CUET UG Applied Mathematics Booster Test 3

🚀

CUET UG Applied Mathematics Booster Test 3 - Index Number Basics

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Applied Mathematics Booster Test 3 - Index Number Basics

1 / 20

QUESTION 1 OF 20

Q1. Concept of Index Number [Area Under Curve]
The price index function is given by:
\(I(x)=100+10x-x^{2}\)
Calculate the area under the index curve between x=0 and x=3:
\(\int_{0}^{3}\,(100+10x-x^{2}) dx\)

Answer Review

1 Q1. Concept of Index Number [Area Under Curve]
The price index function is given by:
\(I(x)=100+10x-x^{2}\)
Calculate the area under the index curve between x=0 and x=3:
\(\int_{0}^{3}\,(100+10x-x^{2}) dx\)

A 300

B 325

C 336

D 350

💡 Short Explanation

Integrate termwise Apply limits 0 to 3 Compute total area

✅ Correct Answer Explanation

Evaluate: \(\int_{0}^{3}\,(100+10x-x^{2})dx\) Integrating: \(100x+5x^{2}-\frac{x^{3}}{3}\) Applying limits: At x = 3: \(100(3)+5(3)^{2}-\frac{{\left(3\right)}^{3}}{3}=300+45-9=336\) At x = 0, value = 0. Hence total area = 336. Therefore, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Ignores quadratic contribution.
Option B → Incorrect limit substitution.
Option D → Arithmetic overestimation.

🎯 Strategy

Used

Substitution

Application:

Integrate the polynomial and substitute upper and lower limits.

Final Logic:

Definite integration gives total area 336.

🧠 Memory Trick

"Integrate → Substitute → Subtract"

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

2 Q2. Index as Measure of Change [Arrange in Order]
Arrange the following index number formulas in increasing order of complexity based on their mathematical formulation:
1. Relative Price Index:
\(\frac{p_{n}}{p_{0}}\times 100\)
1. Fisher's Ideal Index:
\(\sqrt{L\times P}\times 100\)
1. Unweighted Aggregative Index:
\(\frac{\sum p_{n}}{\sum p_{0}}\times 100\)
1. Laspeyres' Index:
\(\frac{\sum p_{n}q_{0}}{\sum p_{0}q_{0}}\times 100\)

A 1, 4, 3, 2

B 3, 1, 4, 2

C 4, 3, 2, 1

D 1, 3, 4, 2

💡 Short Explanation

Start from simplest ratio Weighted methods are more complex Fisher combines two indices

✅ Correct Answer Explanation

Complexity increases as follows: • Relative Price Index → simple ratio of one item • Unweighted Aggregative → sums multiple prices • Laspeyres → weighted aggregate using base quantities • Fisher's Ideal → geometric mean of Laspeyres and Paasche Thus the order is: 1 → 3 → 4 → 2 Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Places Laspeyres before simpler aggregative form.
Option B → Incorrectly places unweighted index first.
Option C → Completely reversed order.

🎯 Strategy

Used

Option Grouping

Application:

Compare formulas from simplest arithmetic form to advanced weighted structure.

Final Logic:

Fisher's formula is mathematically most advanced.

🧠 Memory Trick

"Simple → Aggregate → Weighted → Ideal"

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

📄 Passage

Choice of base period: The base period should be a normal year; meaning that the prices of that period should not be subject to an erratic boom, depression, or the effects of a natural calamity. The base period should not be too short or too long, as prices for a very short period are highly unreliable. Selection of commodities is done by judgement sampling.

According to the passage, why must the base period not be extremely short?

A Prices for a very short period are highly unreliable

B The Fisher's Ideal method becomes undefined

C Vector methods cannot be applied

D Short periods prevent aggregate calculations

💡 Short Explanation

Base year should be stable Very short periods fluctuate heavily Reliability decreases

✅ Correct Answer Explanation

The passage clearly states that very short periods produce highly unreliable price data. An ideal base period should represent stable and normal economic conditions. Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Fisher's formula remains valid.
Option C → Vector methods are unrelated.
Option D → Aggregation is still possible.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Directly identify the statement from the passage.

Final Logic:

Short periods produce unreliable prices.

🧠 Memory Trick

"Stable Base = Reliable Index"

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

📄 Passage

How is the selection of relevant commodities for the data sample primarily conducted?

