UG Applied Mathematics Booster Test 3 - Fundamentals of Integration -

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CUET UG Applied Mathematics Booster Test 3 - Fundamentals of Integration

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Applied Mathematics Booster Test 3 - Fundamentals of Integration

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QUESTION 1 OF 20

By applying the inverse formula derived from the product rule of differentiation, evaluate the integral:
\(\int \left[f^{'}(x)e^{x}+f(x)e^{x}\right] dx\)

Answer Review

1 By applying the inverse formula derived from the product rule of differentiation, evaluate the integral:
\(\int \left[f^{'}(x)e^{x}+f(x)e^{x}\right] dx\)

A \(f(x)e^{x}+C\)

B \(f^{'}(x)e^{x}+C\)

C \(f(x)+e^{x}+C\)

D \([f(x)]^{2}e^{x}+C\)

💡 Short Explanation

Recognize product rule: d/dx [f(\(x)e^{x}\)] Reverse it using integration Direct identification of integrand structure

✅ Correct Answer Explanation

The integrand is: \(f^{'}(x)e^{x}+f(x)e^{x}\) This matches the product rule: \(\frac{d}{dx}[f(x)e^{x}]=f^{'}(x)e^{x}+f(x)e^{x}\) So, \(\int [f^{'}(x)e^{x}+f(x)e^{x}]dx=f(x)e^{x}+C\) Thus option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrect pairing; derivative is not multiplied by e^x alone
Option C → Does not preserve product structure of derivative
Option D → No squaring arises in product rule form

🎯 Strategy

Used: Elimination + Pattern Recognition

Application: Identify expression as exact derivative of product

Final Logic: Match integrand with derivative of f(x)e^x

🧠 Memory Trick

"Product rule reversed = product inside integral"

📚 Class XII – NCERT Mathematics – Chapter: Applications of Derivatives – Topic: Product Rule & Integration by Recognition

2 Match the specific Rational Function in List I with its corresponding Partial Fractions Decomposition in List II.

List I	List II
1. \(\frac{px+q}{\left(x+a)(x+b\right)}\)	a. \(\frac{A}{x+a}+\frac{B}{{\left(x+a\right)}^{2}}\)
2. \(\frac{px+q}{{\left(x+a\right)}^{2}}\)	b. \(\frac{A}{x+a}+\frac{B}{x+b}+\frac{C}{x+c}\)
3. \(\frac{px^{2}+qx+c}{\left(x+a)(x+b)(x+c\right)}\)	c. \(\frac{A}{x+a}+\frac{Bx+C}{x^{2}+b}\)
4. \(\frac{px^{2}+qx+c}{\left(x+a)(x^{2}+b\right)}\)	d. \(\frac{A}{x+a}+\frac{B}{x+b}\)

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

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

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

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

💡 Short Explanation

Match denominator structure with decomposition rules Repeated linear factor → repeated fractions Quadratic factor → linear numerator form

✅ Correct Answer Explanation

(1) two linear factors → A/(x+a)+B/(x+b) → (d) (2) repeated linear factor → A/(x+a)+B/(x+a)² → (a) (3) three linear factors → A/(x+a)+B/(x+b)+C/(x+c) → (b) (4) linear × irreducible quadratic → A/(x+a)+(Bx+C)/(x²+b) → (c) Hence mapping matches option C.

❓ Why Other Options Are Incorrect

Option A → Incorrect structure for all four mappings
Option B → Misplaces repeated and quadratic decomposition
Option D → Completely mismatched pairing logic

🎯 Strategy

Used: Option Grouping

Application: Match denominator pattern types systematically

Final Logic: Structural matching of rational function forms

🧠 Memory Trick

"Linears split simply, quadratics stay linear-in-numerator"

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Partial Fractions Decomposition

3 When using "Integration by Parts" applying the formula ∫ f(x)g(x) dx = f(x)∫ g(x) dx - ∫ [f'(x) ∫ g(x) dx] dx, which of the following rules accurately dictate the choice of the first function f(x)?
1. The function that is easily differentiable is generally taken as the first function.
2. The second function should be chosen such that its integral is readily available.
3. The ILATE rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential) is often used to establish the correct order.
4. The first function must exclusively and always be an exponential function.

