CUET UG Applied Mathematics Booster Test 2

🚀

CUET UG Applied Mathematics Booster Test 2 - 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 2 - Index Number Basics

1 / 20

QUESTION 1 OF 20

Q1. Concept of Index Number [Integral]
If the marginal change of an index number is modeled continuously over time as:
\(\frac{dI}{dt}=6t\)
and the base period index is given by
\(I(0)=100\)
find the index function
\(I(t)\)
using integration.

Answer Review

1 Q1. Concept of Index Number [Integral]
If the marginal change of an index number is modeled continuously over time as:
\(\frac{dI}{dt}=6t\)
and the base period index is given by
\(I(0)=100\)
find the index function
\(I(t)\)
using integration.

A I(t)=3t²

B I(t)=100+3t²

C I(t)=100+6t

D I(t)=100+t³

💡 Short Explanation

Integrate the derivative Add integration constant Apply initial condition

✅ Correct Answer Explanation

Given: \(\frac{dI}{dt}=6t\) Integrating both sides: \(I(t)=\int 6t dt=3t^{2}+C\) Using initial condition: \(I(0)=100\) So, \(100=3(0)^{2}+C\) Thus: \(C=100\) Final equation: \(I(t)=100+3t^{2}\) Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Ignores the initial condition.
Option C → Incorrect integration of 6t.
Option D → Integral of 6t is not t³.

🎯 Strategy

Used

Substitution

Application:

Integrate and apply the initial value directly.

Final Logic:

Correct integration plus constant gives the required function.

🧠 Memory Trick

"Integrate → Add Constant"

📚 Class XII – Mathematics Part II – Chapter: Differential Equations – Topic: Formation by Integration –

2 Q2. Index as Measure of Change [Vector-Based Numerical]
Consider prices represented as a position vector:
\(\vec{r}=P_{0}\hat{i}+P_{n}\hat{j}\)
Given:
\(\vec{r}=30\hat{i}+46\hat{j}\)
representing wheat prices in 1997 and 2001 respectively, the price relative index is defined as:
\(\left(\frac{y-component}{x-component}\right)\times 100\)
Find the value of the index number.

A 130.4

B 146.0

C 153.33

D 160.0

💡 Short Explanation

Use relative index formula Divide current by base price Multiply by 100

✅ Correct Answer Explanation

Given: \(P_{0}=30, P_{n}=46\) Index number: \(\frac{46}{30}\times 100\) Calculating: \(153.33\) Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect division.
Option B → Miscalculated ratio.
Option D → Overestimated value.

🎯 Strategy

Used

Substitution

Application:

Substitute vector components into the index formula.

Final Logic:

Relative price increase equals 153.33.

🧠 Memory Trick

"Current ÷ Base × 100"

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

3 Q3. Relative Comparison Idea [Case/Numerical]
A departmental store spent ₹1.3 lakh in 1990 and ₹2.9 lakh in 2000 on newspaper advertisements. Using 1990 as the base year, find the price index for advertisement prices in 2000.
\(Index=\left(\frac{2.9}{1.3}\right)\times 100\)

A 123

B 167

C 200

D 223

💡 Short Explanation

Divide current by base Multiply by 100 Round appropriately

✅ Correct Answer Explanation

Applying the index formula: \(\frac{2.9}{1.3}\times 100\) Calculation: \(223.08\approx 223\) Therefore, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Too low compared to actual ratio.
Option B → Incorrect computation.
Option C → Assumes doubling only.

🎯 Strategy

Used

Substitution

Application:

Use the standard index number formula directly.

Final Logic:

Advertisement expenditure index equals 223.

🧠 Memory Trick

"Relative Rise × 100"

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

4 Q4. Base and Current Period [Probability]
Out of 5 available years (2000, 2001, 2002, 2003, 2004), a statistician randomly selects one year as the base period and one strictly subsequent year as the current period.
What is the probability that 2000 is chosen as the base period?
\(Total possible pairs=\left(\frac{5}{2}\right)=10\)
\(Favourable cases=4\)
\(P=\frac{4}{10}\)

A 4/10

B 1/5

C 1/2

D 1/4

💡 Short Explanation

Count total valid pairs Count favorable outcomes Use probability formula

✅ Correct Answer Explanation

Possible current years after 2000 are: 2001, 2002, 2003, 2004 Thus favorable cases = 4. Total ordered valid pairs: \(\left(\frac{5}{2}\right)=10\) Therefore: \(P=\frac{4}{10}\) Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrect favorable count.
Option C → Overestimated probability.
Option D → Incorrect denominator.

🎯 Strategy

Used

Substitution

Application:

Apply probability = favorable/total.

Final Logic:

Probability equals 4/10.

