CUET UG Applied Mathematics Booster Test 2 - Geometry, Real-Life, and Economic Models -

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CUET UG Applied Mathematics Booster Test 2 - Geometry, Real-Life, and Economic Models

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Applied Mathematics Booster Test 2 - Geometry, Real-Life, and Economic Models

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QUESTION 1 OF 20

If the given mathematical area of a triangle is 35 sq. units, what must the absolute determinant equation evaluate to before breaking the modulus to solve for an unknown vertex coordinate?

Answer Review

1 If the given mathematical area of a triangle is 35 sq. units, what must the absolute determinant equation evaluate to before breaking the modulus to solve for an unknown vertex coordinate?

A Exactly 35

B Either +35 or -35

C Either +70 or -70

D 0

💡 Short Explanation

�� Area of triangle = ½ × |Determinant| �� Given area = 35 sq. units �� Therefore determinant magnitude = 70

✅ Correct Answer Explanation

The triangle area formula using determinants is Area = ½|D|. If the area is 35, then |D| = 70. Hence the determinant before applying modulus can be either +70 or −70. Option C is correct. Option A ignores the factor ½. Option B gives incorrect magnitude. Option D applies only to collinear points.

❓ Why Other Options Are Incorrect

�� Option A → The determinant itself is not equal to the area. The determinant magnitude must be doubled before comparison.
�� Option B → Although determinant can be positive or negative, its magnitude should be 70, not 35.
�� Option D → A determinant of 0 indicates collinear points and zero area, not area 35.

🎯 Strategy

Used: Substitution

Application: Substitute the given area value into the determinant area formula and solve directly.

Final Logic: 35 = ½|D| ⇒ |D| = 70 ⇒ D = ±70.

🧠 Memory Trick

"Area Half, Determinant Double."

📚 Class XII – Mathematics – Chapter: Determinants – Topic: Area of Triangle Using Determinants

2 Match the determinant equation components with their respective geometric meaning or procedural step.

List I	List II
1. The column of 1s in the triangle determinant	a. Adjusts the standard parallelogram determinant calculation to a triangle.
2. Multiplying the determinant by ½	b. Ensures the matrix is square (3x3) to allow determinant calculation for 2D coordinates.
3. Taking the modulus of the determinant	c. Evaluates if the three coordinate points are completely collinear.
4. Setting the determinant equal to 0	d. Corrects the final output because physical area cannot be a negative quantity.

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

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

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

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

💡 Short Explanation

�� Column of 1s forms a 3×3 determinant �� Half factor gives triangle area �� Modulus removes negative area �� Zero determinant implies collinearity

✅ Correct Answer Explanation

The column of 1s converts coordinate data into a square determinant. Multiplication by ½ converts the parallelogram measure into triangle area. Modulus ensures area remains non-negative. A determinant equal to zero signifies that the three points lie on one straight line. Therefore, option A correctly matches all concepts.

❓ Why Other Options Are Incorrect

�� Option B → Reverses the roles of the column of 1s and the ½ factor. It also mismatches modulus and collinearity.
�� Option C → None of the determinant operations are matched to their actual geometric meanings.
�� Option D → Incorrectly associates modulus, collinearity and matrix structure with unrelated operations.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match each determinant operation with its known geometric interpretation.

Final Logic: Every determinant component corresponds uniquely to one geometric purpose.

🧠 Memory Trick

"1–Half–Mod–Zero → Square–Triangle–Positive–Collinear."

📚 Class XII – Mathematics – Chapter: Determinants – Topic: Geometrical Applications of Determinants

3 For points P(3, −2), Q(8, 8), and R(k, 2) to be collinear, the area determinant equals 0. Which of the following mathematical steps are valid in solving this?
1. (1/2) ×
| 3 −2 1 |
| 8 8 1 | = 0
| k 2 1 |
1. The scalar multiplier (1/2) can be mathematically eliminated by multiplying both sides by 2.
2. Expanding the determinant yields a linear equation in terms of k.
3. The value of k will yield two different answers because area uses a ± modulus.

