UG Applied Mathematics Booster Test 2 - Types, Limitations and Tests -

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UG Applied Mathematics Booster Test 2 - Types, Limitations and Tests

⏱️ 30 Minutes

❓ 20 Questions

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UG Applied Mathematics Booster Test 2 - Types, Limitations and Tests

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QUESTION 1 OF 20

If the average value of inventory in the base year is ₹50,000 and in the current year is ₹75,000, what is the Value Index for the current year?

Answer Review

1 If the average value of inventory in the base year is ₹50,000 and in the current year is ₹75,000, what is the Value Index for the current year?

A 125

B 150

C 175

D 200

💡 Short Explanation

Value Index compares current value with base value Formula uses percentage comparison Increase from 50,000 to 75,000 gives 150

✅ Correct Answer Explanation

Value Index is calculated using: \(Value Index=\frac{Current Year Value}{Base Year Value}\times 100\) Substituting values: \(=\frac{75000}{50000}\times 100=1.5\times 100=150\) Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → 125 is obtained from incorrect percentage calculation.
Option C → 175 overestimates the increase.
Option D → 200 would mean the value doubled.

🎯 Strategy

Used

Substitution

Application:

Directly substitute values into the Value Index formula.

Final Logic:

75,000 is 1.5 times 50,000, so the index is 150.

🧠 Memory Trick

"Current ÷ Base × 100"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Value Index –

2 Which of the following statements about the Quantity Index are correct?
(i) It measures the change in the quantity of goods produced or consumed
(ii) The Index of Industrial Production (IIP) is an example
(iii) It is determined by the area of the region under a moving average curve

A Only (i)

B Only (ii)

C Both (i) and (ii)

D All (i), (ii), and (iii)

💡 Short Explanation

Quantity Index measures physical volume changes IIP is a quantity index Moving average area is unrelated

✅ Correct Answer Explanation

A Quantity Index measures changes in the quantity or physical volume of goods produced, sold, or consumed. The Index of Industrial Production (IIP) is a standard example. Statement (iii) is incorrect because Quantity Index is not based on area under a curve or moving averages. Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Ignores statement (ii), which is also correct.
Option B → Ignores statement (i).
Option D → Statement (iii) is false.

🎯 Strategy

Used

Option Grouping

Application:

Separate valid Quantity Index properties from unrelated mathematical statements.

Final Logic:

Only statements (i) and (ii) correctly define Quantity Index.

🧠 Memory Trick

"IIP = Industrial Quantity"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Quantity Index –

3 Match the price index concepts:

List I (Concepts)	List II (Explanation)
1. Concealed Weights	a. High unit prices dominate the index due to lack of distinct quantity weights
2. Equal Importance	b. Individually calculated relatives disregard absolute quantities
3. Price Dominance Effect	c. All commodities are treated uniformly regardless of their significance
4. Ignoring Quantity Differences	d. Implicit weighting occurs due to differences in price magnitudes

A 1–a, 2–b, 3–c, 4–d

B 1–b, 2–a, 3–d, 4–c

C 1–c, 2–d, 3–a, 4–b

D 1–d, 2–a, 3–b, 4–c

💡 Short Explanation

Base year price = \(p_{0}\) Current year price = \(p_{n}\) Price Relative uses percentage formula

✅ Correct Answer Explanation

Correct matching: Base year price → \(p_{0}\) Current year price → \(p_{n}\) Price Relative → \(\frac{p_{n}}{p_{0}}\times 100\) CPI → Consumer Price Index Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Matches symbols incorrectly.
Option B → Price relative formula is wrongly assigned.
Option C → CPI and symbols are mismatched.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Match standard statistical symbols with their definitions.

Final Logic:

\(p_{0}\), \(p_{n}\), and price relative formulas follow standard notation.

🧠 Memory Trick

"0 = old price, n = new price"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Price Relatives –

4 Assertion (A): The Consumer Price Index (CPI) measures the integral of production volume in the industrial sector.
Reason (R): CPI is specifically classified as a quantity index.

A Both A and R are false

B A is true, R is false

C Both A and R are true, and R is the correct explanation of A

D A is false, R is true

💡 Short Explanation

CPI measures consumer prices CPI is a price index It is not related to industrial production volume

✅ Correct Answer Explanation

Assertion (A) is false because CPI measures consumer price changes, not industrial production volume. Reason (R) is also false because CPI is classified as a price index, not a quantity index. Hence, both statements are false.