A Pure random sampling

B Judgement sampling

C Complete population survey

D Probability density methods

💡 Short Explanation

Commodity selection needs relevance Expert choice is used Judgement sampling is practical

✅ Correct Answer Explanation

The passage explicitly mentions that commodities are selected using judgement sampling. This method ensures representative and economically important commodities are chosen. Therefore, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Pure random selection may ignore relevance.
Option C → Complete surveys are impractical.
Option D → Probability density methods are unrelated.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Match wording directly with the passage statement.

Final Logic:

Commodity selection uses judgement sampling.

🧠 Memory Trick

"Useful Goods → Judgement Choice"

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

5 Match List I with List II based on time series components.

List I	List II
1. Secular Trend	a. Oscillatory movement over more than one year
2. Seasonal Component	b. Smooth, long-term variation
3. Cyclical Component	c. Unpredictable changes (e.g., floods, strikes)
4. Irregular Component	d. Periodic variation within a year

A 1–a, 2–b, 3–c, 4–d

B 1–b, 2–a, 3–d, 4–c

C 1–b, 2–d, 3–a, 4–c

D 1–d, 2–c, 3–a, 4–b

💡 Short Explanation

Trend = long-term movement Seasonal = yearly repetition Irregular = unpredictable events

✅ Correct Answer Explanation

Correct matching: • Secular Trend → smooth long-term variation • Seasonal Component → periodic yearly movement • Cyclical Component → long oscillatory movement • Irregular Component → unexpected fluctuations Thus: 1–b, 2–d, 3–a, 4–c Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Completely mismatched components.
Option B → Swaps seasonal and cyclical parts.
Option D → Incorrect trend mapping.

🎯 Strategy

Used

Option Grouping

Application:

Match each component with its defining characteristic.

Final Logic:

Time series components have distinct meanings.

🧠 Memory Trick

"Trend–Long, Seasonal–Yearly"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Components of Time Series –

6 Q6. Use of Same Base Period [Integral]
An index series has a constant base. The inflation rate is given by:
\(\frac{dI}{dt}=3\sqrt{t}\)
with base index
\(I(0)=100\)
Find the index value at t = 4:
\(I(4)=100+\int_{0}^{4}\,3t^{1/2}dt\)

A 110

B 112

C 114

D 116

💡 Short Explanation

Integrate square-root function Apply limits Add base value

✅ Correct Answer Explanation

Evaluate: \(\int_{0}^{4}\,3t^{1/2}dt\) Integration: \(3⋅\frac{2}{3}t^{3/2}=2t^{3/2}\) Applying limits: \(2(4)^{3/2}=2(8)=16\) Thus: \(I(4)=100+16=116\) Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Underestimates integral value.
Option B → Arithmetic error.
Option C → Incorrect exponent handling.

🎯 Strategy

Used

Substitution

Application:

Integrate and apply boundary conditions directly.

Final Logic:

Total index value equals 116.

🧠 Memory Trick

"√t integrates to t³ᐟ²"

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

7 Q7. CPI and GDP Indices [Multiple Correct]
Which of the following formulas correctly represent weighted aggregate index methods?
(i)
\(\frac{\sum P_{ni}Q_{0i}}{\sum P_{0i}Q_{0i}}\times 100\)
(Laspeyres)
(ii)
\(\frac{\sum P_{ni}Q_{ni}}{\sum P_{0i}Q_{ni}}\times 100\)
(Paasche)
(iii)
\(\frac{\sum P_{n}}{\sum P_{0}}\times 100\)
(Simple Aggregative)

A (i) and (ii)

B (ii) and (iii)

C (i) and (iii)

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

💡 Short Explanation

Weighted methods use quantities Laspeyres uses base weights Paasche uses current weights

✅ Correct Answer Explanation

Statements (i) and (ii) are weighted aggregate methods because they include quantities as weights. Statement (iii) is an unweighted aggregative method since it uses only prices. Thus, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Includes unweighted method.
Option C → Includes unweighted formula.
Option D → Simple aggregative is not weighted.

🎯 Strategy

Used

Elimination

Application:

Remove options containing the unweighted formula.

Final Logic:

Only Laspeyres and Paasche are weighted methods.