🧠 Memory Trick

"Probability = Favorable ÷ Total"

📚 Class XI – Mathematics – Chapter: Probability – Topic: Classical Probability –

5 Q5. Meaning of Index Series [Moving Average/Data]
For a 4-year moving average analysis, the 4-year moving totals are:
1730 and 1772
Find the centered total for these periods:

A 1751

B 875.5

C 437.75

D 443

💡 Short Explanation

Add adjacent totals Divide by 2 Center the moving totals

✅ Correct Answer Explanation

Centered total: \(\frac{1730+1772}{2}\) Calculating: \(\frac{3502}{2}=1751\) Thus, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Divides twice unnecessarily.
Option C → Incorrect averaging.
Option D → Wrong arithmetic.

🎯 Strategy

Used

Substitution

Application:

Compute centered moving total directly.

Final Logic:

Centered value equals 1751.

🧠 Memory Trick

"Centered = Average of Adjacent Totals"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Moving Averages – 296

6 Match List I with List II based on index number formulae.

List I	List II
1. Simple Aggregative	a. (ΣPnQn / ΣP0Qn) × 100
2. Simple Average of Relatives	b. (ΣPn / ΣP0) × 100
3. Laspeyres' Index	c. (ΣPnQ0 / ΣP0Q0) × 100
4. Paasche's Index	d. [Σ((Pn / P0) × 100)] / N

A 1–c, 2–d, 3–b, 4–a

B 1–b, 2–c, 3–a, 4–d

C 1–b, 2–d, 3–c, 4–a

D 1–d, 2–a, 3–c, 4–b

💡 Short Explanation

Identify formulas carefully Laspeyres uses base weights Paasche uses current weights

✅ Correct Answer Explanation

Mappings: • Simple Aggregative → \(\frac{\sum P_{n}}{\sum P_{0}}\times 100\) • Simple Average of Relatives → \(\frac{\sum \left(\frac{P_{n}}{P_{0}},\ 100\right)}{N}\) • Laspeyres → base-period weights: \(\frac{\sum P_{n}Q_{0}}{\sum P_{0}Q_{0}}\times 100\) • Paasche → current-period weights: \(\frac{\sum P_{n}Q_{n}}{\sum P_{0}Q_{n}}\times 100\) Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrectly swaps formulas.
Option B → Misplaces Laspeyres and Paasche.
Option D → Wrong matching structure.

🎯 Strategy

Used

Option Grouping

Application:

Group formulas based on weight usage.

Final Logic:

Base weights → Laspeyres; current weights → Paasche.

🧠 Memory Trick

"L = Old Weights, P = Present Weights"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Weighted Index Numbers – 339

7 Q7. CPI and GDP Indices [Multiple Correct]
Which of the following statements regarding Consumer Price Index (CPI) and GDP are correct?
(i) They are expressed as pure numbers.
(ii) They rely on absolute non-relative metrics.
(iii) They measure changes in general economic variables over time.

A (i) and (ii)

B (ii) and (iii)

C (i) only

D (i) and (iii)

💡 Short Explanation

Index numbers are relative measures Expressed as pure numbers Used for economic comparisons

✅ Correct Answer Explanation

CPI and GDP indices are relative measures expressed as pure numbers or percentages. They measure economic changes over time. Statement (ii) is incorrect because index numbers are based on relative comparison, not absolute values. Thus, statements (i) and (iii) are correct.

❓ Why Other Options Are Incorrect

Option A → Includes incorrect statement (ii).
Option B → Statement (ii) is false.
Option C → Ignores statement (iii).

🎯 Strategy

Used

Elimination

Application:

Remove options containing false statement (ii).

Final Logic:

Only (i) and (iii) remain correct.

🧠 Memory Trick

"Index = Relative Measure"

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

📄 Passage

An index number is a measure that tracks the movement in the general level of prices of consumer goods and services. A collection of index numbers for different years or locations is called an index series. At the National Stock Exchange (NSE) and Bombay Stock Exchange (BSE), market fluctuations are tracked using the Nifty and Sensex indices. The choice of base period should be a normal year, meaning that the prices should not be subject to an erratic boom, depression, or the effects of a natural calamity.

According to the passage, Nifty and Sensex are primarily used to track movements in:

A National and Bombay Stock Exchanges

B International agricultural markets

C Retail sales index tracking

D Gross domestic product calculations

💡 Short Explanation

Nifty belongs to NSE Sensex belongs to BSE Both track stock market movement

✅ Correct Answer Explanation

Nifty and Sensex are stock market indices. They are used to monitor movements in the National Stock Exchange (NSE) and Bombay Stock Exchange (BSE). Therefore, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Unrelated to stock indices.
Option C → Retail sales use separate indices.
Option D → GDP is measured differently.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Use stock market context from the passage.

Final Logic:

Nifty and Sensex belong to NSE and BSE.

🧠 Memory Trick

"Nifty–NSE, Sensex–BSE"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Stock Market Indices –

📄 Passage

Based on the passage, the choice of a base period must avoid years containing:

A Normal economic stability

B Erratic booms or natural calamities

C Average consumer price fluctuations

D Standard living adjustments

💡 Short Explanation

Base year must be normal Avoid abnormal economic conditions Natural calamities distort prices

✅ Correct Answer Explanation

An ideal base year should reflect normal economic conditions. Years with booms, depressions, or natural calamities distort comparisons and are avoided. Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Normal stability is desirable.
Option C → Average fluctuations are acceptable.
Option D → Standard of living changes are measured, not avoided.