❓ Why Other Options Are Incorrect

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

🎯 Strategy

Used

Elimination

Application:

Check truth value of assertion and reason separately.

Final Logic:

CPI relates to prices, not production quantity.

🧠 Memory Trick

"CPI = Consumer Prices Only"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Consumer Price Index –

5 In statistical analysis of index numbers, since data is gathered via sampling rather than a complete population census, the probability of sampling errors existing in the calculated index is:

A Exactly 0

B Always negative

C Greater than 0

D Exactly equal to the integral of the sample size

💡 Short Explanation

Sampling may create errors Errors cannot be fully eliminated Probability is therefore positive

✅ Correct Answer Explanation

Since index numbers are usually based on samples and not the entire population, sampling errors may occur. Therefore, the probability of error is greater than zero. Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Sampling error cannot be completely zero.
Option B → Probability cannot be negative.
Option D → Integral of sample size has no relevance.

🎯 Strategy

Used

Elimination

Application:

Remove mathematically impossible and irrelevant statements.

Final Logic:

Sampling always introduces some chance of error.

🧠 Memory Trick

"Sample → possible error"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Limitations of Index Numbers –

6 For a given dataset, the Laspeyres index is 120 and the Paasche index is 125. The variation in methods yields different values. What is the Fisher's Ideal Index for this data?
\(Fisher Index=\sqrt{Laspeyres\times Paasche}\)

A 120.0

B 122.47

C 125.0

D 144.0

💡 Short Explanation

Fisher Index uses geometric mean Multiply Laspeyres and Paasche Take square root of product

✅ Correct Answer Explanation

Using Fisher's formula: \(F=\sqrt{120\times 125}\) \(F=\sqrt{15000}F\approx 122.47\) Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Equal to Laspeyres only.
Option C → Equal to Paasche only.
Option D → Incorrect square root calculation.

🎯 Strategy

Used

Substitution

Application:

Substitute given indices into Fisher's formula.

Final Logic:

Fisher Index is the geometric mean of Laspeyres and Paasche.

🧠 Memory Trick

"Fisher = √(L × P)"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Fisher's Ideal Index –

7 Identify the incorrect statement regarding the handling of outdated items in index data.

A Including items that are not in trend provides the most accurate and universally valid index analysis

B The selection of commodities should be by judgement sampling to reflect overall changes

C Calculating based on outdated items can create an inaccurate analysis

D The base period should be a normal year to avoid boom or depression distortions

💡 Short Explanation

Outdated items distort analysis Commodity selection should be relevant Base year should be normal

✅ Correct Answer Explanation

Outdated or irrelevant commodities fail to reflect present economic conditions. Hence, including items "not in trend" does not provide accurate analysis. Therefore, Option A is the incorrect statement.

❓ Why Other Options Are Incorrect

Option B → Judgement sampling is commonly used.
Option C → Correct limitation of outdated data.
Option D → Normal base year avoids distortions.

🎯 Strategy

Used

Extreme Word Filter

Application:

Words like "most accurate" and "universally valid" indicate exaggeration.

Final Logic:

Outdated items reduce reliability, not improve it.

🧠 Memory Trick

"Old items = wrong index"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Selection of Commodities –

8 Representation bias in an index number often occurs because:

A The indices act as vectors pointing in opposite directions

B The area under the data curve is not integrated properly

C Every single item in the economy is automatically included

D Samples of representative commodities are deliberated and selected, which can skew outcomes

💡 Short Explanation

Commodity selection uses samples Samples may be biased Bias affects index accuracy

✅ Correct Answer Explanation

Index numbers are usually based on selected representative commodities. Improper or biased selection can skew the results and reduce reliability. Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Vector direction is irrelevant.
Option B → Integration is unrelated.
Option C → Not all items are included in practice.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Connect representation bias with sample selection.

Final Logic:

Bias arises from deliberate commodity selection.

