CUET UG Mathematics Booster Test 1 - Integration as Inverse Process -

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CUET UG Mathematics Booster Test 1 - Integration as Inverse Process

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Mathematics Booster Test 1 - Integration as Inverse Process

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QUESTION 1 OF 20

If the inverse process of differentiation of a function h(x) yields h(x) itself, which of the following is true?

Answer Review

1 If the inverse process of differentiation of a function h(x) yields h(x) itself, which of the following is true?

A h(x)=x²

B h(x)=sin x

C h(x)=eˣ

D h(x)=log x

💡 Short Explanation

Integration reverses differentiation. Function must equal its own derivative. eˣ uniquely satisfies this property.

✅ Correct Answer Explanation

If the inverse process of differentiation returns the same function, then h'(x)=h(x). Among the given options, only eˣ has the property that its derivative equals itself. Therefore ∫eˣdx=eˣ+C. Hence Option C is correct, while the other functions change upon differentiation.

❓ Why Other Options Are Incorrect

Option A → d/dx(x²)=2x, not x².
Option B → d/dx(sin x)=cos x.
Option D → d/dx(log x)=1/x.

🎯 Strategy

Used: Elimination

Application: Differentiate each option and compare with the original function.

Final Logic: Only eˣ remains unchanged after differentiation.

🧠 Memory Trick

"eˣ Stays eˣ."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Integration as Inverse Process of Differentiation

2 The region bounded by y=3x², the x-axis, and x=1 has an area equal to its primitive evaluated at 1 (assuming C=0). What is the primitive function F(x) and the final area?

A F(x)=x², Area=1

B F(x)=3x³, Area=3

C F(x)=x⁴, Area=1

D F(x)=x³, Area=1

💡 Short Explanation

Primitive of 3x² is x³. Evaluate from 0 to 1. Area equals 1.

✅ Correct Answer Explanation

The primitive of 3x² is F(x)=∫3x²dx=x³+C. With C=0, F(x)=x³. The area under the curve from 0 to 1 is [x³]₀¹=1−0=1. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Derivative of x² is 2x, not 3x².
Option B → Derivative of 3x³ is 9x².
Option C → Derivative of x⁴ is 4x³.

🎯 Strategy

Used: Substitution

Application: Apply the power rule and evaluate the definite integral.

Final Logic: ∫3x²dx=x³ and x³(1)=1.

🧠 Memory Trick

"3x² Integrates to x³."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Primitive Functions

3 Assertion (A): The integral of cos(2x) is sin(2x)+C.
Reason (R): Differentiation of sin(2x)+C gives -cos(2x).

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Chain rule must be considered. ∫cos(2x)dx=(1/2)sin(2x)+C. Derivative of sin(2x) is 2cos(2x).

✅ Correct Answer Explanation

The assertion is false because ∫cos(2x)dx=(1/2)sin(2x)+C. The reason is also false because d/dx[sin(2x)+C]=2cos(2x), not −cos(2x). Hence both statements are false and Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Assertion is false.
Option C → Both statements are not true.
Option D → Reason is also false.

🎯 Strategy

Used: Elimination

Application: Verify assertion and reason independently using differentiation.

Final Logic: Both expressions ignore the chain rule.

🧠 Memory Trick

"Inside 2 ⇒ Divide by 2."

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

4 Match the exponential functions with their correct derivatives.

List I	List II
1. d/dx(e³ˣ)	a. -e⁻ˣ
2. d/dx(e⁻ˣ)	b. 3e³ˣ
3. d/dx(eˣ⁄²)	c. (1/2)eˣ⁄²

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

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

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

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

💡 Short Explanation

Apply chain rule. Differentiate exponent first. Multiply by original exponential.

✅ Correct Answer Explanation

Using d/dx(eᵘ)=eᵘ·u', we get: d/dx(e³ˣ)=3e³ˣ (b) d/dx(e⁻ˣ)=−e⁻ˣ (a) d/dx(eˣ⁄²)=(1/2)eˣ⁄² (c) Thus Option B is correct.