A 1, 2, 3

B 1, 3, 4

C 2, 3, 4

D 1, 2, 3, 4

💡 Short Explanation

�� Collinear points have zero area. �� The factor 1/2 can be removed algebraically. �� Determinant expansion produces a linear equation in k.

✅ Correct Answer Explanation

The area of a triangle with vertices ((x_1,y_1)), ((x_2,y_2)), and ((x_3,y_3)) is given by: Area = (1/2) × | x₁ y₁ 1 | | x₂ y₂ 1 | | x₃ y₃ 1 | → For collinear points, the area is zero. Therefore, (1/2) × | 3 −2 1 | | 8 8 1 | = 0 | k 2 1 | Hence, Statement 1 is correct. → Since both sides of the equation are equal to zero, multiplying both sides by 2 does not change the equality. Thus, Statement 2 is correct. → Expanding the determinant gives an equation containing only the variable k raised to the first power. Therefore, Statement 3 is correct. → Statement 4 is incorrect because the ± sign is associated with the absolute value used in the area formula. For collinearity, the determinant itself must be exactly zero, so there is no ± ambiguity and only one value of k satisfies the condition. → Hence, Statements 1, 2, and 3 are correct.

❓ Why Other Options Are Incorrect

�� Option B → Incorrect because it includes Statement 4, which is false.
�� Option C → Incorrect because it also includes Statement 4.
�� Option D → Incorrect because Statement 4 wrongly assumes two solutions due to the ± sign.

🎯 Strategy

Used

Elimination

Application:

�� Verify each statement independently and eliminate every option containing Statement 4.

Final Logic:

�� Collinearity requires the determinant to be exactly zero; removing the constant factor is valid, and determinant expansion gives a linear equation in k.

🧠 Memory Trick

"Collinear ⇒ Zero Area ⇒ Zero Determinant"

📚 | k 2 1 |

4 Identify the mathematically INCORRECT assumption when applying determinant collinearity rules:

A If the expanded determinant evaluates to -10k + 50 = 0, the single valid answer is k = 5.

B Collinear points do not form a triangle, hence their geometric area is null.

C Because collinearity equates to 0, we must evaluate the determinant using both +0 and -0 to find multiple coordinate answers.

D Any row or column expansion of the 3x3 determinant will validly yield the same value for the unknown variable.

💡 Short Explanation

�� Zero has no positive or negative distinction �� Collinearity gives determinant exactly zero �� Only one valid equation is obtained

✅ Correct Answer Explanation

Option C is mathematically incorrect because +0 and −0 represent the same number. Collinearity requires determinant = 0 exactly. Option A correctly solves a linear equation. Option B correctly links collinearity with zero area. Option D is valid because determinant expansion methods produce identical values.

❓ Why Other Options Are Incorrect

�� Option A → Correct algebra gives k = 5.
�� Option B → Collinear points lie on one line and enclose no area.
�� Option D → Determinants have a unique value regardless of expansion method.

🎯 Strategy

Used: Elimination

Application: Identify the statement that violates determinant properties.

Final Logic: Only Option C contradicts basic mathematical treatment of zero.

🧠 Memory Trick

"Zero Has No Sign."

📚 Class XII – Mathematics – Chapter: Determinants – Topic: Collinearity and Area Conditions

5 In a mixture of real-life matrix modeling, shopkeepers A, B, and C sell varied bags. If Shopkeeper A uses 20 polythene, 30 handmade, and 40 newspaper bags, how is this strictly represented in a standard row matrix?

A [20 30 40]

B [20+30+40]

C [20,30,40]^T

D Column: (20, 30, 40)

💡 Short Explanation

�� A row matrix has one row �� Entries are written horizontally �� Data is represented as [20 30 40]

✅ Correct Answer Explanation

A row matrix contains elements arranged in a single horizontal row. Therefore the bag quantities are represented as [20 30 40]. Option B is merely an arithmetic sum. Options C and D describe column representations, not row matrices. Hence Option A correctly represents the data.

