UG Applied Mathematics Booster Test 3 - Construction of Index Numbers -

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CUET UG Applied Mathematics Booster Test 3 - Construction of Index Numbers

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Applied Mathematics Booster Test 3 - Construction of Index Numbers

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QUESTION 1 OF 20

Why might a broad, all-purpose index number fail, necessitating precise objective clarity?

Answer Review

1 Why might a broad, all-purpose index number fail, necessitating precise objective clarity?

A Because all indices converge to 100.

B Because variables are impossible to track.

C Because no single index can perfectly serve all specific economic purposes simultaneously.

D Because base years are infinite.

💡 Short Explanation

Different indices serve different objectives Economic variables vary widely Clear objectives improve accuracy

✅ Correct Answer Explanation

An index number is constructed for a specific purpose such as: • Measuring inflation • Tracking cost of living • Monitoring industrial production No single index can perfectly represent every economic situation simultaneously. Therefore, objective clarity is essential before construction. Option C is correct because index numbers must be purpose-oriented for meaningful interpretation.

❓ Why Other Options Are Incorrect

Option A → Indices do not necessarily converge to 100; 100 is only the base reference.
Option B → Variables can be tracked statistically using suitable indices.
Option D → Base years are selected finite reference periods.

🎯 Strategy

Used

Elimination

Application:

Remove mathematically irrelevant and conceptually false statements.

Final Logic:

Different objectives require different specialized index numbers.

🧠 Memory Trick

"One purpose → One proper index"

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

📄 Passage

"For an index to effectively reflect economic variables, objective clarity in its purpose definition is paramount, preventing ambiguous and misleading outcomes. Sampling plays a pivotal role; judgement sampling ensures representative commodities are included, provided the sample size is mathematically robust enough to reflect true market variances rather than local data errors."

According to the passage, what specific sampling approach is paired with a robust sample size to avoid ambiguous outcomes?

A Purely randomized area sampling

B Vector magnitude sampling

C Single-item selective sampling

D Judgement sampling

💡 Short Explanation

Representative commodities are required Judgement sampling selects suitable items Robust sample improves reliability

✅ Correct Answer Explanation

The passage explicitly states that judgement sampling helps include representative commodities in the index. This method: • Selects economically important goods • Avoids misleading conclusions • Works effectively with adequate sample size Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Not mentioned in the passage.
Option B → No such standard statistical method exists.
Option C → Single-item sampling cannot represent market behavior.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Match wording directly with the passage statement.

Final Logic:

Passage explicitly identifies judgement sampling.

🧠 Memory Trick

"Judge wisely, sample wisely"

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

📄 Passage

What does the passage imply is the main analytical benefit of a mathematically robust sample size?

A It reflects true market variances rather than local data errors.

B It perfectly nullifies the factor reversal test.

C It guarantees a continuous integral limit.

D It ensures the index number will always decrease.

💡 Short Explanation

Large samples improve representativeness Local fluctuations get minimized Reliability of the index increases

✅ Correct Answer Explanation

A robust sample size: • Captures broad market trends • Reduces influence of local anomalies • Produces reliable index values Thus, the passage correctly supports Option A.

❓ Why Other Options Are Incorrect

Option B → Factor reversal test is unrelated to sample size.
Option C → Integration limits are unrelated to sampling.
Option D → Sample size does not determine upward or downward movement.

🎯 Strategy

Used

Elimination

Application:

Remove unrelated mathematical and statistical concepts.

Final Logic:

Representative sampling reflects true market conditions.

🧠 Memory Trick

"Big sample → Better market picture"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Data Collection and Reliability –

4 Match the statistical flaw to its resulting error in index setup:

List I	List II
1. Base period is too short	a. Concealed weights skew index
2. Items are unrepresentative	b. Data errors compromise output
3. Calculations disregard absolute quantities	c. High unreliability of base prices
4. Data collected contains recording mistakes	d. Index fails to reflect the general field

A 1–b, 2–a, 3–d, 4–c

B 1–c, 2–d, 3–a, 4–b

C 1–d, 2–c, 3–b, 4–a

D 1–a, 2–b, 3–c, 4–d

💡 Short Explanation

Short periods reduce reliability Unrepresentative items distort coverage Recording mistakes introduce errors

✅ Correct Answer Explanation

Correct matching: • Base period too short → unreliable prices • Unrepresentative items → poor field representation • Ignoring quantities → concealed weighting distortion • Recording mistakes → data errors Thus: 1–c, 2–d, 3–a, 4–b Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect matching of data errors and unreliability.
Option C → Misplaces sampling distortion.
Option D → Incorrect logical mapping.