❓ Why Other Options Are Incorrect

Option A → First two matches are reversed.
Option C → Incorrect assignment of derivatives.
Option D → Multiple chain-rule errors.

🎯 Strategy

Used: Option Grouping

Application: Differentiate each exponential separately.

Final Logic: Chain rule produces the correct matches.

🧠 Memory Trick

"Differentiate Exponent, Multiply Exponential."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Exponential Functions

5 Which of the following equations demonstrate correct derivative relations for power functions?
1. d/dx(x³/3)=x²
2. d/dx(1/x)=−1/x²
3. d/dx(√x)=1/(2√x)

A 1 and 2 only

B 2 and 3 only

C 1, 2 and 3

D 1 and 3 only

💡 Short Explanation

Apply power rule. Rewrite radicals as powers. All three results are standard.

✅ Correct Answer Explanation

Using d/dx(xⁿ)=nxⁿ⁻¹: 1. d/dx(x³/3)=x² ✓ 2. d/dx(x⁻¹)=−x⁻²=−1/x² ✓ 3. d/dx(x¹ᐟ²)=(1/2)x⁻¹ᐟ²=1/(2√x) ✓ All three statements are correct. Hence Option C.

❓ Why Other Options Are Incorrect

Option A → Omits statement 3.
Option B → Omits statement 1.
Option D → Omits statement 2.

🎯 Strategy

Used: Option Grouping

Application: Test each derivative independently.

Final Logic: All three follow directly from the power rule.

🧠 Memory Trick

"Power Down, Exponent Down by One."

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

6 Find the area under the curve y=x³ from x=0 to x=2 using the standard integral formula for xⁿ.

A 2

B 8

C 6

D 4

💡 Short Explanation

Area equals definite integral. Integrate x³ to x⁴/4. Evaluate between limits.

✅ Correct Answer Explanation

Area =∫₀²x³dx =[x⁴/4]₀² =16/4 =4. Therefore the correct answer is D) 4. The supplied answer B) 8 is incorrect.

❓ Why Other Options Are Incorrect

Option A → Underestimates the integral.
Option B → Ignores division by 4.
Option C → Incorrect evaluation.

🎯 Strategy

Used: Substitution

Application: Apply the power rule and evaluate limits.

Final Logic: x⁴/4 from 0 to 2 equals 4.

🧠 Memory Trick

"x³ → x⁴/4."

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

7 A random variable X has a density function f(x)=e⁻ˣ for x≥0. The probability P(0<X<1) is found using the integral of e⁻ˣ. What is the probability?

A 1−1/e

B 1/e

C 1+1/e

D e−1

💡 Short Explanation

Integrate e⁻ˣ. Apply limits 0 and 1. Simplify result.

✅ Correct Answer Explanation

P(0<X<1) =∫₀¹e⁻ˣdx =[−e⁻ˣ]₀¹ =−1/e+1 =1−1/e. Therefore Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Only the exponential term.
Option C → Exceeds probability limit.
Option D → Wrong expression.

🎯 Strategy

Used: Substitution

Application: Evaluate the definite integral directly.

Final Logic: Probability equals 1−e⁻¹.

🧠 Memory Trick

"Integral of e⁻ˣ = −e⁻ˣ."

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

8 The displacement vector of a particle is given by
s⃗(x)=(∫2ˣdx)î +(d/dx(2ˣ))ĵ.
Assuming the constant of integration is 0, which expression represents s⃗(x)?

A (2ˣln2)î +(2ˣ/ln2)ĵ

B (2ˣ/ln2)î +(2ˣln2)ĵ

C (x·2ˣ⁻¹)î +(2ˣ)ĵ

D (2ˣ)î +(2ˣx)ĵ

💡 Short Explanation

Integral and derivative formulas for aˣ. Apply standard results. Combine into vector form.

✅ Correct Answer Explanation

For aˣ, ∫aˣdx=aˣ/ln(a) and d/dx(aˣ)=aˣln(a). With a=2: ∫2ˣdx=2ˣ/ln2 and d/dx(2ˣ)=2ˣln2. Hence Option B.