🧠 Memory Trick

"Bad sample → biased result"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Representative Commodities –

9 Why is it considered essential to test the adequacy of an index number?

A To calculate the precise probability of market failure

B To convert the index into a vector quantity

C To find the area of the time series graph

D To test the consistency and reliability of a good index number method

💡 Short Explanation

Adequacy tests verify reliability They ensure consistency Good methods should satisfy tests

✅ Correct Answer Explanation

Adequacy tests such as the Unit Test and Time Reversal Test help evaluate whether an index method is reliable and logically consistent. Therefore, they are essential for checking the validity of index numbers. Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Adequacy tests do not predict market failure.
Option B → Index numbers are not vectors.
Option C → Graph area is unrelated.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Link adequacy tests with consistency checking.

Final Logic:

Adequacy tests ensure reliability of the index method.

🧠 Memory Trick

"Test → Trust"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Tests of Adequacy –

10 Which of the following tests are utilized specifically for consistency checking of an index number?
(i) Unit test
(ii) Time reversal test
(iii) Circular test
(iv) Integration test

A (i), (ii), and (iii) only

B (ii), (iii), and (iv) only

C (i) and (iv) only

D All of the above

💡 Short Explanation

Unit test checks unit independence Time reversal checks symmetry Circular test checks multi-period consistency

✅ Correct Answer Explanation

The recognized consistency tests for index numbers include: Unit Test Time Reversal Test Circular Test "Integration test" is not a standard adequacy test in index number theory. Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Includes non-standard "integration test".
Option C → Omits valid tests.
Option D → Integration test is incorrect.

🎯 Strategy

Used

Odd One Out

Application:

Identify the unrelated "integration test".

Final Logic:

Only Unit, Time Reversal, and Circular tests are standard.

🧠 Memory Trick

"UTC = Unit, Time, Circular"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Tests of Adequacy –

11 If commodity A is priced in ₹/kg and commodity B is priced in ₹/litre, the method of index construction should be independent of these units. Which method famously fails this unit independence test?

A Fisher's Ideal Method

B Simple Aggregative Method

C Laspeyres' Method

D Paasche's Method

💡 Short Explanation

Unit Test checks unit independence Simple Aggregative Method depends on units Weighted methods generally satisfy the test

✅ Correct Answer Explanation

The Unit Test requires that the index should remain unaffected by the units in which commodities are measured. The Simple Aggregative Method directly adds prices of commodities without considering unit compatibility. Therefore, changing units like kg to grams or litres to millilitres changes the index value. Hence, it fails the Unit Test. Therefore, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Fisher's Ideal Method generally satisfies important adequacy tests including unit independence.
Option C → Laspeyres' Method uses weights, reducing unit dependence.
Option D → Paasche's Method also uses weighted quantities and is not known for failing the Unit Test.

🎯 Strategy

Used

Odd One Out

Application:

Identify the method that directly sums unlike units without weights.

Final Logic:

Simple Aggregative Method fails because unit conversion changes totals.

🧠 Memory Trick

"Simple Sum → Unit Problem"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Unit Test –

12 When plotting the components of a time series, the moving average smooths out fluctuations. How does this graphical smoothing concept relate to the applicability of the Unit Test in index numbers?

A Moving averages directly prove the unit test

B The unit test uses the area under the moving average graph to verify applicability

C The unit test applies to index methods regardless of graphical smoothing, evaluating whether the formula depends on the physical units of data

D Moving averages always fail the unit test

💡 Short Explanation

Moving averages and unit tests are separate concepts Unit Test checks dependence on units Graph smoothing is unrelated

✅ Correct Answer Explanation

The Unit Test checks whether an index formula changes when units of measurement change. Moving averages are tools for smoothing time series fluctuations and have no direct connection with unit independence. Therefore, the Unit Test applies independently of graphical smoothing methods. Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Moving averages do not prove unit consistency.
Option B → Unit Test does not involve graph area.
Option D → Moving averages are unrelated to passing or failing the Unit Test.

🎯 Strategy

Used

Elimination

Application:

Remove statements incorrectly linking moving averages with adequacy testing.

Final Logic:

Unit Test evaluates formula independence from units only.

