C values give different curves. Derivatives remain unchanged.

✅ Correct Answer Explanation

The constant C acts as a parameter because changing its value produces different members of the anti-derivative family. When an initial condition is supplied, a unique value of C is determined. Statement 3 is false because constants vanish upon differentiation.

❓ Why Other Options Are Incorrect

Option A → Statement 3 is incorrect.
Option C → Statement 1 is also true.
Option D → Statement 3 remains false.

🎯 Strategy

Used: Option Grouping

Application: Evaluate each statement independently.

Final Logic: Only statements 1 and 2 correctly describe C.

🧠 Memory Trick

"C Chooses the Curve."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Arbitrary Constant

18 The integral
∫2x dx = x² + C
geometrically represents a family of parabolas. How do these parabolas relate to each other on a graph?

A They share the same roots.

B They are shifted horizontally.

C They are shifted vertically along the y-axis.

D They intersect at the origin.

💡 Short Explanation

Constant C changes only y-values. Shape of parabola remains unchanged. Vertical translations are produced.

✅ Correct Answer Explanation

Adding different constants to x² shifts the graph upward or downward without affecting shape. Thus all curves x²+C are vertical translations of one another. They are not horizontal shifts, do not necessarily share roots, and do not always pass through the origin.

❓ Why Other Options Are Incorrect

Option A → Roots change with C.
Option B → Horizontal shifts require changes inside x.
Option D → Most family members do not pass through the origin.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Recall the geometric effect of adding a constant.

Final Logic: +C creates vertical displacement.

🧠 Memory Trick

"+C = Vertical Shift."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Family of Curves

📄 Passage

There is a connection, known as the Fundamental Theorem of Calculus, between indefinite integral and definite integral which makes the definite integral a practical tool for science and engineering. The definite integral is also used to solve many interesting problems from various disciplines like economics, finance and probability. The development of integral calculus arises out of the efforts of solving problems such as finding a function whenever its derivative is given and finding the area bounded by the graph of a function.

Question:
Based on the passage's application to finance, if the marginal cost function of producing an item is
MC = 2x + 5,
find the total accumulated cost to produce 10 units (from x=0) by integration.

A 100

B 50

C 125

D 150

💡 Short Explanation

Total cost equals integral of marginal cost. Integrate from 0 to 10. Evaluate the definite integral.

✅ Correct Answer Explanation

Total accumulated cost = ∫₀¹⁰(2x+5)dx = [x²+5x]₀¹⁰ = (100+50)−0 = 150. Hence Option D is correct. The remaining options result from incomplete integration or incorrect evaluation of the limits.

❓ Why Other Options Are Incorrect

Option A → Omits part of the integrated expression.
Option B → Uses only the constant contribution.
Option C → Results from arithmetic error.

🎯 Strategy

Used: Substitution

Application: Integrate the marginal cost and evaluate limits.

Final Logic: x²+5x evaluated from 0 to 10 equals 150.

🧠 Memory Trick

"Marginal Cost → Integrate."

📚 Class XII – NCERT Mathematics – Chapter: Definite Integrals – Topic: Applications of Integration

📄 Passage

Question:
According to the passage, which theorem explicitly makes the definite integral a practical tool for science, engineering, and probability?

A Fundamental Theorem of Calculus

B Mean Value Theorem

C Chain Rule Theorem

D L'Hôpital's Principle

💡 Short Explanation

Passage explicitly names the theorem. It connects differentiation and integration. It enables practical computation of definite integrals.

✅ Correct Answer Explanation

The passage directly states that the Fundamental Theorem of Calculus connects indefinite and definite integrals and makes definite integrals practical tools in science, engineering, economics, finance, and probability. Therefore Option A is correct. The other theorems serve different purposes in calculus.

❓ Why Other Options Are Incorrect

Option B → Concerns average rates and function behavior.
Option C → Deals with derivatives of composite functions.
Option D → Helps evaluate certain limits, not definite integrals.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Locate the theorem explicitly mentioned in the passage.

Final Logic: The passage directly names the Fundamental Theorem of Calculus.

🧠 Memory Trick

"FTC Connects All."

📚 Class XII – NCERT Mathematics – Chapter: Definite Integrals – Topic: Fundamental Theorem of Calculus

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