🧠 Memory Trick

"Weights Need Quantities"

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

8 Q8. Stock Market Indices [Graph-Based]
In technical analysis of a stock time series, a smooth trend-cycle curve is plotted over raw daily data.
Which mathematical technique is used to generate this smooth curve?

A Definite integration

B Moving averages method

C Unit test of adequacy

D Absolute summation method

💡 Short Explanation

Moving averages smooth fluctuations Trend becomes visible Used in stock analysis

✅ Correct Answer Explanation

The moving averages method smooths short-term fluctuations and highlights the long-term trend-cycle movement. This is commonly used in stock market analysis. Therefore, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Integration measures area, not smoothing.
Option C → Unit test checks adequacy.
Option D → Not a recognized smoothing technique.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Identify the method associated with trend smoothing.

Final Logic:

Moving averages create smooth curves.

🧠 Memory Trick

"Moving Average = Smooth Trend"

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

9 Q9. Price Level Tracking [Moving Average]
Given rainfall data (in mm):
1.2, 1.9, 2.0, 1.4, 2.1
Using a 3-year moving average, find the value corresponding to the year with rainfall 2.0:

A 1.50

B 1.65

C 1.76

D 1.83

💡 Short Explanation

Take three consecutive observations Compute average Center around middle value

✅ Correct Answer Explanation

For rainfall value 2.0, use: 1.9, 2.0, 1.4 Moving average: \(\frac{1.9+2.0+1.4}{3}\) Calculation: \(\frac{5.3}{3}=1.76\) Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Too low.
Option B → Incorrect averaging.
Option D → Overestimated average.

🎯 Strategy

Used

Substitution

Application:

Insert nearby values into moving average formula.

Final Logic:

Centered average equals 1.76.

🧠 Memory Trick

"Middle Year Uses Neighboring Data"

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

10 Q10. Living Standard Measure [Case/Numerical]
Using Fisher's Ideal Index:
\(F=\sqrt{L\times P}\)
Given:
\(L=400, P=361\)
Compute the Fisher's price index.

A 360

B 370

C 375

D 380

💡 Short Explanation

Multiply L and P Take square root Obtain Fisher index

✅ Correct Answer Explanation

Using Fisher's formula: \(F=\sqrt{400\times 361}\) Calculation: \(F=\sqrt{144400}=380\) Thus, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Too small after square root.
Option B → Arithmetic error.
Option C → Incorrect square root value.

🎯 Strategy

Used

Substitution

Application:

Substitute values into Fisher's formula directly.

Final Logic:

Square root of 144400 equals 380.

🧠 Memory Trick

"Fisher = √(L×P)"

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

11 Q11. Business Forecasting [Probability]
A dataset consists of n = 5 years of time series data. If the method of least squares is applied to fit the trend line
\(y=a+bx\)
what is the probability that
\(\sum X=0\)
when the origin is randomly assigned to any one of the 5 years?

A 1/5

B 2/5

C 3/5

D 4/5

💡 Short Explanation

Central year gives ΣX = 0 Only one suitable origin exists Probability = favorable/total

✅ Correct Answer Explanation

For an odd number of observations (n = 5), the least squares method sets the middle year as origin so that: \(\sum X=0\) Out of 5 years, only the central year satisfies this condition. Thus: \(P=\frac{1}{5}\) Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Assumes two middle years exist.
Option C → Overcounts favorable cases.
Option D → Incorrect probability calculation.

🎯 Strategy

Used

Substitution

Application:

Identify the single middle year among five observations.

Final Logic:

One favorable origin out of five gives 1/5.

🧠 Memory Trick

"Odd n → One middle origin"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Method of Least Squares –

12 Q12. Economic Planning [Vectors]
For calculating ΣXY using the least squares method, consider the vectors
\(\vec{X}=(-2,-1,0,1,2)\)
\(\vec{Y}=(140,150,190,200,220)\)
The value of ΣXY is equal to the dot product
\(\vec{X}⋅\vec{Y}\)
Compute this value.

A 190

B 210

C 240

D 250

💡 Short Explanation

Multiply corresponding terms Add all products Obtain dot product

✅ Correct Answer Explanation

Compute: \(\left(-2)(140)+(-1)(150)+(0)(190)+(1)(200)+(2)(220\right)\) Simplifying: \(-280-150+0+200+440=210\) Thus: \(\sum XY=210\) Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Misses some vector products.
Option C → Arithmetic error.
Option D → Incorrect total summation.