🧠 Memory Trick

"Units, not graphs"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Unit Test –

13 Assertion (A): Laspeyres' method flawlessly satisfies the time-reversal test due to its time symmetry.
Reason (R): The time-reversal test states that
\(p_{01}+p_{10}=0\)

A Both A and R are false

B A is true, R is false

C Both A and R are true, and R is the correct explanation of A

D A is false, R is true

💡 Short Explanation

Laspeyres fails Time Reversal Test Correct condition is product equals 1 Addition formula is incorrect

✅ Correct Answer Explanation

Assertion (A) is false because Laspeyres' Index does not satisfy the Time Reversal Test. The correct Time Reversal condition is: \(p_{01}\times p_{10}=1\) The Reason (R) is false because the formula given using addition is incorrect. Hence, both A and R are false.

❓ Why Other Options Are Incorrect

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

🎯 Strategy

Used

Elimination

Application:

Verify the standard Time Reversal formula.

Final Logic:

Time Reversal uses multiplication, not addition.

🧠 Memory Trick

"Reverse → Product = 1"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Time Reversal Test –

14 What mathematical property defines the time-reversal test for an index \(p_{01}\)?

A \(p_{01}=\int p_{10} dx\)

B \(p_{01}\)and \(p_{10}\)are orthogonal vectors

C \(p_{01}+p_{10}=1\)

D \(p_{01}\times p_{10}=1\)

💡 Short Explanation

Forward and backward indices are reciprocals Product must equal 1 Ensures time symmetry

✅ Correct Answer Explanation

The Time Reversal Test states that if the time periods are reversed, the resulting indices should be reciprocals of each other. Thus: \(p_{01}\times p_{10}=1\) Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Integration is unrelated.
Option B → Orthogonal vectors are irrelevant.
Option C → Addition is not the condition.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Match the standard adequacy test formula.

Final Logic:

Reciprocal relation implies product equals 1.

🧠 Memory Trick

"Forward × Backward = 1"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Time Reversal Test –

15 Given that
\(\sum p_{1}Q_{0}=435,\sum p_{0}Q_{0}=359,\sum p_{1}Q_{1}=623,\sum p_{0}Q_{1}=515\)
calculate the approximate value of Fisher's Price Index.

A 115.4

B 119.8

C 121.2

D 124.5

💡 Short Explanation

Compute Laspeyres Index Compute Paasche Index Fisher is geometric mean

✅ Correct Answer Explanation

Laspeyres Index: \(L=\frac{435}{359}\times 100\approx 121.17\) Paasche Index: \(P=\frac{623}{515}\times 100\approx 120.97\) Fisher Index: \(F=\sqrt{121.17\times 120.97}\) \(F\approx 121.2\) Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Too low compared to computed mean.
Option B → Calculation error in geometric mean.
Option D → Overestimation.

🎯 Strategy

Used

Substitution

Application:

Substitute values into Laspeyres, Paasche, and Fisher formulas.

Final Logic:

Fisher Index is approximately 121.2.

🧠 Memory Trick

"Fisher = √(L×P)"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Fisher's Ideal Index –

16 Match the method to its defining characteristic regarding validity tests.

List I	List II
1. Fisher's Ideal	a. Fails time reversal, uses current year quantities
2. Laspeyres	b. Fails time reversal, uses base year quantities
3. Simple Aggregative	c. Fails unit test
4. Paasche	d. Satisfies time reversal test

A 1–b, 2–d, 3–a, 4–c

B 1–d, 2–b, 3–c, 4–a

C 1–c, 2–a, 3–d, 4–b

D 1–a, 2–c, 3–b, 4–d

💡 Short Explanation

Fisher satisfies Time Reversal Test Laspeyres uses base quantities Paasche uses current quantities

✅ Correct Answer Explanation

Correct matching: Fisher's Ideal → satisfies Time Reversal Test Laspeyres → uses base year quantities and fails time reversal Simple Aggregative → fails Unit Test Paasche → uses current year quantities and fails time reversal Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrectly swaps Fisher and Laspeyres.
Option C → Incorrect method-characteristic pairing.
Option D → Paasche does not satisfy Time Reversal Test.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Match each index with its defining property.

Final Logic:

Fisher alone satisfies Time Reversal among these methods.