❓ Why Other Options Are Incorrect

�� Option B → Represents addition rather than matrix structure.
�� Option C → Represents a column vector, not a row matrix.
�� Option D → Explicitly states a column arrangement.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Identify whether the data arrangement is horizontal or vertical.

Final Logic: One row of entries means a row matrix.

🧠 Memory Trick

"Row Runs Right."

📚 Class XI – Mathematics – Chapter: Matrices – Topic: Row Matrix

6 Under matrix algebra constraints, if a sales data matrix S is structured as a 3x2 matrix (3 shops, 2 item types), and the profit per item matrix P is 2x1, what is the defined order of the resultant total profit matrix?

A 2x3

B 3x1

C 2x1

D The multiplication is strictly undefined.

💡 Short Explanation

�� Matrix multiplication rule applies �� Inner dimensions match (2 = 2) �� Result order is 3×1

✅ Correct Answer Explanation

For matrix multiplication, the number of columns of the first matrix must equal the number of rows of the second matrix. Here S is 3×2 and P is 2×1. Therefore multiplication is defined and the resulting matrix has order 3×1. Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Reverses the resulting dimensions.
�� Option C → Ignores the row count of the first matrix.
�� Option D → Multiplication is valid because inner dimensions are equal.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Apply matrix order multiplication rule.

Final Logic: (3×2)(2×1) ⇒ 3×1.

🧠 Memory Trick

"Outer Dimensions Survive."

📚 Class XI – Mathematics – Chapter: Matrices – Topic: Matrix Multiplication

7 Formulate the EMI (Equated Matrix Integer) equation to find the individual bag costs. If the augmented matrix maps Row 1 as [20, 30, 40 | 250], what is the simplified linear equation representing this?

A 2x + 3y + 4z = 25

B 2x + 3y = 25

C 20x + 30y + 40z = 0

D x + y + z = 250

💡 Short Explanation

�� Divide all coefficients by 10 �� Equation remains equivalent �� Simplified form is 2x+3y+4z=25

✅ Correct Answer Explanation

The augmented matrix row corresponds to 20x + 30y + 40z = 250. Dividing every term by 10 gives 2x + 3y + 4z = 25. This preserves equality while simplifying coefficients. Therefore Option A is the correct simplified equation.

❓ Why Other Options Are Incorrect

�� Option B → Omits variable z.
�� Option C → Changes the right-hand side incorrectly to zero.
�� Option D → Does not represent the original coefficients.

🎯 Strategy

Used: Substitution

Application: Translate the augmented row into a linear equation and simplify.

Final Logic: Divide every term by the common factor 10.

🧠 Memory Trick

"Matrix Row → Equation → Simplify."

📚 Class XII – Mathematics – Chapter: Matrices – Topic: Augmented Matrix and Linear Equations

8 If an organization tracks a moving promotional campaign, calculating matrix X (quantities) multiplied by matrix C (costs), what mathematical condition must be true for the campaign costs to be successfully computed?

A Matrix X and Matrix C must be perfectly identical in order.

B The number of columns in X must equal the number of rows in C.

C Both matrices must strictly be singular.

D The determinant of X must equal 1.

💡 Short Explanation

�� Matrix multiplication follows dimension rules �� Inner dimensions must match �� Otherwise multiplication is undefined

✅ Correct Answer Explanation

The essential condition for matrix multiplication is compatibility of dimensions. The number of columns of matrix X must equal the number of rows of matrix C. Identical orders are unnecessary. Singularity and determinant values are unrelated to multiplication feasibility. Hence Option B is correct.

❓ Why Other Options Are Incorrect

�� Option A → Same order is not required.
�� Option C → Singular matrices can still participate in multiplication.
�� Option D → Determinant 1 is unrelated to multiplication compatibility.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Check matrix multiplication compatibility.

Final Logic: Matching inner dimensions guarantees multiplication.

🧠 Memory Trick

"Columns First = Rows Second."

📚 Class XI – Mathematics – Chapter: Matrices – Topic: Matrix Multiplication Conditions

9 If an economic system's input-output matrix yields a determinant |I - A| that evaluates precisely to -3/12, what is the probability that the system is viable according to Hawkins-Simon?

