CUET UG Mathematics Booster Test 2 - Integration as Inverse Process -

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CUET UG Mathematics Booster Test 2 - 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 2 - Integration as Inverse Process

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QUESTION 1 OF 20

If f(x) is a continuous function such that
∫f(x) dx = e^x cos x + C,
then what is the numerical value of f(0)?

Answer Review

1 If f(x) is a continuous function such that
∫f(x) dx = e^x cos x + C,
then what is the numerical value of f(0)?

A 0

B -1

C 2

D 1

💡 Short Explanation

f(x) is the derivative of e^x cos x. Apply product rule. Evaluate at x=0.

✅ Correct Answer Explanation

Since ∫f(x)dx = e^x cos x + C, we have f(x)=d/dx(e^x cos x) = e^x(cos x−sin x). At x=0: f(0)=1(1−0)=1. Therefore Option D is correct. Options A, B, and C do not satisfy the derivative evaluation.

❓ Why Other Options Are Incorrect

Option A → Ignores derivative value at x=0.
Option B → Incorrect sign obtained from differentiation.
Option C → Product rule does not yield 2 at x=0.

🎯 Strategy

Used: Substitution

Application: Differentiate the primitive and substitute x=0.

Final Logic: f(0)=e^0(cos0−sin0)=1.

🧠 Memory Trick

"Integral given → Differentiate back."

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

2 Consider the primitive function
F(x)=x^3/3−x.
Which of the following are true about its corresponding integrand f(x)?
1. f(x)=x²−1
2. f(x) has roots at x=±1
3. f(x) is an odd function

A 1 and 2 only

B 2 and 3 only

C 1 and 3 only

D 1, 2 and 3

💡 Short Explanation

Differentiate F(x). Solve f(x)=0. Check symmetry.

✅ Correct Answer Explanation

Differentiating, F'(x)=x²−1=f(x). Thus statement 1 is true. Roots: x²−1=0 ⇒ x=±1. Statement 2 is true. Since f(−x)=x²−1=f(x), f(x) is even, not odd. Hence statement 3 is false and Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Statement 3 is false.
Option C → Statement 2 is also true.
Option D → Includes incorrect statement 3.

🎯 Strategy

Used: Option Grouping

Application: Verify each statement independently.

Final Logic: Only statements 1 and 2 are true.

🧠 Memory Trick

"x² terms create even functions."

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

3 Evaluate the integral
∫sin²(x/2) dx
using analytical trigonometric identities.

A (x−cos x)/2 + C

B (x−sin x)/2 + C

C (x+sin x)/2 + C

D x/2−sin x + C

💡 Short Explanation

Use half-angle identity. Integrate term-wise. Simplify result.

✅ Correct Answer Explanation

Using sin²(x/2)=(1−cos x)/2, ∫sin²(x/2)dx = 1/2∫(1−cos x)dx = (x−sin x)/2 + C. Hence Option B is correct. The remaining options arise from incorrect integration of cos x.

❓ Why Other Options Are Incorrect

Option A → cos x appears instead of sin x.
Option C → Wrong sign before sin x.
Option D → Missing factor 1/2.

🎯 Strategy

Used: Substitution

Application: Convert to a simpler trigonometric identity.

Final Logic: Half-angle identity directly leads to Option B.

🧠 Memory Trick

"sin²(x/2) → (1−cos x)/2."

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

4 A mathematical force vector field is given by
F⃗(x)=d/dx(xe^x)î +∫(xe^x+e^x)dx ĵ.
Find the vector F⃗(1), assuming C=0.

A eî + eĵ

B eî +2eĵ

C 2eî +eĵ

D 2eî +2eĵ

💡 Short Explanation

Differentiate xe^x. Integrate xe^x+e^x. Evaluate at x=1.

