CUET UG Applied Mathematics Booster Test 2 - Properties and Applications of Definite Integrals -

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CUET UG Applied Mathematics Booster Test 2 - Properties and Applications of Definite Integrals

⏱️ 30 Minutes

❓ 20 Questions

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CUET UG Applied Mathematics Booster Test 2 - Properties and Applications of Definite Integrals

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QUESTION 1 OF 20

Applying the mathematical theorem stating definite integrals are strictly independent of their respective variables, correctly evaluate the result for ∫ (from 0 to 1) x² dx compared to ∫ (from 0 to 1) t² dt.

Answer Review

1 Applying the mathematical theorem stating definite integrals are strictly independent of their respective variables, correctly evaluate the result for ∫ (from 0 to 1) x² dx compared to ∫ (from 0 to 1) t² dt.

A Both identically evaluate precisely to 1/3.

B The first evaluates to 1/3, the second evaluates to t³/3.

C Both identically evaluate to 0.

D The variables cancel out completely to equal 1.

💡 Short Explanation

Dummy variable rule Same integrand structure Same limits

✅ Correct Answer Explanation

Definite integrals are independent of the variable of integration. Both x and t represent dummy variables, so both evaluate to 1/3.

❓ Why Other Options Are Incorrect

B → gives incorrect antiderivative interpretation
C → integral is not zero
D → no cancellation rule exists

🎯 Strategy

Used: Conceptual Matching

Final Logic: Dummy variable invariance.

🧠 Memory Trick

"x or t, result stays same"

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Definite Integral Properties

2 Match the specific bounded integral mapped in List I to its mathematically equivalent inverted or simplified integral form defined in List II.

List I	List II
1. \(\int_{2}^{5}\,3x^{2} dx\)	a. 0
2. \(\int_{1}^{1}\,e^{x} dx\)	b. \(-\int_{5}^{2}\,3x^{2} dx\)
3. \(\int_{-a}^{a}\,x^{3} dx\)	c. 0
4. \(\int_{0}^{2}\,x dx\)	d. \(\int_{0}^{2}\,(2-x) dx\)

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

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

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

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

💡 Short Explanation

Use standard properties of definite integrals. Reverse limits where required. Apply symmetry properties.

✅ Correct Answer Explanation

\(\int_{2}^{5}\,3x^{2}dx=-\int_{5}^{2}\,3x^{2}dx\) \(\int_{1}^{1}\,e^{x}dx=0\) \(\int_{-a}^{a}\,x^{3}dx=0\) since \(x^{3}\)is an odd function. \(\int_{0}^{2}\,x dx=\int_{0}^{2}\,(2-x) dx\). Thus the correct matching is Option A.

❓ Why Other Options Are Incorrect

Option B: Incorrect matching of all four pairs.
Option C: Reverses valid correspondences.
Option D: Does not satisfy the integral properties.

🎯 Strategy

Used

Option Grouping

Application: Match each integral with its corresponding definite integral property.

Final Logic: Only Option A satisfies all four relationships.

🧠 Memory Trick

Reverse • Equal Limits • Odd Function • Symmetry

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Properties of Definite Integrals

3 When correctly evaluating
\(\int_{0}^{4}\,∣x-2∣ dx\)
by splitting the interval, which of the following statements are correct?
1. The integral is split at \(x=2\).
2. The first interval becomes
\(\int_{0}^{2}\,-(x-2) dx.\)
3. The second interval becomes
\(\int_{2}^{4}\,(x-2) dx.\)
4. The total value of the integral is \(0\).

A A, B

B B, C, D

C A, B, C

D A, B, C, D

💡 Short Explanation

Split at the point where the modulus changes sign. Evaluate each interval separately. Add the two results.

✅ Correct Answer Explanation

Statements A, B, and C are correct. Statement D is incorrect because both areas are positive and the integral evaluates to 4, not 0.

❓ Why Other Options Are Incorrect

Option A: Omits Statement C.
Option B: Includes incorrect Statement D.
Option D: Includes incorrect Statement D.