A 1, 2, 4

B 1, 2

C 1, 2, 3

D 1, 2, 3, 4

💡 Short Explanation

Variable substitution changes all components Differential transformation is required Final integral form must adjust completely

✅ Correct Answer Explanation

1 is correct: variable changes from x → t 2 is correct: dx = g'(t)dt 3 is correct: integral becomes ∫ f(g(t))g'(t)dt 4 is false: constant C is still required in indefinite integrals So 1, 2, 3 are correct → Option C.

❓ Why Other Options Are Incorrect

Option A → misses correct transformed integral form
Option B → incomplete transformation
Option D → wrongly removes constant of integration

🎯 Strategy

Used: Elimination

Application: Check correctness of each transformation step

Final Logic: Only D violates integration rule

🧠 Memory Trick

"Substitute everything except the constant"

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Substitution Method

4 Which statement is INCORRECT regarding definite integrals?

A The function f must be continuous across the entire closed interval [a, b].

B The value of the definite integral is fundamentally evaluated as F(b) - F(a).

C The constant of integration 'C' must be explicitly calculated and added to the final numerical answer of the definite integral.

D F(x) represents a valid anti-derivative of the function f(x).

💡 Short Explanation

Definite integrals eliminate constant of integration Evaluation uses limits Only antiderivative difference matters

✅ Correct Answer Explanation

In definite integrals: \(\int_{a}^{b}\,f(x) dx=F(b)-F(a)\) The constant cancels out, so adding C is unnecessary. Hence C is incorrect.

❓ Why Other Options Are Incorrect

A → required continuity condition
B → correct fundamental theorem
D → correct definition of antiderivative

🎯 Strategy

Used: Extreme Word Filter

Application: Identify absolute incorrect requirement

Final Logic: Definite integrals do not use constant C

🧠 Memory Trick

"Limits kill constants"

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

5 For an indefinite integral evaluated as F(x) + C, if F(x) = log|x|, the graphical family of curves generated translates to what specific visual transformation as C changes?

A Shifting the identical logarithmic curve up or down vertically along the y-axis.

B Shifting the curve horizontally left or right along the x-axis.

C Modifying the mathematical base of the logarithm.

D Transforming the logarithmic curve into a standard parabola.

💡 Short Explanation

Adding C shifts graph vertically Shape remains unchanged Only position changes

✅ Correct Answer Explanation

y = log|x| + C shifts the graph up/down depending on C, forming a family of curves.

❓ Why Other Options Are Incorrect

B → horizontal shift is not caused by +C
C → logarithmic base does not change
D → function type remains logarithmic

🎯 Strategy

Used: Conceptual Matching

Application: Interpret effect of constant C

Final Logic: Constant shifts only vertically

🧠 Memory Trick

"C means Curve shifts"

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

6 What geometric constraint does a definite integral inherently solve that an indefinite integral is unable to?

A It finds a single bounded, finite area under the curve between ordinates \(x=a\) and \(x=b\), rather than representing an infinite unbounded family of curves.

B It calculates the linear distance between the roots of a parabola.

C It creates a curve that is strictly constrained to the first coordinate quadrant.

D It bypasses the requirement for using standard area formulas.

💡 Short Explanation

A definite integral has fixed limits of integration. It gives a unique numerical value representing the area (or signed area). An indefinite integral represents a family of antiderivatives and includes an arbitrary constant \(C\).

✅ Correct Answer Explanation

A definite integral evaluates the accumulated quantity between two specified limits, \(x=a\) and \(x=b\). Geometrically, it determines the finite bounded area (or signed area) under a curve between these limits. Since the limits are fixed, the result is a single numerical value rather than a family of functions. An indefinite integral, on the other hand, gives the general antiderivative of a function and is expressed with an arbitrary constant \(C\). It does not specify any interval and therefore cannot determine a unique bounded area unless limits are provided. Thus, Option A correctly distinguishes the geometric significance of a definite integral from an indefinite integral.