❓ Why Other Options Are Incorrect

Option A → Integral and derivative are interchanged.
Option C → Not a standard derivative/integral result.
Option D → Incorrect vector components.

🎯 Strategy

Used: Substitution

Application: Use standard exponential formulas.

Final Logic: Integral divides by ln2; derivative multiplies by ln2.

🧠 Memory Trick

"Integral Divide Log, Derivative Multiply Log."

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

9 Arrange the definite integrals of 1/x from x=1 to x=k for k=2,4,e,3 in ascending order of their mathematical values.

A 1, 2, 3, 4

B 1, 4, 3, 2

C 1, 3, 4, 2

D 3, 1, 4, 2

💡 Short Explanation

Integral equals ln(k). Compare logarithms. Arrange increasing values.

✅ Correct Answer Explanation

Values are: k=2 → ln2≈0.693 k=e → 1 k=3 → ln3≈1.099 k=4 → ln4≈1.386 Ascending order: 2, e, 3, 4 i.e. 1 → 3 → 4 → 2. Hence Option C.

❓ Why Other Options Are Incorrect

Option A → Places ln4 before ln3.
Option B → Incorrect ordering of logarithms.
Option D → Begins with a larger value.

🎯 Strategy

Used: Substitution

Application: Convert each integral into ln(k).

Final Logic: Compare logarithmic values numerically.

🧠 Memory Trick

"Integral of 1/x = ln x."

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

10 In an electronic circuit, the charge q(t) satisfies
dq/dt=5/t, t>0.
If q(1)=2, find the numerical value of q(e).

A 3

B 5

C 10

D 7

💡 Short Explanation

Integrate 5/t. Use initial condition. Evaluate at t=e.

✅ Correct Answer Explanation

Integrating: q(t)=5ln(t)+C. Given q(1)=2: 2=5ln(1)+C ⇒ C=2. Therefore q(e)=5ln(e)+2 =5+2 =7. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Ignores coefficient 5.
Option B → Omits constant 2.
Option C → Adds incorrectly.

🎯 Strategy

Used: Substitution

Application: Integrate and use the initial condition.

Final Logic: q(e)=5+2=7.

🧠 Memory Trick

"ln1=0, lne=1."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Integral of 1/x

11 Evaluate the indefinite integral
∫(sin x + cos x) dx.

A -cos x + sin x + C

B cos x + sin x + C

C -sin x - cos x + C

D cos x - sin x + C

💡 Short Explanation

Integrate each term separately. ∫sin x dx = -cos x. ∫cos x dx = sin x.

✅ Correct Answer Explanation

Using linearity of integration: ∫(sin x + cos x)dx = ∫sin x dx + ∫cos x dx = -cos x + sin x + C. Option A is correct. Options B, C, and D do not differentiate back to sin x + cos x.

❓ Why Other Options Are Incorrect

Option B → Derivative gives -sin x + cos x.
Option C → Derivative gives -cos x + sin x.
Option D → Derivative gives -sin x - cos x.

🎯 Strategy

Used: Substitution

Application: Apply standard trigonometric integral formulas term-by-term.

Final Logic: Integrate sin and cos separately, then combine.

🧠 Memory Trick

"sin → -cos, cos → sin."

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

12 The rate of data transfer over a channel is given by
R(x)=sec x tan x.
The total data transferred between x=0 and x=π/3 is the integral of R(x). Find this total data amount.

A 0

B 1

C 2

D 3

💡 Short Explanation

Integral of sec x tan x is sec x. Evaluate between limits. Result equals 2.

✅ Correct Answer Explanation

The provided answer is incorrect. Since ∫sec x tan x dx = sec x, the required value is [sec x]₀^(π/3) = sec(π/3) − sec(0) = 2 − 1 = 1. Wait: evaluating carefully gives 1. Therefore the correct answer is B) 1.