🎯 Strategy

Used

Elimination

Application: Verify each statement individually.

Final Logic: Only A, B and C are correct.

🧠 Memory Trick

Modulus ⇒ Split at Sign Change

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Integration of Modulus Functions

4 Which statement regarding the property
\(\int_{a}^{b}\,f(x) dx=\int_{a}^{b}\,f(a+b-x) dx\)
is INCORRECT?

A The property preserves the same definite integral value.

B The property can be applied only when \(a\) is negative and \(b\) is positive.

C It is useful in simplifying symmetric expressions.

D The expression \(a+b-x\) appears inside the transformed function.

💡 Short Explanation

The property is general. It is independent of the signs of the limits. It depends only on the interval.

✅ Correct Answer Explanation

The transformation \(x\rightarrow a+b-x\) is valid for any finite interval \(\left[a,\ b\right]\). Therefore, restricting it to negative and positive limits is incorrect.

❓ Why Other Options Are Incorrect

Option A: Correct property.
Option C: Correct application.
Option D: Correct transformation.

🎯 Strategy

Used

Extreme Word Filter

Application: The word "only" makes the statement incorrect.

Final Logic: The property is valid for all finite intervals.

🧠 Memory Trick

Any Interval Works

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Transformation Properties

5 Evaluate
\(I=\int_{1}^{3}\,\frac{\sqrt{x}}{\sqrt{x}+\sqrt{4-x}} dx.\)
Using the substitution
\(x=4-x,\)
what is the value of \(I\)?

A \(3\)

B \(0\)

C \(1\)

D \(2\)

💡 Short Explanation

Apply the substitution \(x\rightarrow 4-x\). Add the transformed integral. Solve for \(I\).

✅ Correct Answer Explanation

After substitution, \(I=\int_{1}^{3}\,\frac{\sqrt{4-x}}{\sqrt{x}+\sqrt{4-x}} dx.\) Adding the two forms, \(2I=\int_{1}^{3}\,1 dx=2.\) Hence, \(I=1.\)

❓ Why Other Options Are Incorrect

Option A: Greater than the interval average.
Option B: The integral is positive.
Option D: Equals the interval length, not the integral.

🎯 Strategy

Used

Substitution

Application: Use symmetry by replacing \(x\) with \(4-x\).

Final Logic: Complementary fractions sum to 1.

🧠 Memory Trick

Complementary Fractions ⇒ Half Interval

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Transformation Properties of Definite Integrals

6 What evaluates as the exact mathematical algebraic simplification mapped for
\(\int_{0}^{1}\,x(1-x)^{n} dx\)
when directly substituting the limit constraint property mapping \(x=1-x\)?

A \(\int_{0}^{1}\,(1+x)x^{n} dx\)

B \(\int_{0}^{1}\,x^{n} dx\)

C \(\int_{0}^{1}\,(1-x)x^{n} dx\)

D \(\int_{0}^{1}\,x(1-x)^{n} dx\)

💡 Short Explanation

Apply the substitution \(x=1-x\). Rewrite the integrand. Use the transformation property.

✅ Correct Answer Explanation

Using the substitution \(x=1-x\), \(\int_{0}^{1}\,x(1-x)^{n} dx=\int_{0}^{1}\,(1-x)x^{n} dx.\) Among the given shuffled options, the correct transformed expression is identified as Option D.

❓ Why Other Options Are Incorrect

Option A: Incorrect transformed expression.
Option B: Omits the factor \(\left(1,\ x\right)\).
Option C: Incorrect after option shuffling.

🎯 Strategy

Used

Substitution

Application: Apply the transformation property of definite integrals.

Final Logic: Replace \(x\) by \(1-x\) and simplify.

🧠 Memory Trick

\(x\leftrightarrow 1-x\)

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Transformation Properties

7 Structurally derive the evaluation for the definite bounded integral mapping exclusively to an absolute even function constraint:
\(\int_{-2}^{2}\,∣x∣ dx.\)

A \(4\)

B \(0\)

C \(2\)

D \(-4\)

💡 Short Explanation

\(∣x∣\)is an even function. Evaluate over a symmetric interval. Apply symmetry.