❓ Why Other Options Are Incorrect

Option B → Definite integration is not used to calculate the linear distance between the roots of a parabola. Root separation is determined using algebraic methods.
Option C → A definite integral does not restrict a curve to the first quadrant. Curves may lie in any quadrant depending on the function.
Option D → Definite integration does not eliminate the need for geometric area concepts or standard formulas. It provides a calculus-based method for finding areas, especially for irregular regions.

🎯 Strategy

Used

Elimination

Application:

Eliminate options that describe concepts unrelated to definite integration (such as root distance, quadrant restriction, or bypassing geometry). The remaining option correctly states the defining geometric property of a definite integral.

Final Logic:

A definite integral produces a unique bounded area between specified limits, whereas an indefinite integral represents a family of antiderivatives.

🧠 Memory Trick

"Limits → One Value; No Limits → Many Curves."

📚 Class XII – Mathematics Part II – Chapter: Integrals – Topic: Definite Integral as the Limit of a Sum and Area Under a Curve

7 Evaluate
\(\int \frac{x^{3}+x^{2}+x+1}{x+1} dx\)

A \(\frac{x^{3}}{3}+x+C\)

B \(\frac{x^{2}}{2}+x+C\)

C \(\frac{x^{4}}{4}+x+C\)

D \(\frac{x^{3}}{3}+x^{2}+C\)

💡 Short Explanation

Simplify by division Cancel (x+1) Integrate termwise

✅ Correct Answer Explanation

\((x^{3}+x^{2}+x+1)/(x+1)=x^{2}+1\) Then: \(\int (x^{2}+1)dx=x^{3}/3+x+C\)

❓ Why Other Options Are Incorrect

B → incorrect algebra simplification
C → wrong degree integration
D → mismatch with division result

🎯 Strategy

Used: Substitution/Simplification

Application: Reduce rational expression first

Final Logic: Simplify then integrate

🧠 Memory Trick

"Divide first, integrate next"

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Algebraic Simplification

8 Evaluate the logarithmic function integral meticulously using integration by parts:
\(\int xlog⁡(x) dx\)

A \(x^{2}log⁡(x)-\frac{x^{2}}{4}+C\)

B \(\frac{x^{2}log⁡(x)}{2}-\frac{x^{2}}{4}+C\)

C \(log⁡(x)-\frac{x^{2}}{2}+C\)

D \(xlog⁡(x)-x+C\)

💡 Short Explanation

Use integration by parts Choose log x as second function Apply uv - ∫vdu

✅ Correct Answer Explanation

Let u = log x, dv = x dx Then: \(\int xlogxdx=(x^{2}/2)logx-\int x^{2}/2*1/xdx=(x^{2}/2)logx-x^{2}/4+C\)

❓ Why Other Options Are Incorrect

A → missing correct coefficient
C → incorrect structure
D → wrong integration form

🎯 Strategy

Used: Integration by Parts

Application: ILATE rule selection

Final Logic: Log chosen as u

🧠 Memory Trick

"LIATE: Log goes first"

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Integration by Parts

9 By applying the algebraic method of completing the square, evaluate the integral:
\(\int \frac{1}{\sqrt{5+4x+x^{2}}} dx\)

A \(log⁡∣x+2+\sqrt{5+4x+x^{2}}∣+C\)

B \(log⁡∣x+\sqrt{5+4x+x^{2}}∣+C\)

C \(\frac{1}{2}log⁡∣\frac{x+2}{x-2}∣+C\)

D \({sin⁡}^{-1}\left(\frac{x+2}{1}\right)+C\)

💡 Short Explanation

Complete square Recognize log form Use standard identity

✅ Correct Answer Explanation

5+4x+x² = (x+2)²+1 So result: \(log⁡∣x+2+\sqrt{x^{2}+4x+5}∣+C\)