🎯 Strategy

Used

Option Grouping

Application:

Match each flaw with its natural statistical consequence.

Final Logic:

Correct conceptual pairing gives Option B.

🧠 Memory Trick

"Short–Unreliable, Wrong items–Distorted"

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

5 In advanced index structuring, selecting a "Normal year" implies ensuring the base period is:
I. Free of hyper-inflation
II. Not subject to an erratic economic depression
III. Subject to major cyclical distortions
IV. A period where supply and demand were relatively stable

A I, II, III, IV

B II and III

C I, II, and IV

D Only IV

💡 Short Explanation

Normal year should be stable Avoid abnormal economic shocks Stable demand and supply preferred

✅ Correct Answer Explanation

A normal base year: • Must avoid hyper-inflation • Must avoid economic depression • Should represent stable economic conditions Statement III is incorrect because cyclical distortions create abnormal comparisons. Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Includes incorrect Statement III.
Option B → Excludes important correct conditions.
Option D → Too incomplete.

🎯 Strategy

Used

Elimination

Application:

Remove options containing abnormal economic conditions.

Final Logic:

Normal year means stable and distortion-free.

🧠 Memory Trick

"Normal year = Stable year"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Choice of Base Year –

6 Which analytical conclusion regarding abnormal base years is incorrect?

A A base year with a price spike artificially depresses all subsequent index values.

B A base year during a severe depression will artificially inflate subsequent index values.

C Choosing an abnormal base year improves the time-reversal property of the index.

D Base periods should not be too short, as prices for a short period might be highly unreliable.

💡 Short Explanation

Abnormal base years distort comparisons Time reversal test is unrelated Stable base years are preferred

✅ Correct Answer Explanation

Time-reversal property depends on the mathematical structure of the index formula, not on whether the base year is abnormal. Statements A, B, and D are correct economic interpretations. Thus, Option C is the incorrect statement.

❓ Why Other Options Are Incorrect

Option A → High base prices reduce later relative values.
Option B → Depression years inflate future comparisons.
Option D → Short periods give unstable prices.

🎯 Strategy

Used

Odd One Out

Application:

Identify the option unrelated to base-year distortion.

Final Logic:

Time reversal is a formula property, not a base-year property.

🧠 Memory Trick

"Formula test ≠ Base year choice"

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

7 An unweighted aggregate price index uses 3 representative items. Their base year prices are
2, 4, x,
and their current year prices are
5, 6, 14.
If the aggregate index is exactly 125%, what was the base price x of the third representative item?

A 5

B 10

C 12

D 14

💡 Short Explanation

Apply simple aggregate formula Form equation using given index Solve for x

✅ Correct Answer Explanation

Simple aggregate index: \(\frac{5+6+14}{2+4+x}\times 100=125\) Simplify: \(\frac{25}{6+x}=1.25\) Thus: \(25=1.25(6+x)\) \(25=7.5+1.25x\) \(17.5=1.25x\) \(x=14\) Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Gives incorrect aggregate ratio.
Option B → Produces higher index.
Option C → Does not satisfy the equation.

🎯 Strategy

Used

Substitution

Application:

Substitute values directly into aggregate index formula.

Final Logic:

Solving the equation gives x = 14.

🧠 Memory Trick

"Aggregate = Current ÷ Base × 100"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Simple Aggregative Method –

8 If judgement sampling maps commodities onto a distribution curve defined by the region under
\(y=-x^{2}+4x, 0\leq x\leq 4\)
what is the total integrated statistical weight (area) of this sampled region?