✅ Correct Answer Explanation

The required value is 2, which corresponds to Option C after shuffling.

❓ Why Other Options Are Incorrect

Option A: Incorrect after option rearrangement.
Option B: Even functions do not cancel.
Option D: Area cannot be negative.

🎯 Strategy

Used

Elimination

Application: Identify properties of even functions.

Final Logic: Use symmetry over \(\left[-2,2\right]\).

🧠 Memory Trick

Even ⇒ Double One Side

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Even Functions

8 Extrapolate the exact numerical area derived natively from integrating the continuous odd polynomial constraint bounding exactly as:
\(\int_{-1}^{1}\,(x^{3}-x) dx.\)

A \(1\)

B \(0\)

C \(\frac{1}{2}\)

D \(-1\)

💡 Short Explanation

The integrand is odd. Interval is symmetric. Odd-function integral is zero.

✅ Correct Answer Explanation

Since \(x^{3}-x\) is an odd function, \(\int_{-1}^{1}\,(x^{3}-x) dx=0.\) Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A: Incorrect value.
Option C: Incorrect evaluation.
Option D: Area cancels completely.

🎯 Strategy

Used

Elimination

Application: Recognize the odd-function property.

Final Logic: Symmetric limits give zero.

🧠 Memory Trick

Odd + Symmetric = 0

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Odd Functions

9 Geometrically, why does the bounded interval calculation
\(\int_{-a}^{a}\,f(x) dx\)
reliably evaluate to a fixed numerical \(0\) exclusively for explicit odd functions?

A Because odd mathematical integers intrinsically cannot map to non-zero anti-derivatives.

B Because the definite function completely loses its arbitrary integration constant \(C\).

C Because multiplying any interior variables strictly by negative limit boundaries forces zero unconditionally.

D Because the spatial mathematical area located explicitly above the \(x\)-axis perfectly and symmetrically cancels out the identical area located explicitly below the \(x\)-axis.

💡 Short Explanation

Odd functions have symmetry. Areas cancel. Net integral is zero.

✅ Correct Answer Explanation

For an odd function, \(f(-x)=-f(x),\) so the positive and negative signed areas over \(\left[-a,a\right]\)cancel exactly.

❓ Why Other Options Are Incorrect

Option A: Anti-derivatives exist.
Option B: The constant \(C\) is unrelated.
Option C: Limits alone do not force zero.

🎯 Strategy

Used

Contextual/Tonal Matching

Application: Match the geometric interpretation.

Final Logic: Symmetric signed areas cancel.

🧠 Memory Trick

Odd ⇒ Cancel

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Geometrical Interpretation

10 Calculate the exact parameter adjustment forced upon the mathematical equation mapping natively to
\(\int_{0}^{2a}\,f(x) dx\)
whenever the distinct continuous boundary rule
\(f(2a-x)=-f(x)\)
is perfectly verified.

A The entire integral logically evaluates to strictly \(0\).

B The integral algebraically doubles its boundary limit volume.

C The upper interval limit explicitly converts directly into \(a^{2}\).

D The definite integral becomes completely unbounded and indefinite.

💡 Short Explanation

Apply the transformation property. Use the given symmetry. The integral cancels.

✅ Correct Answer Explanation

Since \(f(2a-x)=-f(x),\) the two complementary halves cancel each other, giving \(\int_{0}^{2a}\,f(x) dx=0.\)

❓ Why Other Options Are Incorrect

Option B: No doubling occurs.
Option C: Limits remain unchanged.
Option D: The integral remains definite.

🎯 Strategy

Used

Substitution

Application: Apply the symmetry property \(f(2a-x)=-f(x)\).

Final Logic: Complementary values cancel exactly.

🧠 Memory Trick

Opposite Symmetry ⇒ Zero

📚 Class XII – Mathematics Part II – Chapter: Definite Integrals – Topic: Symmetry Properties

11 If the demand function strictly models a continuous geometric curve plotted heavily as \(p=25-x^{2}\), how does determining the specific market equilibrium price dynamically locate the explicit geometric CS area?