❓ Why Other Options Are Incorrect

B → missing shift
C → wrong form
D → inverse trig not applicable

🎯 Strategy

Used: Pattern Recognition

Application: Convert to standard identity

Final Logic: Match log form integral

🧠 Memory Trick

"Square inside → log outside"

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

10 Identify the accurately computed result for the specific rational integral:
\(\int \frac{1}{x^{2}-a^{2}} dx\)

A \(\frac{1}{2a}log⁡∣\frac{x-a}{x+a}∣+C\)

B \(\frac{1}{2a}log⁡∣\frac{a+x}{a-x}∣+C\)

C \(log⁡∣x+\sqrt{x^{2}-a^{2}}∣+C\)

D \(\frac{1}{a}{tan⁡}^{-1}\left(\frac{x}{a}\right)+C\)

💡 Short Explanation

Use partial fractions Split into linear factors Apply log formula

✅ Correct Answer Explanation

\(1/(x^{2}-a^{2})=1/(2a)[1/(x-a)-1/(x+a)]\) Integrating gives: \(\frac{1}{2a}log⁡∣\frac{x-a}{x+a}∣+C\)

❓ Why Other Options Are Incorrect

B → reversed fraction form
C → wrong identity type
D → trig form not applicable

🎯 Strategy

Used: Elimination

Application: Factor denominator

Final Logic: Partial fraction identity

🧠 Memory Trick

"Difference of squares → log difference"

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Rational Function Integration

11 By using the linear property of substitution where
\(\int f(ax+b) dx=\frac{1}{a}F(ax+b)+C,\)
correctly evaluate:
\(\int e^{2x+3} dx\)

A \(2e^{2x+3}+C\)

B \(\frac{1}{2}e^{2x+3}+C\)

C \(e^{2x+3}+C\)

D \(\frac{1}{3}e^{2x+3}+C\)

💡 Short Explanation

Use substitution u = 2x + 3 du = 2 dx Apply standard exponential integral

✅ Correct Answer Explanation

Let u = 2x + 3, so du = 2 dx ⇒ dx = du/2 → ∫ e^(2x+3) dx = (1/2) ∫ e^u du → = ½ e^u + C = ½ e^(2x+3) + C

❓ Why Other Options Are Incorrect

Option A → missing correct scaling factor inversion
Option C → ignores substitution factor ½
Option D → incorrect coefficient

🎯 Strategy

Used: Substitution

Application: Transform exponential inner function linearly

Final Logic: Chain rule reversal introduces 1/2 factor

🧠 Memory Trick

"Inside derivative comes out inverted"

📚 Class XII – NCERT Mathematics Part II – Chapter 7: Integrals – Substitution Rule

12 Evaluate
\(\int \frac{2x-3}{x^{2}-3x-18} dx\)
by observing the exact proportional relationship between the numerator and the derivative of the denominator.

A \(log⁡∣x^{2}-3x-18∣+C\)

B \(\frac{1}{2}log⁡∣x^{2}-3x-18∣+C\)

C \((x^{2}-3x-18)^{2}+C\)

D \(2xlog⁡∣x^{2}-3x-18∣+C\)

💡 Short Explanation

Numerator matches derivative of denominator Direct log integral form Apply ∫ f'/f rule

✅ Correct Answer Explanation

d/dx(x² - 3x - 18) = 2x - 3 → Integral becomes ∫ f'/f dx → Result = ln|x² - 3x - 18| + C

❓ Why Other Options Are Incorrect

Option B → wrong scaling factor
Option C → incorrect algebraic transformation
Option D → invalid logarithmic structure

🎯 Strategy

Used: Elimination

Application: Identify derivative-over-function pattern

Final Logic: f'/f → ln|f(x)|

🧠 Memory Trick

"Derivative over function → log result"

📚 Class XII – NCERT Mathematics Part II – Chapter 7: Integrals – Logarithmic Form

13 Using the method of partial fractions, decompose
\(\frac{x}{\left(x+1)(x+4\right)}\)
and accurately evaluate the integral.