✅ Correct Answer Explanation

Derivative: d/dx(xe^x)=e^x+xe^x=e^x(x+1). At x=1 ⇒ 2e. Integral: ∫(xe^x+e^x)dx =∫d(xe^x) =xe^x. At x=1 ⇒ e. Wait carefully: xe^x+e^x=d/dx(xe^x). Therefore integral = xe^x+C. At x=1 ⇒ e. Thus vector=(2e,e), giving Option C. Hence the provided answer is actually correct.

❓ Why Other Options Are Incorrect

Option A → Both components underestimated.
Option B → First component incorrect.
Option D → Second component incorrectly doubled.

🎯 Strategy

Used: Elimination

Application: Treat derivative and integral separately.

Final Logic: F⃗(1)=(2e,e).

🧠 Memory Trick

"xe^x differentiates to e^x(x+1)."

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

5 Arrange the derivatives of the following functions evaluated at x=2 in strictly descending numerical order.
1. x⁴/4
2. x³/3
3. x²/2
4. x

A 4, 3, 2, 1

B 1, 3, 2, 4

C 2, 1, 4, 3

D 1, 2, 3, 4

💡 Short Explanation

Differentiate each function. Evaluate at x=2. Arrange largest to smallest.

✅ Correct Answer Explanation

Derivatives at x=2: 1 → x³ = 8 2 → x² = 4 3 → x = 2 4 → 1 Descending order: 8 > 4 > 2 > 1 Therefore order is 1,2,3,4 and Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Gives ascending order.
Option B → Places values incorrectly.
Option C → Largest derivative omitted first.

🎯 Strategy

Used: Substitution

Application: Evaluate derivatives numerically.

Final Logic: 8>4>2>1.

🧠 Memory Trick

"Higher power → Larger derivative at x=2."

📚 Class XI – NCERT Mathematics – Chapter: Derivatives – Topic: Power Functions

6 The velocity of an object moving in a straight line is
v(t)=3t².
If its initial position s(0)=4, what is the exact position at t=3?

A 31

B 27

C 30

D 23

💡 Short Explanation

Integrate velocity. Apply initial condition. Evaluate at t=3.

✅ Correct Answer Explanation

Integrating: s(t)=∫3t²dt=t³+C. Given s(0)=4, C=4. Thus s(3)=27+4=31. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Ignores initial position.
Option C → Arithmetic error.
Option D → Incorrect integration.

🎯 Strategy

Used: Substitution

Application: Integrate velocity to obtain position.

Final Logic: s(t)=t³+4.

🧠 Memory Trick

"Velocity integrates to position."

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

7 The moving average over an interval for the continuous function f(x)=e^x is calculated by
1/2 ∫₀² e^x dx.
What is this exact average value?

A (e²+1)/2

B (e²−1)/2

C e²−1

D (e−1)/2

💡 Short Explanation

Integrate e^x. Evaluate limits. Multiply by 1/2.

✅ Correct Answer Explanation

∫₀²e^x dx =[e^x]₀² =e²−1. Average: (1/2)(e²−1) =(e²−1)/2. Hence Option B.

❓ Why Other Options Are Incorrect

Option A → Wrong sign.
Option C → Missing averaging factor.
Option D → Uses wrong upper limit.

🎯 Strategy

Used: Substitution

Application: Evaluate definite integral first.

Final Logic: Average=(Integral)/(Length).

🧠 Memory Trick

"Average = Area ÷ Interval Length."

📚 Class XII – NCERT Mathematics – Chapter: Definite Integrals – Topic: Average Value

8 Assertion (A):
∫3^x dx = 3^x/ln3 + C.
Reason (R):
d/dx(3^x)=3^x ln3.

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Standard exponential integral. Uses derivative of a^x. Reason directly justifies assertion.

✅ Correct Answer Explanation

Since d/dx(a^x)=a^x lna, the anti-derivative is ∫a^xdx=a^x/lna+C. For a=3, ∫3^xdx=3^x/ln3+C. Thus both assertion and reason are true, and the reason explains the assertion.

❓ Why Other Options Are Incorrect

Option A → Both statements are correct.
Option B → Reason is true.
Option D → Assertion is true.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match derivative and anti-derivative formulas.