A 100%

B 50%

C 0%

D Cannot be determined

💡 Short Explanation

�� Hawkins-Simon requires positive determinant �� Determinant is negative �� Viability condition fails immediately

✅ Correct Answer Explanation

The first Hawkins-Simon condition requires |I−A| > 0. Here the determinant equals −3/12, which is negative. Therefore the system fails the viability test. Option C correctly states that viability is impossible under this condition. Options A and B incorrectly assume success, while D is unnecessary.

❓ Why Other Options Are Incorrect

�� Option A → A negative determinant cannot satisfy the condition.
�� Option B → No partial viability exists in this criterion.
�� Option D → The determinant value already provides sufficient information.

🎯 Strategy

Used: Elimination

Application: Compare determinant sign with Hawkins-Simon requirement.

Final Logic: Negative determinant ⇒ condition violated.

🧠 Memory Trick

"Positive Determinant = Possible Economy."

📚 Class XII – Economics/Applied Mathematics – Topic: Leontief Input-Output Analysis – Hawkins-Simon Condition

10 In the Leontief model, the "Total Output" vector represents the sum of which two distinct mathematical components?

A The inputs consumed internally plus the scalar identity.

B The inputs consumed by interconnected sectors plus the external market demand.

C The adjoint matrix plus the determinant matrix.

D The minor arrays and the cofactor arrays.

💡 Short Explanation

�� Total output satisfies X = AX + D �� AX represents inter-industry demand �� D represents final external demand

✅ Correct Answer Explanation

The Leontief equation is X = AX + D. Here AX denotes goods consumed internally by interconnected industries, while D denotes external or final demand. Total output equals the sum of these two components. Therefore Option B is correct. The remaining options involve unrelated matrix concepts.

❓ Why Other Options Are Incorrect

�� Option A → Identity matrix is not part of output composition.
�� Option C → Adjoint and determinant are algebraic tools, not economic output components.
�� Option D → Minors and cofactors are determinant concepts unrelated to production output.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Recall the fundamental Leontief equation and identify its components.

Final Logic: Total Output = Internal Consumption + Final Demand.

🧠 Memory Trick

"X = AX + D."

📚 Class XII – Applied Mathematics – Chapter: Input-Output Model – Topic: Total Output Equation

11 To properly map the fractional "area" of the Technology Matrix A, if Sector 1 produces 10 total units and consumes 2 itself, and Sector 2 produces 20 total units and consumes 5 from Sector 1, what constitutes the first row of matrix A?

A [2/10, 5/20]

B [2/10, 5/10]

C [10/2, 20/5]

D [1/5, 1/4]

💡 Short Explanation

�� Technology coefficients are input-output ratios �� Each coefficient equals input consumed ÷ total output �� The first row is represented as [2/10, 5/20]

✅ Correct Answer Explanation

In a technology matrix, each coefficient shows the amount of input required from a sector per unit of output of the consuming sector. Thus, the first-row coefficients are 2/10 and 5/20. Option A correctly follows this definition. Although Option D simplifies these fractions, it is not the row as directly constructed from the given data.

❓ Why Other Options Are Incorrect

�� Option B → Uses 5/10 as the second coefficient, whereas the denominator should be Sector 2's total output (20).
�� Option C → Reverses the technology coefficient formula by placing outputs in the numerator and inputs in the denominator.
�� Option D → [1/5, 1/4] is the simplified form of Option A, but the question asks what constitutes the row from the given production and consumption data; therefore the unsimplified coefficient form in Option A is the standard construction.

🎯 Strategy

Used: Substitution

Application: Apply the technology coefficient formula directly to each sector using the given input and output values.

Final Logic: Technology coefficient = Input Consumed ÷ Total Output.

🧠 Memory Trick

"Input Over Output."

📚 Class XII – Applied Mathematics – Chapter: Input-Output Model – Topic: Technology Matrix Coefficients

12 Evaluate the integral stability matrix (I - A) if the Technology Matrix A evaluates to Row 1: [0.2, 0.3] and Row 2: [0.4, 0.1]. What is the calculated value of (I - A)?