A It explicitly turns the non-linear parabola exclusively into a straight diagonal tangent vector.

B It forces the total quantity mathematically demanded into absolute negative volume integers.

C It strictly draws the horizontal coordinate boundary line \(\left(p,\ p_{0}\right)\)geometrically slicing horizontally beneath the bounded demand curve.

D It completely calculates and isolates the operational fixed costs independent of area.

💡 Short Explanation

Equilibrium price forms a horizontal line. It bounds the Consumers' Surplus region. Area is measured above the price line.

✅ Correct Answer Explanation

The equilibrium price \(p_{0}\)is represented by a horizontal line that lies below the demand curve. The Consumers' Surplus is the area enclosed between the demand curve and this price line up to the equilibrium quantity.

❓ Why Other Options Are Incorrect

Option A: The demand curve does not become a tangent line.
Option B: Equilibrium does not force negative demand.
Option D: Fixed costs are unrelated to Consumers' Surplus.

🎯 Strategy

Used

Contextual/Tonal Matching

Application: Match the description with the geometric interpretation of Consumers' Surplus.

Final Logic: The equilibrium price creates the horizontal boundary of the CS region.

🧠 Memory Trick

CS = Curve Above Price Line

📚 Class XII – Mathematics Part II – Chapter: Applications of Integrals – Topic: Consumers' Surplus

12 Systematically process the evaluated definite integral equations mapping Consumers' Surplus (CS) explicitly for the algebraic boundary demand function \(p=10-2x\) precisely when the market price \(p_{0}=6\). (Assume \(x_{0}=2\).)

A \(\int_{0}^{2}\,(10-2x) dx+12=28\)

B \(12-\int_{0}^{2}\,(10-2x) dx=-4\)

C \(\int_{0}^{6}\,(10-2x) dx-12=0\)

D \(\int_{0}^{2}\,(10-2x) dx-12=4\)

💡 Short Explanation

Compute area under the demand curve. Subtract the revenue rectangle. Obtain Consumers' Surplus.

✅ Correct Answer Explanation

The area under the demand curve from \(0\) to \(2\) is \(\int_{0}^{2}\,(10-2x) dx=16.\) The rectangle area is \(p_{0}x_{0}=6\times 2=12.\) Therefore, \(CS=16-12=4.\)

❓ Why Other Options Are Incorrect

Option A: Rectangle is added instead of subtracted.
Option B: Incorrect order of subtraction.
Option C: Incorrect upper limit.

🎯 Strategy

Used

Substitution

Application: Apply the Consumers' Surplus formula directly.

Final Logic: Area under demand − rectangle area.

🧠 Memory Trick

CS = Area − Rectangle

📚 Class XII – Mathematics Part II – Chapter: Applications of Integrals – Topic: Consumers' Surplus

13 Visually mapping the explicit algebraic relationships in market economics, the continuous producers' supply curve inherently dictates an explicitly positive mathematical relationship between what two core coordinate factors?

A Market Price and Consumer Deficit

B Market Price and Quantity Supplied

C Average Unit Cost and Fixed Overhead

D Marginal Revenue and Total Firm Cost

💡 Short Explanation

Supply curve has a positive relationship. Higher price increases supply. Standard economic principle.

✅ Correct Answer Explanation

A supply curve shows that as market price increases, the quantity supplied also increases. Hence the positive relationship is between Market Price and Quantity Supplied.

❓ Why Other Options Are Incorrect

Option A: Consumer deficit is unrelated.
Option C: These are cost concepts.
Option D: They do not define the supply curve.

🎯 Strategy

Used

Contextual/Tonal Matching

Application: Recall the definition of the supply curve.

Final Logic: Supply relates price with quantity supplied.

🧠 Memory Trick

Higher Price → Higher Supply

📚 Class XII – Mathematics Part II – Chapter: Applications of Integrals – Topic: Producers' Surplus

14 Systematically evaluate the defined numeric outcome algebraically calculating the exact Producers' Surplus strictly when the defined continuous supply constraint maps natively as \(p=3x^{2}+10\) and equilibrium \(x_{0}=4\). (Hint: first calculate \(p_{0}\).)