❓ Why Other Options Are Incorrect

Option A → Integral over a positive interval is not zero.
Option C → Equals sec(π/3) only; lower limit omitted.
Option D → Not obtained from definite integration.

🎯 Strategy

Used: Substitution

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

Final Logic: sec(π/3)−sec(0)=2−1=1.

🧠 Memory Trick

"sec tan → sec."

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

13 Which statement is incorrect regarding the integral of
1/√(1-x²)?

A Its integral evaluates to sin⁻¹x + C.

B Its integral can be equivalently expressed as -cos⁻¹x + C.

C The integrand is defined for all real numbers x.

D The integral relates to inverse trigonometric forms.

💡 Short Explanation

Denominator must remain real. 1−x² ≥ 0 is required. Domain is limited to |x|≤1.

✅ Correct Answer Explanation

The function 1/√(1−x²) is defined only when 1−x²>0, i.e., |x|<1. Therefore it is not defined for all real numbers. Options A, B, and D are standard inverse trigonometric integration results. Hence Option C is the incorrect statement.

❓ Why Other Options Are Incorrect

Option A → Standard integral formula.
Option B → Since d/dx(cos⁻¹x)=−1/√(1−x²).
Option D → Integral belongs to inverse trigonometric forms.

🎯 Strategy

Used: Extreme Word Filter

Application: The phrase "all real numbers" signals a domain issue.

Final Logic: Square root requires 1−x²>0.

🧠 Memory Trick

"Inverse Sine Lives Inside ±1."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Inverse Trigonometric Integrals

14 What is the integral of
1/(1+x²)
with respect to x?

A cot⁻¹x + C

B sin⁻¹x + C

C cos⁻¹x + C

D tan⁻¹x + C

💡 Short Explanation

Standard inverse trigonometric integral. Derivative of tan⁻¹x is 1/(1+x²). Reverse differentiation applies.

✅ Correct Answer Explanation

Since d/dx(tan⁻¹x)=1/(1+x²), its anti-derivative is ∫dx/(1+x²)=tan⁻¹x+C. Therefore Option D is correct. The remaining inverse trigonometric functions have different derivatives.

❓ Why Other Options Are Incorrect

Option A → Derivative is −1/(1+x²).
Option B → Derivative is 1/√(1−x²).
Option C → Derivative is −1/√(1−x²).

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Recall the standard inverse trigonometric formula.

Final Logic: tan⁻¹x differentiates to 1/(1+x²).

🧠 Memory Trick

"1+x² → tan⁻¹."

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

15 Assertion (A): ∫2x dx = x³.
Reason (R): The integration constant is not required for indefinite integrals.

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

∫2x dx = x² + C. Constant is required. Both statements are false.

✅ Correct Answer Explanation

Integrating 2x gives ∫2x dx = x² + C, not x³. Hence the assertion is false. The reason is also false because indefinite integrals must include an arbitrary constant C. Therefore both statements are false.

❓ Why Other Options Are Incorrect

Option B → Assertion is false.
Option C → Neither statement is true.
Option D → Reason is false.

🎯 Strategy

Used: Elimination

Application: Check assertion and reason independently.

Final Logic: Both statements contradict integration rules.

🧠 Memory Trick

"2x Integrates to x²."

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

16 The family of curves
y=x²+C
represents infinite solutions to dy/dx=2x. Graphically, what does changing the value of C do to the region bounded by the curve?

A Rotates the curve.

B Translates the curve vertically, shifting the region up or down.

C Translates the curve horizontally.

D Reflects the curve across the x-axis.

💡 Short Explanation

C affects only y-values. Shape remains unchanged. Vertical shift occurs.

✅ Correct Answer Explanation

Changing C in y=x²+C adds a constant to every y-coordinate. The parabola keeps the same shape and orientation but moves upward or downward. Hence Option B is correct. No rotation, horizontal translation, or reflection occurs.

❓ Why Other Options Are Incorrect

Option A → Constant does not rotate graphs.
Option C → Horizontal shifts require x replacement.
Option D → Reflection needs multiplication by −1.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Analyze the graphical effect of adding a constant.