Final Logic: Integration reverses differentiation.

🧠 Memory Trick

"Integral of a^x divides by ln a."

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

9 The geometric area bounded by y=1/x, the x-axis, and x=1 to x=e² is represented by
∫₁^(e²) 1/x dx.
Find the exact numerical area.

A 1

B e

C 1/e

D 2

💡 Short Explanation

Integral of 1/x is ln x. Evaluate limits. Simplify logarithm.

✅ Correct Answer Explanation

∫₁^(e²)1/x dx =[lnx]₁^(e²) =ln(e²)−ln1 =2−0 =2. Hence Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Equals ln e.
Option B → Confuses logarithm with exponential.
Option C → Incorrect reciprocal.

🎯 Strategy

Used: Substitution

Application: Apply standard logarithmic integral.

Final Logic: ln(e²)=2.

🧠 Memory Trick

"1/x integrates to ln x."

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

10 If a variable x is uniformly distributed in the interval [1,e³], what is the probability that the definite integral
∫₁ˣ 1/t dt
evaluates to less than 1?

A (e−1)/(e³−1)

B 1/e

C 1/e²

D e/(e³−1)

💡 Short Explanation

Integral equals ln x. Solve ln x < 1. Use uniform probability.

✅ Correct Answer Explanation

∫₁ˣ1/t dt =lnx. Condition: lnx<1 ⇒ x<e. Since x is uniformly distributed on [1,e³], Probability =(e−1)/(e³−1). Therefore Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Not based on interval length.
Option C → Uses incorrect probability model.
Option D → Wrong numerator.

🎯 Strategy

Used: Substitution

Application: Convert the integral into a logarithmic inequality.

Final Logic: P(x<e)=Length favourable/Length total.

🧠 Memory Trick

"lnx<1 ⇒ x<e."

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

11 Find the definite integral value of the curve y = sin x bounded from x = 0 to x = 2π.

A 2

B 0

C 4

D 1

💡 Short Explanation

Evaluate the definite integral. Positive and negative areas cancel. Net integral equals zero.

✅ Correct Answer Explanation

The required value is \(\int_{0}^{2\pi }\,sin⁡x dx=[-cos⁡x]_{0}^{2\pi }=-1-(-1)=0.\) Over one complete cycle, the positive area from \(0\) to \(\pi\) equals the negative area from \(\pi\) to \(2\pi\). Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Represents only part of the cycle.
Option C → Ignores negative contribution.
Option D → Not obtained from evaluation.

🎯 Strategy

Used: Substitution

Application: Use the standard integral of sin x and evaluate limits.

Final Logic: One full sine cycle has zero net integral.

🧠 Memory Trick

"Full sine cycle ⇒ net area 0."

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

12 Match the following trigonometric functions to their correct indefinite integrals.

List I	List II
1. sec²x	a. -cot x + C
2. sec x tan x	b. tan x + C
3. csc²x	c. sec x + C

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

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

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

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

💡 Short Explanation

Use standard trigonometric integrals. Match derivative–antiderivative pairs. Compare with options.

✅ Correct Answer Explanation

Standard results are: \(\int {sec⁡}^{2}x dx=tan⁡x+C\int sec⁡xtan⁡x dx=sec⁡x+C\int {csc⁡}^{2}x dx=-cot⁡x+C\) Hence matching is 1-b, 2-c, 3-a, giving Option C.

❓ Why Other Options Are Incorrect

Option A → Incorrectly matches sec²x.
Option B → sec xtan x does not integrate to −cot x.
Option D → sec²x does not integrate to sec x.

🎯 Strategy

Used: Option Grouping

Application: Recall standard trigonometric integration formulas.

Final Logic: Match each integrand with its known anti-derivative.

🧠 Memory Trick

"sec²→tan, sec tan→sec, csc²→−cot."

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

13 Which statement about the integral
\(\int \frac{-1}{\sqrt{1-x^{2}}} dx\)
is structurally incorrect?

A The integral evaluates to cos⁻¹x + C.