🎯 Strategy

Used

Substitution

Application:

Directly substitute vector values into dot product computation.

Final Logic:

Sum of corresponding products equals 210.

🧠 Memory Trick

"Dot Product = Multiply then Add"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Least Squares Method –

13 Q13. Time Comparison [Assertion–Reason]
Assertion (A): Fisher's Ideal Index is considered the ideal index number.
Reason (R): Fisher's index satisfies both the time-reversal test and the factor-reversal test, whereas Laspeyres and Paasche indices do not.

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

Fisher satisfies adequacy tests Other indices fail some tests Hence called ideal index

✅ Correct Answer Explanation

Fisher's Ideal Index is called "ideal" because it satisfies: • Time Reversal Test • Factor Reversal Test Laspeyres and Paasche fail at least one adequacy condition. Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion. Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Both statements are actually true.
Option B → Reason is also true.
Option D → Assertion is true.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Connect "ideal" terminology with adequacy tests.

Final Logic:

Fisher is ideal because it passes both tests.

🧠 Memory Trick

"Fisher Passes Both Tests"

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

14 Q14. Place Comparison [Incorrect Statement]
When comparing the cost of living across different places, various tests of adequacy are used to verify index consistency. Identify the incorrect statement.

A The Unit Test ensures that the method is independent of physical units such as kilograms or litres.

B The Simple Aggregative Method successfully satisfies the Unit Test.

C The Time Reversal Test verifies whether time subscripts are interchangeable.

D Fisher's Ideal Index satisfies the Time Reversal Test.

💡 Short Explanation

Simple aggregative depends on units Unit test checks unit independence Fisher satisfies reversal test

✅ Correct Answer Explanation

The Simple Aggregative Method fails the Unit Test because changing units (kg to grams, litres to ml) changes the index value. Thus Option B is incorrect. Other statements are conceptually correct. Hence, Option B is correct as the incorrect statement.

❓ Why Other Options Are Incorrect

Option A → Correct definition of Unit Test.
Option C → Correct meaning of Time Reversal Test.
Option D → Fisher satisfies the Time Reversal Test.

🎯 Strategy

Used

Elimination

Application:

Verify which statement violates standard adequacy properties.

Final Logic:

Simple aggregative method fails the Unit Test.

🧠 Memory Trick

"Simple Aggregate → Unit Sensitive"

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

15 Q15. Poverty Measurement [Area]
Using the Marshall–Edgeworth method, the numerator is given by
\(\sum P_{n}(Q_{0}+Q_{n})\)
If
\(\sum P_{n}Q_{0}=120\)
and
\(\sum P_{n}Q_{n}=150\)
find the total value of the numerator.

A 270

B 300

C 18000

D 1.25

💡 Short Explanation

Add both summations Apply Marshall–Edgeworth numerator Obtain total weighted value

✅ Correct Answer Explanation

Using: \(\sum P_{n}(Q_{0}+Q_{n})=\sum P_{n}Q_{0}+\sum P_{n}Q_{n}\) Substitute values: \(120+150=270\) Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrect addition.
Option C → Multiplication used incorrectly.
Option D → Represents a ratio, not total.

🎯 Strategy

Used

Substitution

Application:

Insert given totals directly into formula.

Final Logic:

Numerator total equals 270.

🧠 Memory Trick

"Marshall = Add Both Weighted Parts"

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

16 Q16. Prosperity Indicators [Arrange in Order]
Arrange the following steps involved in calculating the Laspeyres Index (L) in the correct sequential order:
1. Compute Σp₀q₀ = 3380
2. Multiply by 100 to obtain the final index I = 152.07
3. Compute Σpₙq₀ = 5140
4. Divide Σpₙq₀ by Σp₀q₀

A 1, 2, 3, 4

B 1, 3, 4, 2

C 3, 1, 2, 4

D 4, 3, 1, 2

💡 Short Explanation

Compute denominator first Compute numerator next Divide and multiply by 100

✅ Correct Answer Explanation

Laspeyres formula: \(L=\frac{\sum p_{n}q_{0}}{\sum p_{0}q_{0}}\times 100\) Correct sequence: • Compute denominator • Compute numerator • Divide numerator by denominator • Multiply by 100 Thus: 1 → 3 → 4 → 2 Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Final multiplication occurs too early.
Option C → Wrong computational sequence.
Option D → Division cannot occur before totals are computed.