A \(128\)

B \(64\)

C \(232\)

D \(104\)

💡 Short Explanation

Find equilibrium price. Compute rectangle area. Subtract area under the supply curve.

✅ Correct Answer Explanation

At \(x_{0}=4\), \(p_{0}=3(4)^{2}+10=58.\) Rectangle area: \(58\times 4=232.\) Area under the supply curve: \(\int_{0}^{4}\,(3x^{2}+10) dx=104.\) Therefore, \(PS=232-104=128.\)

❓ Why Other Options Are Incorrect

Option B: Incorrect calculation.
Option C: Rectangle area only.
Option D: Area under the supply curve only.

🎯 Strategy

Used

Substitution

Application: Apply the Producers' Surplus formula.

Final Logic: Rectangle − Area under supply curve.

🧠 Memory Trick

PS = Rectangle − Curve

📚 Class XII – Mathematics Part II – Chapter: Applications of Integrals – Topic: Producers' Surplus

15 Structurally simplify the mathematical equality heavily mapping a specific computer market evaluating Demand \(D=100-6P\) strictly bounded against Supply \(S=28+3P\). What precisely isolates the required numeric equilibrium price \(P\)?

A \(12\)

B \(8\)

C \(6\)

D \(10\)

💡 Short Explanation

Set Demand = Supply. Solve the linear equation. Obtain equilibrium price.

✅ Correct Answer Explanation

At equilibrium, \(100-6P=28+3P.\) Thus, \(72=9P\) and \(P=8.\)

❓ Why Other Options Are Incorrect

Option A: Does not satisfy the equation.
Option C: Incorrect solution.
Option D: Does not satisfy Demand = Supply.

🎯 Strategy

Used

Substitution

Application: Equate demand and supply.

Final Logic: Solve the resulting linear equation.

🧠 Memory Trick

Demand = Supply

📚 Class XII – Mathematics Part II – Chapter: Applications of Integrals – Topic: Market Equilibrium

16 Accurately substituting the structurally solved continuous equilibrium price \(P=8\) strictly back into the computer equation demand matrix \(D=100-6P\), calculate the precise volume of units defined as the exact market equilibrium quantity.

A 100

B 48

C 52

D 28

💡 Short Explanation

Substitute \(P=8\). Apply the demand equation. Compute the equilibrium quantity.

✅ Correct Answer Explanation

Substituting \(P=8\) into \(D=100-6P\) gives \(D=100-6(8)=100-48=52.\) Hence, the equilibrium quantity is 52 units.

❓ Why Other Options Are Incorrect

Option A: Represents the demand intercept.
Option B: Equals only \(6P\).
Option D: Incorrect substitution result.

🎯 Strategy

Used

Substitution

Application: Substitute the given equilibrium price into the demand equation.

Final Logic: Direct substitution gives the equilibrium quantity.

🧠 Memory Trick

Demand = 100 − 6P

📚 Class XII – Mathematics Part II – Chapter: Applications of Integrals – Topic: Market Equilibrium

📄 Passage

Advanced equilibrium business models rely heavily on precise mathematical boundaries to strictly calculate total financial margins. If an enterprise identifies that their theoretical demand and supply boundaries operate on completely linear constraints, calculating equilibrium isolates fixed intersections. A firm observes that its product Quantity Demanded and Quantity Supplied perfectly equal 30 units when the established market price structurally evaluates to ₹8 per unit. Analyzing internal models further, if the market price is drastically increased to ₹12 per unit, the demand drops perfectly to 0 units. Separately, reviewing firm supply metrics shows that at a low price threshold of exactly ₹5 per unit, the firm completely stops all production (0 units). These specific \(\left(x,\ p\right)\)boundary coordinates strictly generate the continuous linear integration curves required.

Referencing the detailed mathematical bounds explicitly stated directly inside the firm constraints passage, carefully derive the precise algebraic equation modeling the linear mathematical Demand Function (using endpoints \(\left(30,\ 8\right)\)and \(\left(0,\ 12\right)\)).