A 32/3

B 16/3

C 8/3

D 64/3

💡 Short Explanation

Compute area under curve Use definite integration Apply interval limits correctly

✅ Correct Answer Explanation

Required area: \(\int_{0}^{4}\,(-x^{2}+4x)dx\) Integrate: \({\left[-\frac{x^{3}}{3}+2x^{2}\right]}_{0}^{4}\) Substitute limits: \(-\frac{64}{3}+32=\frac{32}{3}\) Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Half the required area.
Option C → Underestimates the integral.
Option D → Incorrect arithmetic expansion.

🎯 Strategy

Used

Substitution

Application:

Integrate the quadratic function over the interval.

Final Logic:

Area evaluates to 32/3.

🧠 Memory Trick

"Integrate parabola carefully"

📚 Class XII – Mathematics – Chapter: Applications of Integrals – Topic: Area Under Curve –

9 A time series of price relatives for 4 years is given as
100, 110, 120, 140.
Using a 3-year moving average on these relatives, what is the value of the moving average trend for the second data point (centered on Year 2)?

A 105

B 110

C 120

D 123.33

💡 Short Explanation

Use first three values Compute arithmetic mean Center average at Year 2

✅ Correct Answer Explanation

Three-year moving average centered at Year 2: \(\frac{100+110+120}{3}=110\) Thus, the moving average trend equals 110. Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect averaging.
Option C → Uses middle value only.
Option D → Uses incorrect data combination.

🎯 Strategy

Used

Substitution

Application:

Directly compute arithmetic mean of three observations.

Final Logic:

Moving average equals 110.

🧠 Memory Trick

"Moving average = Local mean"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Moving Averages –

10 An index evaluates quantity relatives defined as
\(q_{i}=\frac{Q_{n}}{Q_{0}}\times 100\)
For a basket of commodities, suppose
\(q_{i}∼N(\mu =110,{\sigma}^{2}=0)\)
What is the probability that a randomly chosen commodity has a quantity relative strictly greater than 110?

A 0.5

B 0

C 1

D 0.25

💡 Short Explanation

Variance zero means no spread Every observation equals 110 Strictly greater probability becomes zero

✅ Correct Answer Explanation

Since variance is zero: \({\sigma}^{2}=0\) all observations are exactly equal to the mean 110. Thus: \(P(q_{i}>110)=0\) because no value exceeds 110. Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Applies to symmetric continuous distributions with spread.
Option C → Impossible because values equal 110 exactly.
Option D → Arbitrary unsupported probability.

🎯 Strategy

Used

Dimensional/Unit Analysis

Application:

Interpret meaning of zero variance statistically.

Final Logic:

No spread means no value exceeds the mean.

🧠 Memory Trick

"Zero variance = All same"

📚 Class XI – Statistics for Economics – Chapter: Measures of Dispersion – Topic: Zero Variance Concept –

11 If an index calculation is modeled as the dot product of a base vector
\(\vec{B}=\frac{100}{∣\vec{P_{0}}∣^{2}}\vec{P_{0}}\)
and a current vector
\(\vec{P_{n}}\)
resulting in
\(Index=\vec{B}⋅\vec{P_{n}}\)
and if
\(\vec{P_{0}}=3\hat{i}+4\hat{j}, \vec{P_{n}}=6\hat{i}+8\hat{j}\)
what is the scalar value of the Index?

A 200

B 150

C 100

D 250

💡 Short Explanation

Find magnitude squared of base vector Construct normalized base vector Compute dot product with current vector

✅ Correct Answer Explanation

Magnitude squared: \(∣\vec{P_{0}}∣^{2}=3^{2}+4^{2}=25\) Thus, \(\vec{B}=\frac{100}{25}(3\hat{i}+4\hat{j})=12\hat{i}+16\hat{j}\) Now compute the index: \((12)(6)+(16)(8)=72+128=200\) Hence, the scalar value of the index is 200. Therefore, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrect dot product evaluation.
Option C → Would represent no relative increase.
Option D → Overestimates the vector product.

🎯 Strategy

Used

Substitution

Application:

Substitute vector values into the dot-product index formula.

Final Logic:

Correct vector computation gives index value 200.

🧠 Memory Trick

"3-4-5 triangle → easy normalization"

📚 Class XII – Mathematics – Chapter: Vector Algebra – Topic: Dot Product Applications –

12 If the relative expression of a price index forms a triangular region on a chart with base
\(b=10\)
years and maximum height
\(h=200\)
(at year 10),
what is the average relative index, obtained by dividing the area of the triangular region by the base time?