B The integral evaluates to -sin⁻¹x + C.

C The derivative of cos⁻¹x is -1/√(1-x²).

D The integral is geometrically valid for all real numbers x.

💡 Short Explanation

Domain restriction exists. |x|≤1 is required. Not valid for all real x.

✅ Correct Answer Explanation

The integrand contains \(\sqrt{1-x^{2}},\) which is real only when \(∣x∣\leq 1\). Therefore the expression is not defined for all real numbers. Options A, B, and C are standard inverse trigonometric relationships. Hence Option D is the incorrect statement.

❓ Why Other Options Are Incorrect

Option A → Correct because derivative of cos⁻¹x is negative.
Option B → Equivalent anti-derivative form.
Option C → Standard differentiation formula.

🎯 Strategy

Used: Extreme Word Filter

Application: Focus on the phrase "all real numbers."

Final Logic: Domain restrictions invalidate Option D.

🧠 Memory Trick

"Inverse sine/cosine live inside ±1."

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

14 The analytical derivative of tan⁻¹x + C is exactly equal to:

A 1/(1+x²)

B -1/(1+x²)

C 1/√(1-x²)

D 1/(1-x²)

💡 Short Explanation

Differentiate tan⁻¹x. Constant disappears. Standard formula applies.

✅ Correct Answer Explanation

The derivative formula is \(\frac{d}{dx}({tan⁡}^{-1}x)=\frac{1}{1+x^{2}}.\) The constant C differentiates to zero. Hence the derivative of tan⁻¹x + C is exactly \(1/(1+x^{2})\). Therefore Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Corresponds to derivative of −tan⁻¹x.
Option C → Derivative of sin⁻¹x.
Option D → Not a standard inverse trigonometric derivative.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Recall the standard derivative formula.

Final Logic: tan⁻¹x differentiates to \(1/(1+x^{2})\).

🧠 Memory Trick

"tan⁻¹ → 1/(1+x²)."

📚 Class XII – NCERT Mathematics – Chapter: Inverse Trigonometric Functions – Topic: Derivatives

15 Evaluate the indefinite integral
\(\int (2x+1) dx.\)
The constant of integration must be explicitly shown.

A x²+2x+C

B x²+x+C

C 2x²+x+C

D x²−x+C

💡 Short Explanation

Integrate term by term. Apply power rule. Add constant C.

✅ Correct Answer Explanation

Using linearity: \(\int (2x+1) dx=\int 2x dx+\int 1 dx=x^{2}+x+C.\) Therefore Option B is correct. The remaining options contain coefficient or sign errors.

❓ Why Other Options Are Incorrect

Option A → Integral of 1 is x, not 2x.
Option C → Integral of 2x is x², not 2x².
Option D → Wrong sign for ∫1 dx.

🎯 Strategy

Used: Substitution

Application: Integrate each term independently.

Final Logic: \(x^{2}+x+C\) differentiates back to \(2x+1\).

🧠 Memory Trick

"Integrate separately, then add C."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Basic Integration Rules

16 Assertion (A): Differential equations of the form dy/dx=f(x) have infinitely many solutions.
Reason (R): The arbitrary constant C in integration indicates infinitely many parallel curves.

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Integration introduces C. Different C values give different solutions. Family of curves results.

✅ Correct Answer Explanation

Solving \(\frac{dy}{dx}=f(x)\) gives \(y=\int f(x) dx=F(x)+C.\) Different values of C generate infinitely many solution curves. Thus both statements are true and the reason correctly explains the assertion.

❓ Why Other Options Are Incorrect

Option A → Both statements are true.
Option B → Reason is also true.
Option D → Assertion is true.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Connect differential equations with indefinite integration.

Final Logic: Arbitrary constant creates infinitely many solutions.

🧠 Memory Trick

"+C ⇒ Infinite Family."