A Row 1: [0.8, -0.3], Row 2: [-0.4, 0.9]

B Row 1: [1.2, 0.3], Row 2: [0.4, 1.1]

C Row 1: [-0.2, -0.3], Row 2: [-0.4, -0.1]

D Row 1: [0.8, 0.7], Row 2: [0.6, 0.9]

💡 Short Explanation

�� Identity matrix = [[1,0],[0,1]] �� Subtract A entry-wise �� Result is [[0.8,-0.3],[-0.4,0.9]]

✅ Correct Answer Explanation

For a 2×2 matrix, I = [[1,0],[0,1]]. Subtracting A entry-wise gives [1−0.2, 0−0.3; 0−0.4, 1−0.1]. This equals [0.8, −0.3; −0.4, 0.9]. Therefore, Option A is correct. Other options either add values or perform incorrect subtraction.

❓ Why Other Options Are Incorrect

�� Option B → Adds instead of subtracting matrix entries.
�� Option C → Treats all entries as negatives of A.
�� Option D → Incorrectly subtracts off-diagonal entries from one.

🎯 Strategy

Used: Substitution

Application: Subtract corresponding entries of A from the identity matrix.

Final Logic: (I − A) is found by entry-wise subtraction.

🧠 Memory Trick

"Identity Minus Technology."

📚 Class XII – Applied Mathematics – Chapter: Input-Output Model – Topic: Matrix (I − A)

13 Deriving from the fundamental Leontief equation D = X - AX, factoring out the requirement output matrix X mathematically results in which intermediate expression?

A X

B D = (I -

C D = X / A

D D = I(X -

💡 Short Explanation

�� D = X − AX �� Factor X on the right �� D = (I − A)X

✅ Correct Answer Explanation

Starting with D = X − AX, write X as IX. Then D = IX − AX. Factoring X from the right gives D = (I − A)X. This is the standard Leontief form used for solving output requirements. Other options either factor incorrectly or are mathematically invalid.

❓ Why Other Options Are Incorrect

�� Option A → Incorrect order and sign arrangement.
�� Option C → Division of matrices is not defined this way.
�� Option D → Does not represent proper matrix factorization.

🎯 Strategy

Used: Substitution

Application: Rewrite X as IX and factor out X.

Final Logic: IX − AX = (I − A)X.

🧠 Memory Trick

"IX Minus AX = (I−A)X."

📚 Class XII – Applied Mathematics – Chapter: Input-Output Model – Topic: Leontief Equation

14 When predicting future market stability, if the external Demand matrix (D) changes, how is the new Requirement Output (X) strictly calculated?

A ^-1 inverse matrix by the new Demand matrix D.

B By multiplying the previously verified (I -

C By recalculating the entire determinant |A|.

D By setting D to a null matrix.

💡 Short Explanation

�� Leontief solution is X = (I−A)^−1D �� New demand changes output requirement �� Inverse matrix remains applicable

✅ Correct Answer Explanation

After obtaining X = (I − A)^−1D, any change in demand D leads to a new output vector through matrix multiplication with the inverse matrix. Option B directly applies the standard Leontief solution. Options A, C and D do not produce the required output vector correctly.

❓ Why Other Options Are Incorrect

�� Option A → Demand and technology matrices cannot simply be added.
�� Option C → Determinant recalculation alone does not produce X.
�� Option D → Setting demand to zero changes the problem entirely.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Recall the final Leontief output formula.

Final Logic: New output equals inverse matrix times demand vector.

🧠 Memory Trick

"Inverse × Demand = Output."

📚 Class XII – Applied Mathematics – Chapter: Input-Output Model – Topic: Output Determination

15 If evaluating the Hawkins-Simon conditions yields |I - A| = 19/90, what is the immediate mathematical and economic conclusion regarding this specific condition?

A The condition fails because it is a fraction.

B The condition is successfully verified because 19/90 is strictly positive (>0).