🎯 Strategy

Used

Option Grouping

Application:

Follow the logical order of formula evaluation.

Final Logic:

Formula sequence matches Option B.

🧠 Memory Trick

"Denominator → Numerator → Divide → ×100"

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

17 Q17. Trend Understanding [Integral]
A continuous straight-line trend of sales (in lakhs) is given by
\(Y(x)=5.9+0.13x\)
The total aggregate expected sales between x = 0 and x = 10 is represented by
\(\int_{0}^{10}\,(5.9+0.13x)dx\)
Compute this value.

A 55.4

B 60.5

C 65.5

D 72.0

💡 Short Explanation

Integrate linear trend Apply upper and lower limits Compute total sales area

✅ Correct Answer Explanation

Integrate: \(\int (5.9+0.13x)dx=5.9x+0.065x^{2}\) Applying limits: At x = 10: \(5.9(10)+0.065(10)^{2}=59+6.5=65.5\) At x = 0, value = 0. Hence total expected sales = 65.5. Therefore, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Omits quadratic contribution.
Option B → Arithmetic error.
Option D → Overestimation.

🎯 Strategy

Used

Substitution

Application:

Integrate function and evaluate at boundaries.

Final Logic:

Total area under trend line equals 65.5.

🧠 Memory Trick

"Integrate Line → Total Trend"

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

18 Q18. Market Analysis [Multiple Correct]
Which of the following are components of a time series used in market data analysis?
(i) Secular Trend Component
(ii) Seasonal Component
(iii) Cyclical Component
(iv) Irregular Component

A (i) and (ii) only

B (ii) and (iii) only

C (i), (ii), and (iv) only

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

💡 Short Explanation

Time series has four components Includes regular and irregular movement Used for trend analysis

✅ Correct Answer Explanation

A complete time series consists of: • Secular Trend • Seasonal Component • Cyclical Component • Irregular Component All four are standard components in market analysis. Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Omits cyclical and irregular components.
Option B → Omits trend and irregular parts.
Option C → Omits cyclical component.

🎯 Strategy

Used

Option Grouping

Application:

Recall the standard four-part decomposition of time series.

Final Logic:

All four components are essential.

🧠 Memory Trick

"Trend–Seasonal–Cycle–Irregular"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Components of Time Series – 286

19 Q19. Policy Formulation [Moving Average/Data]
For a 4-year moving average, the first two 4-year moving totals are 1730 and 1762. The corresponding 4-year moving averages are 432.5 and 440.5. Determine the final centered moving average.

A 436.5

B 437.75

C 438.5

D 440.0

💡 Short Explanation

Centering averages two moving averages Add neighboring averages Divide by 2

✅ Correct Answer Explanation

Centered moving average: \(\frac{432.5+440.5}{2}\) Calculation: \(\frac{873}{2}=436.5\) Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrect averaging.
Option C → Arithmetic mistake.
Option D → Uses one average only.

🎯 Strategy

Used

Substitution

Application:

Apply centered moving average formula directly.

Final Logic:

Average of adjacent moving averages equals 436.5.

🧠 Memory Trick

"Center = Average of Two Averages"

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

20 Q20. Decision Making [Case/Numerical]
An analyst applies the Method of Least Squares for an odd number of years (n = 7), with central year A = 2017. Given that
\(\sum Y=1065\)
compute the value of the trend constant a.

A 16.6

B 152.14

C 201.9

D 465

💡 Short Explanation

For odd years, ΣX = 0 Constant term equals mean of Y Divide total by n

✅ Correct Answer Explanation

In least squares fitting: \(a=\frac{\sum Y}{n}\) Substitute values: \(a=\frac{1065}{7}=152.14\) Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect division.
Option C → Arithmetic overestimation.
Option D → Uses total instead of average.

🎯 Strategy

Used

Substitution

Application:

Directly apply least squares constant formula.

Final Logic:

Trend constant equals average of Y values.

🧠 Memory Trick

"a = Average of Y"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Least Squares Method –

CUET UG Applied Mathematics Booster Test 3 - Index Number Basics

Test Complete!

Answer Review