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

17 If
\(\frac{d}{dx}[F(x)]=f(x)\)
for all x in an interval I, what mathematically represents the entire class of anti-derivatives?

A F'(x)

B f(x)

C F(x)

D F(x)+C

💡 Short Explanation

Anti-derivatives differ by constants. One primitive is not the whole family. Add arbitrary C.

✅ Correct Answer Explanation

If F'(x)=f(x), then every anti-derivative of f(x) has the form \(F(x)+C.\) The arbitrary constant accounts for all functions with the same derivative. Therefore Option D correctly represents the entire family.

❓ Why Other Options Are Incorrect

Option A → Represents derivative only.
Option B → Represents integrand.
Option C → Only one member of the family.

🎯 Strategy

Used: Elimination

Application: Identify the expression containing the arbitrary constant.

Final Logic: Entire family requires +C.

🧠 Memory Trick

"Primitive Family = F+C."

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

18 The relation
\(\int 2x dx=x^{2}+C\)
yields a family of parabolas. What is the fundamental geometrical property of this family specifically at x=1?

A They all have the identical slope of 2.

B They all intersect precisely at (1,1).

C They all have highly differing slopes depending on C.

D They are all orthogonal to each other.

💡 Short Explanation

Derivative determines slope. Constant C does not affect slope. Evaluate derivative at x=1.

✅ Correct Answer Explanation

For \(y=x^{2}+C,\) the slope is \(\frac{dy}{dx}=2x.\) At x=1, \(\frac{dy}{dx}=2.\) Since C disappears during differentiation, every member of the family has the same slope at x=1. Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Different C values give different y-values.
Option C → Slope does not depend on C.
Option D → Family members are parallel shifts, not orthogonal.

🎯 Strategy

Used: Elimination

Application: Differentiate and evaluate at x=1.

Final Logic: C affects position, not slope.

🧠 Memory Trick

"C Shifts, Slope Stays."

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

📄 Passage

In applications, it is often necessary to satisfy an additional condition which then determines a specific value of C, giving a unique anti-derivative of the given function. For instance, if F is an anti-derivative of f defined by f(x)=4x³−6, we write F(x)=x⁴−6x+C. If we are given an initial condition such as F(0)=3, we can substitute x=0 to find C=3, yielding a unique function.

What defining role does the additional boundary condition play in finding the applied anti-derivative?

A It changes the core degree of the polynomial.

B It determines the specific numerical value of the constant C.

C It alters the derivative structure of the function.

D It makes the integral mathematically undefined.

💡 Short Explanation

Boundary condition fixes C. Unique function is obtained. Derivative remains unchanged.

✅ Correct Answer Explanation

The passage explicitly states that an additional condition such as F(0)=3 determines the value of the arbitrary constant C. Once C is known, a unique anti-derivative is obtained. Hence Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Degree remains unchanged.
Option C → Derivative structure is unaffected.
Option D → Boundary conditions do not cause undefined integrals.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Use the information stated directly in the passage.

Final Logic: Boundary conditions determine C.

🧠 Memory Trick

"Condition Fixes C."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Determination of Constant

📄 Passage

Based purely on the passage logic, if
f(x)=4x³−6
and we are given F(1)=0, what is the value of C for the unique anti-derivative?

A 3

B 0

C 5

D -5

💡 Short Explanation

Use anti-derivative form. Substitute x=1. Solve for C.

✅ Correct Answer Explanation

Given \(F(x)=x^{4}-6x+C,\) and \(F(1)=0,\) we get \(1-6+C=0C=5.\) Therefore Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Does not satisfy F(1)=0.
Option B → Gives F(1)=−5.
Option D → Gives F(1)=−10.

🎯 Strategy

Used: Substitution

Application: Substitute the given condition into the anti-derivative.

Final Logic: 1−6+C=0 ⇒ C=5.

🧠 Memory Trick

"Plug Condition, Solve C."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Unique Anti-derivative from Initial Conditions

CUET UG Mathematics Booster Test 2 - Integration as Inverse Process

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