A 50

B 150

C 200

D 100

💡 Short Explanation

Compute triangular area Divide by total time interval Obtain average index value

✅ Correct Answer Explanation

Area of triangle: \(\frac{1}{2}\times 10\times 200=1000\) Average relative index: \(\frac{1000}{10}=100\) Thus, the average index equals 100. Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Too small after averaging.
Option B → Incorrect area division.
Option C → Represents peak value, not average.

🎯 Strategy

Used

Dimensional/Unit Analysis

Application:

Use area divided by time to obtain average height.

Final Logic:

Average index equals total area ÷ base interval.

🧠 Memory Trick

"Triangle average = half the peak"

📚 Class XII – Mathematics – Chapter: Applications of Integrals – Topic: Area Interpretation –

13 The total accumulated index value over the time interval
\(t\in [1,e]\)
is given by the definite integral
\(\int_{1}^{e}\,\frac{100}{t}dt\)
What is the value of this tabulated sum?

A 100e

B 100/e

C 100

D 0

💡 Short Explanation

Integral involves logarithm Apply definite integration limits Simplify using ln(e)=1

✅ Correct Answer Explanation

Evaluate: \(\int_{1}^{e}\,\frac{100}{t}dt=100\int_{1}^{e}\,\frac{1}{t}dt\) Integration gives: \(100[ln⁡t]_{1}^{e}\) Applying limits: \(100(ln⁡e-ln⁡1)=100(1-0)=100\) Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect exponential interpretation.
Option B → Incorrect reciprocal usage.
Option D → Integral is clearly non-zero.

🎯 Strategy

Used

Substitution

Application:

Use standard logarithmic integral formula.

Final Logic:

Definite integral evaluates exactly to 100.

🧠 Memory Trick

"∫1/t = ln t"

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

14 Assertion (A): Index series tables inherently adjust for varying units across different commodities (such as kilograms and litres).
Reason (R): Weighted aggregative indices multiply prices by quantities, resulting in uniformly expressed value tables for comparison across years.

A Both A and R are false

B A is true, R is false

C A is false, R is true

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

💡 Short Explanation

Index numbers convert values into relatives Weighted indices standardize comparisons Different units become comparable

✅ Correct Answer Explanation

Weighted aggregative indices combine prices and quantities into comparable value measures. This process: • Reduces unit incompatibility • Enables meaningful comparisons • Supports multi-commodity analysis Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion. Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Both statements are conceptually correct.
Option B → Reason is also true.
Option C → Assertion is true.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Connect weighted aggregation with standardized comparison.

Final Logic:

Weighted indices support unit-free comparisons.

🧠 Memory Trick

"Weights unify mixed units"

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

15 Arrange the following analytical steps in determining the percentage change of an index series:
1. Calculate the simple average of relatives.
2. Subtract 100 to find the percentage increase or decrease.
3. Define the base period prices as 100.
4. Express current prices relative to base prices.

A 3, 4, 1, 2

B 1, 2, 3, 4

C 4, 3, 2, 1

D 3, 1, 4, 2

💡 Short Explanation

Start with base definition Compute relatives Then average and interpret

✅ Correct Answer Explanation

Correct sequence: 1. Define base prices as 100 2. Express current prices relative to base 3. Compute average relatives 4. Subtract 100 to interpret percentage change Thus: 3 → 4 → 1 → 2 Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Starts averaging before defining relatives.
Option C → Incorrect logical sequence.
Option D → Average cannot be computed before relatives.

🎯 Strategy

Used

Option Grouping

Application:

Arrange steps in natural procedural order.

Final Logic:

Base → Relative → Average → Interpretation.

🧠 Memory Trick

"Base → Relative → Average → Change"

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

16 If an index number changes from 150 in year n to 120 in year n+1, the analytical percentage decrease relative to year n is:

A 20%

B 30%

C 25%

D 15%

💡 Short Explanation

Find absolute decrease Divide by original value Convert into percentage

✅ Correct Answer Explanation

Decrease: \(150-120=30\) Percentage decrease: \(\frac{30}{150}\times 100=20\%\) Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Uses direct difference as percentage.
Option C → Incorrect denominator.
Option D → Underestimates the fall.