A \(-\frac{1}{3}log⁡∣x+1∣+\frac{4}{3}log⁡∣x+4∣+C\)

B \(\frac{1}{3}log⁡∣x+1∣+\frac{4}{3}log⁡∣x+4∣+C\)

C \(log⁡∣x+1∣+log⁡∣x+4∣+C\)

D \(\frac{4}{3}log⁡∣x+1∣-\frac{1}{3}log⁡∣x+4∣+C\)

💡 Short Explanation

Use partial fractions Split rational expression Integrate logarithmically

✅ Correct Answer Explanation

x/((x+1)(x+4)) = -1/3(1/(x+1)) + 4/3(1/(x+4)) → Integrating gives logarithmic terms → Final result matches option A

❓ Why Other Options Are Incorrect

Option B → incorrect coefficient signs
Option C → ignores decomposition
Option D → swapped coefficients

🎯 Strategy

Used: Option Grouping

Application: Compare coefficients after decomposition

Final Logic: Correct split determines log structure

🧠 Memory Trick

"Split → log → coefficients matter"

📚 Class XII – NCERT Mathematics Part II – Chapter 7: Integrals – Partial Fractions

14 Utilizing substitution techniques, evaluate the definite integral:
\(\int_{0}^{1}\,xe^{x} dx\)

A \(1\)

B \(e\)

C \(0\)

D \(e-1\)

💡 Short Explanation

Use integration by parts Evaluate definite limits Simplify expression

✅ Correct Answer Explanation

∫ x e^x dx = x e^x - ∫ e^x dx → = x e^x - e^x → Evaluate from 0 to 1 gives 1

❓ Why Other Options Are Incorrect

Option B → incorrect evaluation
Option C → ignores non-zero area
Option D → incomplete simplification

🎯 Strategy

Used: Substitution

Application: Apply integration by parts method

Final Logic: Boundary evaluation gives final numeric result

🧠 Memory Trick

"x e^x → parts → plug limits"

📚 Class XII – NCERT Mathematics Part II – Chapter 7: Integrals – Definite Integrals

15 If the Marginal Cost is modeled as
\(MC=\frac{x}{\sqrt{2500+x^{2}}}\)
and the fixed cost of producing \(0\) units is ₹1000, calculate the particular Cost function \(C(x)\).

A \(C(x)=\sqrt{2500+x^{2}}+950\)

B \(C(x)=\sqrt{2500+x^{2}}+1000\)

C \(C(x)=x\sqrt{2500+x^{2}}+950\)

D \(C(x)=\frac{1}{\sqrt{2500+x^{2}}}+1000\)

💡 Short Explanation

Recognize derivative form Integrate MC Apply fixed cost condition

✅ Correct Answer Explanation

MC = x/√(2500+x²) = d/dx(√(2500+x²)) → So C(x) = √(2500+x²) + C → Using C(0)=1000 gives constant adjustment → 950 in expression form

❓ Why Other Options Are Incorrect

Option B → incorrect constant adjustment
Option C → wrong integration structure
Option D → not integrated form

🎯 Strategy

Used: Substitution

Application: Recognize derivative of square root expression

Final Logic: Reverse chain rule integration

🧠 Memory Trick

"x over root → root result"

📚 Class XII – NCERT Mathematics Part II – Chapter 7: Integrals – Applications in Economics

16 The marginal cost of producing \(x\) units is given as
\(MC=x.\)
If the total cumulative cost of producing 3 units is ₹7800, isolate the exact cost function \(C(x)\).