Final Logic: Adding C causes vertical translation.

🧠 Memory Trick

"+C Moves Up/Down."

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

17 Match the relation definitions correctly based on standard integration notation.

List I	List II
1. f(x)	a. Constant of integration
2. F(x)	b. Integrand
3. C	c. Anti-derivative

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

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

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

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

💡 Short Explanation

f(x) is integrated. F(x) is the anti-derivative. C is arbitrary constant.

✅ Correct Answer Explanation

In ∫f(x)dx = F(x)+C, f(x) is the integrand, F(x) is an anti-derivative, and C is the constant of integration. Therefore the correct matching is 1-b, 2-c, 3-a.

❓ Why Other Options Are Incorrect

Option A → Assigns incorrect roles.
Option B → Swaps integrand and anti-derivative.
Option D → Misidentifies F(x).

🎯 Strategy

Used: Option Grouping

Application: Recall standard notation components.

Final Logic: Integrand → Anti-derivative → Constant.

🧠 Memory Trick

"f, F, C = Integrand, Primitive, Constant."

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

18 The expression
∫f(x)dx = F(x)+C
implies which of the following analytical truths?
1. It represents the entire class of anti-derivatives.
2. F'(x)=f(x).
3. C is a fixed, non-variable parameter that cannot change across different curve families.

A 1 and 3 only

B 2 and 3 only

C 1, 2 and 3

D 1 and 2 only

💡 Short Explanation

Anti-derivatives form a family. F'(x)=f(x). C can vary.

✅ Correct Answer Explanation

Statements 1 and 2 are true because F(x)+C represents all anti-derivatives and F'(x)=f(x). Statement 3 is false because different values of C generate different members of the family. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Includes false statement 3.
Option B → Omits true statement 1.
Option C → Statement 3 is incorrect.

🎯 Strategy

Used: Option Grouping

Application: Verify each statement separately.

Final Logic: Only statements 1 and 2 hold.

🧠 Memory Trick

"C Changes, Family Changes."

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

📄 Passage

In practice, we normally do not mention the interval over which the various functions are defined. However, in any specific problem one has to keep it in mind. The process of differentiation and integration are inverses of each other in the sense that
d/dx [∫f(x) dx]=f(x)
and
∫f'(x) dx=f(x)+C,
where C is any arbitrary constant. Thus, functions with the same derivatives differ by a constant.

According to the passage, what is the result of differentiating the integral of f(x)?

A f(x)

B f'(x)

C f(x)+C

D 0

💡 Short Explanation

Differentiation and integration are inverse processes. Differentiating an integral restores the function. Constant disappears.

✅ Correct Answer Explanation

The passage explicitly states d/dx[∫f(x)dx]=f(x). Therefore differentiating the integral of f(x) returns the original function. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Gives derivative instead of original function.
Option C → Constant is removed by differentiation.
Option D → Result is not zero.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Directly apply the statement given in the passage.

Final Logic: Differentiate the integral → recover f(x).

🧠 Memory Trick

"Derivative Cancels Integral."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Inverse Relationship

📄 Passage

If you integrate the derivative of f(x) according to the passage, the result yields:

A f'(x)+C

B f(x)+C

C f(x)

D f''(x)

💡 Short Explanation

Integration reverses differentiation. Arbitrary constant appears. Original function is restored.

✅ Correct Answer Explanation

The passage explicitly gives ∫f'(x)dx=f(x)+C. Thus integrating the derivative of a function restores the original function along with an arbitrary constant. Therefore Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Returns derivative, not anti-derivative.
Option C → Omits the constant C.
Option D → Represents second derivative.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Apply the exact formula from the passage.

Final Logic: ∫f'(x)dx=f(x)+C.

🧠 Memory Trick

"Integrate Derivative → Function + C."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Inverse Relationship of Differentiation and Integration

CUET UG Mathematics Booster Test 1 - Integration as Inverse Process

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