C The condition fails because it is less than 1.

D The technology matrix is singular.

💡 Short Explanation

�� Hawkins-Simon requires positivity �� 19/90 > 0 �� First viability condition is satisfied

✅ Correct Answer Explanation

The first Hawkins-Simon condition requires the determinant of (I−A) to be positive. Since 19/90 is greater than zero, the determinant condition is satisfied. The determinant need not exceed 1. Fractions are acceptable positive values. Therefore Option B correctly states the conclusion.

❓ Why Other Options Are Incorrect

�� Option A → Fractional values are allowed.
�� Option C → The criterion is positivity, not being greater than one.
�� Option D → A positive determinant implies the matrix is non-singular.

🎯 Strategy

Used: Elimination

Application: Compare the determinant directly with the positivity condition.

Final Logic: Positive determinant ⇒ condition satisfied.

🧠 Memory Trick

"Positive, Not Necessarily Large."

📚 Class XII – Applied Mathematics – Chapter: Input-Output Model – Topic: Hawkins-Simon Condition

16 To verify the first Hawkins-Simon rule on a 2x2 (I - A) matrix configured as Row 1: [3/5, -5/8] and Row 2: [-1/5, 3/8], you calculate the determinant as:
**Correct Answer: B) (3/5 * 3/8) - (-5/8 * -1/5)**

A (3/5 * -1/5) - (-5/8 * 3/8)

B (3/5 * 3/8) - (-5/8 * -1/5)

C (3/5 + 3/8)

D (3/5 * 3/8) + (-5/8 * -1/5)

💡 Short Explanation

�� Determinant of 2×2 matrix = ad − bc �� Apply the standard formula �� Option B follows ad − bc

✅ Correct Answer Explanation

For matrix [[a,b],[c,d]], determinant equals ad − bc. Here a=3/5, b=−5/8, c=−1/5 and d=3/8. Thus determinant is (3/5×3/8) − (−5/8×−1/5). Option B exactly matches the determinant formula and is therefore correct.

❓ Why Other Options Are Incorrect

�� Option A → Uses incorrect multiplication pairings.
�� Option C → Adds diagonal elements instead of finding determinant.
�� Option D → Uses addition instead of subtraction.

🎯 Strategy

Used: Substitution

Application: Apply the standard determinant formula directly.

Final Logic: Determinant = ad − bc.

🧠 Memory Trick

"Diagonal Product Minus Cross Product."

📚 Class XII – Mathematics – Chapter: Determinants – Topic: Determinant of 2×2 Matrix

17 During the verification of economic stability, you observe the matrix (I - A) has principal diagonal elements of 4/5 and 4/5. According to the second Hawkins-Simon condition, what does this signify?

A The diagonal condition fails because they are not integers.

B The diagonal condition is satisfied because both elements are strictly positive.

C The matrix is skew-symmetric.

D The economy is unstable.

💡 Short Explanation

�� Hawkins-Simon requires positive diagonal entries �� Both values are positive fractions �� Condition is satisfied

✅ Correct Answer Explanation

The second Hawkins-Simon condition requires principal diagonal elements of (I−A) to be positive. Since both entries are 4/5, they are positive and satisfy the condition. The entries need not be integers. Therefore Option B correctly describes the stability requirement.

❓ Why Other Options Are Incorrect

�� Option A → Positivity matters, not integrality.
�� Option C → Positive diagonals do not imply skew-symmetry.
�� Option D → The given information supports stability, not instability.

🎯 Strategy

Used: Elimination

Application: Check whether the diagonal entries satisfy positivity.

Final Logic: Positive diagonal entries satisfy the second condition.

🧠 Memory Trick

"Positive Diagonal = Good Signal."

📚 Class XII – Applied Mathematics – Chapter: Input-Output Model – Topic: Hawkins-Simon Diagonal Condition

18 If a Leontief input-output calculation reveals that |I - A| evaluates to exactly -1/10, what is the ultimate conclusion about the system's viability?

A The system is perfectly viable.