🧠 Memory Trick

"Fisher passes, Simple fails"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Tests of Adequacy – –334

17 Which statement is false regarding tests of adequacy such as the Factor Reversal Test and Circular Test?

A The Circular Test is an extension of the time-reversal test

B The simple aggregative method easily passes the Factor Reversal Test because it uses integration

C These tests verify consistency

D Fisher's Ideal Index generally satisfies both Factor Reversal and Time Reversal tests

💡 Short Explanation

Factor Reversal checks consistency Fisher satisfies major tests Simple Aggregative does not use integration

✅ Correct Answer Explanation

The statement in Option B is false because the Simple Aggregative Method neither uses integration nor satisfies the Factor Reversal Test. The Circular Test is indeed related to repeated time-reversal consistency, and Fisher's Ideal Index satisfies both major adequacy tests. Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Correct description of Circular Test.
Option C → Adequacy tests check consistency.
Option D → Fisher satisfies both important tests.

🎯 Strategy

Used

Odd One Out

Application:

Identify the statement containing incorrect conceptual terminology.

Final Logic:

Simple Aggregative Method does not rely on integration.

🧠 Memory Trick

"Fisher clears both tests"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Factor and Circular Tests –

18 Arrange the terms to complete the formula for the Circular Test across three years (0, 1, 2):
1. \(p_{12}\)2. \(p_{20}\)3. \(p_{01}\)4. \(=1\)

A 3, 1, 2, 4

B 1, 2, 3, 4

C 2, 3, 1, 4

D 3, 2, 1, 4

💡 Short Explanation

Circular Test uses chained multiplication Product across periods equals 1 Sequence follows cyclic order

✅ Correct Answer Explanation

The Circular Test condition is: \(p_{01}\times p_{12}\times p_{20}=1\) Thus the correct order is: 3 → 1 → 2 → 4 Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrect cyclic arrangement.
Option C → Formula order disturbed.
Option D → Does not follow proper circular progression.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Use the standard Circular Test identity.

Final Logic:

Circular progression must return to the starting year.

🧠 Memory Trick

"0 → 1 → 2 → back to 0"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Circular Test –

19 In judging the accuracy of an aggregative index, why is a weighted method preferred over an unweighted one?

A Unweighted methods require finding the area of complex vectors

B Weighted methods give equal importance to every commodity

C Unweighted methods have zero probability of sampling errors

D Unweighted approaches are heavily influenced by commodities with large per-unit pricing, concealing true weights

💡 Short Explanation

Unweighted methods hide real importance High-priced items dominate results Weighted methods improve accuracy

✅ Correct Answer Explanation

In unweighted aggregative methods, commodities with high prices automatically exert more influence, even if they are less important economically. Weighted methods assign proper significance using quantities or value weights. Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Vector area has no role.
Option B → Equal importance is actually a flaw of unweighted methods.
Option C → Sampling errors can still occur.

🎯 Strategy

Used

Elimination

Application:

Remove irrelevant mathematical statements.

Final Logic:

Weighted methods correct hidden price dominance.

🧠 Memory Trick

"Weight removes bias"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Weighted Index Numbers –

20 When evaluating and comparing Laspeyres' index \(\left(I_{L}\right)\)and Paasche's index \(\left(I_{P}\right)\), how is the Marshall–Edgeworth index \(\left(I_{ME}\right)\)structurally evaluated?

A \(I_{ME}=\sqrt{I_{L}\times I_{P}}\)

B \(I_{ME}=I_{L}+I_{P}\)

C \(\frac{\sum p_{n}(Q_{0}+Q_{n})}{\sum p_{0}(Q_{0}+Q_{n})}\times 100\)

D \(I_{ME}=\int (I_{L}-I_{P}) dt\)

💡 Short Explanation

Marshall–Edgeworth uses combined quantities Base and current quantities are added It is a weighted aggregative method

✅ Correct Answer Explanation

The Marshall–Edgeworth Index uses average weights by combining base and current year quantities. Its formula is: \(I_{ME}=\frac{\sum p_{n}(Q_{0}+Q_{n})}{\sum p_{0}(Q_{0}+Q_{n})}\times 100\) Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → This is Fisher's Ideal formula.
Option B → Simple addition is incorrect.
Option D → Integration is unrelated to Marshall–Edgeworth.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Match the formula with the correct weighted index method.

Final Logic:

Marshall–Edgeworth combines both period quantities.

🧠 Memory Trick

"ME = Mixed Quantities"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Marshall–Edgeworth Index –

UG Applied Mathematics Booster Test 2 - Types, Limitations and Tests

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