A \(C(x)=\frac{x^{2}}{2}+7795.5\)

B \(C(x)=\frac{x^{2}}{2}+7800\)

C \(C(x)=x^{2}+7795.5\)

D \(C(x)=\frac{x^{3}}{3}+7791\)

💡 Short Explanation

Integrate MC Use given condition Solve constant

✅ Correct Answer Explanation

C(x) = ∫ x dx = x²/2 + C → C(3) = 9/2 + C = 7800 → C = 7795.5 → Final expression matches A

❓ Why Other Options Are Incorrect

Option B → incorrect constant handling
Option C → wrong integration
Option D → incorrect power rule

🎯 Strategy

Used: Substitution

Application: Apply definite condition to find constant

Final Logic: Integration + boundary condition

🧠 Memory Trick

"Integrate → plug → adjust C"

📚 Class XII – NCERT Mathematics Part II – Chapter 7: Integrals – Cost Functions

📄 Passage

At market equilibrium, the exact quantity of a product that consumers want to buy is mathematically equal to the amount producers are willing to sell, creating a balanced state denoted by the equation xd = xs. The price at this precise intersection is known as the equilibrium price (p₀), and the corresponding volume is the equilibrium quantity (x₀). By comprehending this equilibrium point, businesses can subsequently calculate the Producers' Surplus (PS), which theoretically represents the economic gain to producers who would have been willing to supply the product at a lower price than p₀. Mathematically, the Producers' Surplus is determined by evaluating a definite integral. It is computed as the total area of the revenue rectangle (p₀ × x₀) minus the total area located strictly under the supply curve function g(x) bounded between x = 0 and x = x₀.

Before one can calculate the surplus, what specific algebraic action must be taken to solve for the particular equilibrium price \(\left(p_{0}\right)\)?

A Evaluate the unbounded definite integral of the demand curve.

B Algebraically set the demand curve function equal to the supply curve function \(\left(x_{d},\ x_{s}\right)\).

C Set the marginal cost equal to absolute zero.

D Substitute \(x=0\) into both the demand and supply equations simultaneously.

💡 Short Explanation

Equilibrium occurs when demand equals supply. Solve \(x_{d}=x_{s}\)to determine the equilibrium point. Surplus is calculated only after finding equilibrium.

✅ Correct Answer Explanation

The passage states that market equilibrium is achieved when the quantity demanded equals the quantity supplied, represented by \(x_{d}=x_{s}.\) Solving this equation gives the equilibrium quantity \(\left(x_{0}\right)\), from which the equilibrium price \(\left(p_{0}\right)\)is obtained using either the demand or supply function. Since producer surplus depends on the equilibrium price and quantity, finding the equilibrium point is the essential first step before evaluating the definite integral. Therefore, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → A definite integral is used to calculate producer surplus after equilibrium has been determined.
Option C → Marginal cost has no role in determining the equilibrium between demand and supply.
Option D → Substituting \(x=0\) only gives intercept values and does not determine the equilibrium point.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

The passage explicitly states that equilibrium is created when \(x_{d}=x_{s}\). Matching this statement directly with the options identifies the correct answer.

Final Logic:

The passage directly defines equilibrium by the equation \(x_{d}=x_{s}\), making Option B the correct choice.

🧠 Memory Trick

"Demand = Supply = Equilibrium."

📚 Class XII – Mathematics Part II – Chapter: Application of Integrals – Topic: Consumer Surplus, Producer Surplus and Market Equilibrium

📄 Passage

At market equilibrium, the exact quantity of a product that consumers want to buy is mathematically equal to the amount producers are willing to sell, creating a balanced state denoted by the equation \(x_{d}=x_{s}\). The price at this precise intersection is known as the equilibrium price \(\left(p_{0}\right)\), and the corresponding volume is the equilibrium quantity \(\left(x_{0}\right)\). By comprehending this equilibrium point, businesses can subsequently calculate the Producers' Surplus (PS), which theoretically represents the economic gain to producers who would have been willing to supply the product at a lower price than \(p_{0}\). Mathematically, the Producers' Surplus is determined by evaluating a definite integral. It is computed as the total area of the revenue rectangle \(\left(p_{0},\ x_{0}\right)\)minus the total area located strictly under the supply curve function \(g(x)\)bounded between \(x=0\) and \(x=x_{0}\).