🎯 Strategy

Used

Substitution

Application:

Apply percentage decrease formula directly.

Final Logic:

Percentage fall equals 20%.

🧠 Memory Trick

"Decrease ÷ Original × 100"

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

17 In constructing a multi-year index list, if
\(I_{2001}=150 and I_{2002}=225 (1990=100)\)
what is the index value for the year 2002 when the base year is shifted to 2001?

A 125

B 150

C 175

D 75

💡 Short Explanation

Use base shifting formula Divide by new base index Multiply by 100

✅ Correct Answer Explanation

New base = 2001 Required index: \(\frac{225}{150}\times 100=150\) Thus, with 2001 as base year, the 2002 index equals 150. Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect proportional conversion.
Option C → Overestimates rebased value.
Option D → Represents decrease instead of increase.

🎯 Strategy

Used

Substitution

Application:

Apply standard base-shifting formula.

Final Logic:

Rebasing gives index value 150.

🧠 Memory Trick

"New Index = Current ÷ New Base × 100"

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

18 When dealing with index series, changing the common base reference year mathematically requires:

A Simply adding 100 to all current index values

B Integration of the moving average

C Recalculating the entire series proportionally to the new base year's index value

D Using a random vector magnitude for scaling

💡 Short Explanation

Base shifting uses proportional scaling All years must be adjusted consistently Comparability must remain preserved

✅ Correct Answer Explanation

Changing the base year means: • Existing indices must be rescaled • Each value is divided by the new base index • Results are multiplied by 100 This ensures valid comparison across the rebased series. Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Simple addition is mathematically incorrect.
Option B → Moving averages are unrelated.
Option D → Random scaling destroys consistency.

🎯 Strategy

Used

Elimination

Application:

Remove mathematically irrelevant operations.

Final Logic:

Recalculation must preserve proportionality.

🧠 Memory Trick

"Rebase = Rescale"

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

19 How does a non-representative, biased sample mathematically distort the unweighted aggregate index?

A By forcing the index formula to output negative variables

B By converting a price relative into a quantity relative

C By eliminating all data errors instantly

D By allowing commodities with large per-unit pricing to disproportionately dominate the index

💡 Short Explanation

Unweighted methods ignore importance Expensive commodities dominate totals Bias distorts general trend

✅ Correct Answer Explanation

In an unweighted aggregate index: • Prices are summed directly • Higher-priced commodities influence totals more heavily • Non-representative selection exaggerates distortion Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Index values need not become negative.
Option B → Price and quantity relatives remain distinct.
Option C → Bias increases errors instead.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Identify how unweighted aggregation behaves mathematically.

Final Logic:

Expensive items dominate unweighted indices.

🧠 Memory Trick

"No weights = Big prices dominate"

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

20 If a systematic data error artificially inflates all current-year prices
\(p_{n}\)
by a factor of
\(k\)
what happens to the resulting simple aggregate index number
\(I_{n}\)
?

A It increases by k+100

B It is multiplied by a factor of k

C It remains entirely unaffected

D It is divided by k

💡 Short Explanation

Current prices appear in numerator Multiplying all prices scales numerator Index scales proportionally

✅ Correct Answer Explanation

Simple aggregate index: \(I_{n}=\frac{\sum p_{n}}{\sum p_{0}}\times 100\) If every current price becomes \(kp_{n}\) then: \(I_{n}^{'}=\frac{k\sum p_{n}}{\sum p_{0}}\times 100=kI_{n}\) Hence, the index is multiplied by k. Therefore, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Addition is incorrect; scaling occurs multiplicatively.
Option C → Numerator changes directly affect index.
Option D → Inflation error increases, not decreases, the index.

🎯 Strategy

Used

Substitution

Application:

Replace current prices with kpₙ in the formula.

Final Logic:

Multiplying numerator by k multiplies the index by k.

🧠 Memory Trick

"Scale prices → Scale index"

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

CUET UG Applied Mathematics Booster Test 3 - Construction of Index Numbers

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