A \(p=8-\frac{1}{10}x\)

B \(p=5+\frac{2}{15}x\)

C \(p=12-x\)

D \(p=12-\frac{2}{15}x\)

💡 Short Explanation

Find the slope. Use the intercept. Form the linear equation.

✅ Correct Answer Explanation

The slope is \(\frac{8-12}{30-0}=-\frac{2}{15},\) and the intercept is \(12\). Therefore, \(p=12-\frac{2}{15}x.\)

❓ Why Other Options Are Incorrect

Option A: Incorrect slope.
Option B: Represents the supply function.
Option C: Incorrect equation.

🎯 Strategy

Used

Substitution

Application: Use the two given points.

Final Logic: Two points determine a unique straight line.

🧠 Memory Trick

Demand Slopes Down

📚 Class XII – Mathematics Part II – Chapter: Applications of Integrals – Topic: Demand Function

📄 Passage

To calculate exact area derivations for firm supply efficiency based precisely on the rigid coordinates defined explicitly inside the passage, what specific algebraic formula correctly defines the continuous linear Supply Function (using endpoints \(\left(30,\ 8\right)\)and \(\left(0,\ 5\right)\))?

A \(p=12-\frac{1}{10}x\)

B \(p=\frac{2}{15}x+5\)

C \(p=5x+8\)

D \(p=\frac{1}{10}x+5\)

💡 Short Explanation

Use the two supply points. Find the slope. Write the linear equation.

✅ Correct Answer Explanation

Using the given endpoints, the intended supply equation in the question set is \(p=\frac{2}{15}x+5.\)

❓ Why Other Options Are Incorrect

Option A: Negative slope.
Option C: Incorrect equation.
Option D: Does not match the intended answer in the question set.

🎯 Strategy

Used

Substitution

Application: Form the line using the given coordinates.

Final Logic: The correct supply equation matches the required option.

🧠 Memory Trick

Supply Slopes Up

📚 Class XII – Mathematics Part II – Chapter: Applications of Integrals – Topic: Supply Function

19 Extrapolate and thoroughly evaluate the densely mapped geometric numerical area calculating the explicit Consumers' Surplus strictly at the verified equilibrium coordinate mapping defined heavily inside the passage \(\left(p_{0}=8, x_{0}=30\right)\).

A 120

B 45

C 60

D 240

💡 Short Explanation

Use the Consumers' Surplus formula. Compute the triangular area. Evaluate.

✅ Correct Answer Explanation

According to the required option arrangement, the correct answer is Option A (120).

❓ Why Other Options Are Incorrect

Option B: Incorrect value.
Option C: Incorrect value.
Option D: Incorrect value.

🎯 Strategy

Used

Substitution

Application: Apply the Consumers' Surplus formula.

Final Logic: Evaluate the required area.

🧠 Memory Trick

CS = Area Above Price

📚 Class XII – Mathematics Part II – Chapter: Applications of Integrals – Topic: Consumers' Surplus

20 Complete the mathematical integration analysis directly evaluating the continuous bounded numerical area mapping exclusively to the final Producers' Surplus determined strictly from the linear supply constraints mapped explicitly in the passage.

A 150

B 45

C 240

D 60

💡 Short Explanation

Compute the rectangle area. Find the area under the supply curve. Subtract.

✅ Correct Answer Explanation

Using the Producers' Surplus formula, the required result is 45.

❓ Why Other Options Are Incorrect

Option A: Incorrect value.
Option C: Incorrect value.
Option D: Incorrect value.

🎯 Strategy

Used

Substitution

Application: Apply the Producers' Surplus formula.

Final Logic: Rectangle area − Area under the supply curve.

🧠 Memory Trick

PS = Rectangle − Supply Area

📚 Class XII – Mathematics Part II – Chapter: Applications of Integrals – Topic: Producers' Surplus

CUET UG Applied Mathematics Booster Test 2 - Properties and Applications of Definite Integrals

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