Identify the precise mathematical formula utilized for determining the Producers' Surplus (PS) as a definitive value.

A \(PS=\int_{0}^{x_{0}}\,f(x) dx-p_{0}x_{0}\)

B \(PS=p_{0}x_{0}-\int_{0}^{x_{0}}\,g(x) dx\)

C \(PS=\int_{0}^{x_{0}}\,[f(x)-g(x)] dx\)

D \(PS=p_{0}x_{0}+\int_{0}^{x_{0}}\,g(x) dx\)

💡 Short Explanation

Producer surplus is the area above the supply curve. It lies below the equilibrium price line. It equals the revenue rectangle minus the area under the supply curve.

✅ Correct Answer Explanation

The passage clearly states that the Producer Surplus (PS) is obtained by subtracting the area under the supply curve from the total revenue rectangle. Thus, \(PS=p_{0}x_{0}-\int_{0}^{x_{0}}\,g(x) dx.\) Here, \(p_{0}x_{0}\)represents the area of the rectangle formed by the equilibrium price and quantity. \(\int_{0}^{x_{0}}\,g(x) dx\) represents the area under the supply curve. Their difference gives the producer surplus. Therefore, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → This represents the reverse order and uses the demand function instead of the supply function.
Option C → This formula represents the area between demand and supply curves, not the producer surplus.
Option D → Producer surplus is obtained by subtracting, not adding, the area under the supply curve.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

The passage explicitly provides the producer surplus formula. Matching the stated formula with the options immediately identifies the correct answer.

Final Logic:

Producer surplus equals the revenue rectangle minus the area under the supply curve, which is exactly Option B.

🧠 Memory Trick

"Producer Surplus = Rectangle − Supply Area."

📚 Class XII – Mathematics Part II – Chapter: Application of Integrals – Topic: Producer Surplus Formula and Market Equilibrium

19 Evaluate the definite integral using the property
\(\int_{a}^{b}\,f(x) dx=\int_{a}^{b}\,f(a+b-x) dx,\)
to carefully evaluate
\(\int_{1}^{3}\,\frac{\sqrt[3]{x}}{\sqrt[3]{x}+\sqrt[3]{ 4-x }} dx.\)

A \(1\)

B \(2\)

C \(3\)

D \(0\)

💡 Short Explanation

Use symmetry property Apply substitution x → 4-x Average integrals

✅ Correct Answer Explanation

Let I = integral → Using symmetry, I + I = ∫1 to 3 1 dx → So 2I = 2 ⇒ I = 1

❓ Why Other Options Are Incorrect

Option B → overestimation
Option C → incorrect scaling
Option D → ignores symmetry

🎯 Strategy

Used: Option Grouping

Application: Symmetry-based integral simplification

Final Logic: Symmetry reduces integral to constant

🧠 Memory Trick

"Symmetry → halve total"

📚 Class XII – NCERT Mathematics Part II – Chapter 7: Integrals – Definite Integral Properties

20 The market supply function for a specific commodity is modeled as
\(p=x^{2}+4x+5.\)
Determine the numerical Producer's Surplus when the fixed market price is
\(p_{0}=10.\)

A \(\frac{8}{3}\)

B \(10\)

C \(\frac{22}{3}\)

D \(1\)

💡 Short Explanation

Find equilibrium point Apply PS formula Evaluate integral

✅ Correct Answer Explanation

Solve x² + 4x + 5 = 10 ⇒ x₀ = 1 → PS = p₀x₀ − ∫₀¹ (x²+4x+5)dx → = 10 − (1/3 + 2 + 5) = 8/3

❓ Why Other Options Are Incorrect

Option B → incorrect subtraction
Option C → wrong integration
Option D → ignores area concept

🎯 Strategy

Used: Substitution

Application: Apply equilibrium + integral definition

Final Logic: Surplus = rectangle − curve area

🧠 Memory Trick

"Price area minus curve area"

📚 Class XII – NCERT Mathematics Part II – Chapter 7: Integrals – Economic Applications

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