B The system is viable but requires a demand shift.

C The first condition of Hawkins-Simon is false; therefore, the system is strictly NOT viable.

D The system has infinite solutions.

💡 Short Explanation

�� Determinant is negative �� Hawkins-Simon positivity condition fails �� Economic viability is rejected

✅ Correct Answer Explanation

A valid Leontief system must satisfy |I−A| > 0. Since the determinant equals −1/10, the first Hawkins-Simon condition fails. Therefore the economy cannot be considered viable under the model assumptions. Hence Option C correctly states the economic conclusion.

❓ Why Other Options Are Incorrect

�� Option A → Contradicts the negative determinant.
�� Option B → Demand changes cannot repair a failed determinant condition.
�� Option D → Negative determinant does not imply infinite solutions.

🎯 Strategy

Used: Elimination

Application: Compare determinant sign with viability criterion.

Final Logic: Negative determinant ⇒ Not viable.

🧠 Memory Trick

"Negative Determinant, Negative Verdict."

📚 Class XII – Applied Mathematics – Chapter: Input-Output Model – Topic: Hawkins-Simon Viability Test

📄 Passage

In a two-variable economy comprising the Food Industry (FI) and the Transport Industry (TI), viability testing dictates the mathematical health of the interlinked sectors. The process begins by constructing the input-output coefficient matrix (A) from internal consumption and total production data. To test viability using the Hawkins-Simon constraints, the matrix (I - A) is generated. The system's lifeblood depends completely on two factors: the calculated determinant |I - A| must mathematically sit above zero, and the principal diagonal elements mapping the remaining fractional self-reliance of the industries must also be strictly positive integers or fractions.

Question:
According strictly to the passage, the principal diagonal elements of the (I - A) matrix conceptually represent what economic reality?

A The total market demand.

B The remaining fractional self-reliance of the industries.

C The external debt of the sectors.

D The sum of all outputs combined.

💡 Short Explanation

�� Passage explicitly defines diagonal elements �� They indicate self-reliance after internal consumption �� Option B directly matches the passage

✅ Correct Answer Explanation

The passage clearly states that principal diagonal elements represent the remaining fractional self-reliance of industries. This interpretation is directly provided and requires no additional inference. Therefore Option B is correct. The passage does not associate these elements with demand, debt or total output.

❓ Why Other Options Are Incorrect

�� Option A → Demand is represented separately in Leontief models.
�� Option C → No reference to debt appears in the passage.
�� Option D → Diagonal entries do not represent total production.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match the wording of the option with the exact passage statement.

Final Logic: Option B repeats the passage definition verbatim.

🧠 Memory Trick

"Diagonal = Self-Reliance."

📚 Class XII – Applied Mathematics – Chapter: Input-Output Model – Topic: Interpretation of (I − A) Matrix

📄 Passage

Question:
If the determinant |I - A| crashes strictly below zero, what is the consequence for the interlinked sectors according to the passage's stated constraints?

A The system mathematically passes the viability test.

B The system's lifeblood fails, proving the economic interlinks are not viable.

C The industries become perfectly independent.

D The demand matrix must be doubled.

💡 Short Explanation

�� Passage requires determinant above zero �� Negative determinant violates viability condition �� Economic system fails the test

✅ Correct Answer Explanation

The passage explicitly states that viability depends on the determinant |I−A| being greater than zero. If it falls below zero, the required condition fails and the economic structure is not viable. Therefore, Option B accurately reflects the consequence described in the passage.

❓ Why Other Options Are Incorrect

�� Option A → Directly contradicts the stated condition.
�� Option C → Negative determinant does not imply independence.
�� Option D → The passage gives no such corrective rule.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Use the exact viability condition stated in the passage.

Final Logic: Determinant below zero ⇒ viability fails.

🧠 Memory Trick

"Below Zero, System No."

📚 Class XII – Applied Mathematics – Chapter: Input-Output Model – Topic: Hawkins-Simon Viability Conditions

CUET UG Applied Mathematics Booster Test 2 - Geometry, Real-Life, and Economic Models

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