🎯 Strategy

Used

Extreme Word Filter

Application:

Identify abnormal conditions mentioned in the passage.

Final Logic:

Abnormal years are unsuitable as base years.

🧠 Memory Trick

"Base Year = Normal Year"

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

10 Q10. Living Standard Measure [Assertion–Reason]
Assertion (A): The Laspeyres index may overestimate the rise in the cost of living.
Reason (R): The Laspeyres index uses base-period weights, ignoring the possibility that consumers may reduce consumption of items that become relatively more expensive.

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 uses old weights Consumer substitution ignored Cost rise may be overstated

✅ Correct Answer Explanation

The Laspeyres Index uses base-year quantities as weights. Consumers often reduce consumption of goods that become expensive, but Laspeyres ignores this substitution effect. Therefore, it may overestimate the increase in cost of living. Hence, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

❓ 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:

Link the substitution effect with Laspeyres bias.

Final Logic:

Base weights cause overestimation.

🧠 Memory Trick

"Laspeyres Uses Old Habits"

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

11 Arrange the steps involved in forecasting using the method of least squares for fitting a straight line:
\(Y_{t}=a+bX\)
1. Compute
\(a=\frac{\sum Y}{n}\)
2. Set the midpoint of time as origin so that
\(\sum X=0\)
3. Compute
\(b=\frac{\sum XY}{\sum X^{2}}\)
4. Substitute a and b into
\(Y=a+bX\)
\(a\)
\(b\)

A 1, 2, 3, 4

B 2, 4, 1, 3

C 3, 1, 4, 2

D 2, 1, 3, 4

💡 Short Explanation

Start by setting origin Then compute constants Finally substitute into trend equation

✅ Correct Answer Explanation

In the least squares method: 1. First set the midpoint as origin so that \(\sum X=0\) 1. Then calculate \(a=\frac{\sum Y}{n}\) 1. Next calculate slope: \(b=\frac{\sum XY}{\sum X^{2}}\) 1. Finally substitute into: \(Y=a+bX\) \(a\) \(b\) Thus, the correct order is 2, 1, 3, 4.

❓ Why Other Options Are Incorrect

Option A → Midpoint origin must be set before calculations.
Option B → Substitution cannot occur before finding constants.
Option C → Sequence is logically incorrect.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Arrange steps according to the standard forecasting procedure.

Final Logic:

Constants are computed only after defining the origin.

🧠 Memory Trick

"Origin → a → b → Equation"

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

12 An economic plan requires the total planned output over 4 years. If the planned output index is given by:
\(y=50x\)
find the total output represented by:
\(\int_{0}^{4}\,50x dx\)

A 400

B 200

C 800

D 600

💡 Short Explanation

Integrate the function Apply upper and lower limits Definite integral gives total output

✅ Correct Answer Explanation

Integrating: \(\int 50x dx=25x^{2}\) Applying limits: \(25(4)^{2}-25(0)^{2}\) Thus: \(25\times 16=400\) Therefore, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrect integration.
Option C → Overestimation.
Option D → Arithmetic error.

🎯 Strategy

Used

Substitution

Application:

Apply definite integration directly.

Final Logic:

Area under the output curve equals 400.

🧠 Memory Trick

"Integral = Total Accumulation"

📚 Class XII – Mathematics Part II – Chapter: Integrals – Topic: Definite Integrals – 353

13 Which of the following is an incorrect statement regarding the time-reversal test for index numbers?
\(I_{01}\times I_{10}=1\)

A Fisher's Ideal Index satisfies the time-reversal test.

B Laspeyres method satisfies the time-reversal test.

C The test requires that

D Paasche method does not satisfy the time-reversal test.

💡 Short Explanation

Fisher satisfies the test Laspeyres does not satisfy it Reciprocal condition defines the test

✅ Correct Answer Explanation

The time-reversal test requires: \(I_{01}\times I_{10}=1\) Fisher's Ideal Index satisfies this property. Laspeyres and Paasche indices do not satisfy the time-reversal test. Hence, Option B is the incorrect statement.

❓ Why Other Options Are Incorrect

Option A → Fisher's Ideal Index correctly satisfies the test.
Option C → This is the exact mathematical condition.
Option D → Paasche indeed fails the test.

🎯 Strategy

Used

Elimination

Application:

Identify which statement contradicts standard index theory.

Final Logic:

Only Fisher satisfies the time-reversal test.

🧠 Memory Trick

"Fisher Passes Time Test"

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

14 Consider the cost of living indices for two places:
• Place A:
\(y=120\)
• Place B:
\(y=110+2x\)
Find the value of x (in years) at which the index of Place B intersects and begins to exceed that of Place A.

A x = 2

B x = 3

C x = 5

D x = 10

💡 Short Explanation

Equate both indices Solve linear equation Intersection gives threshold year

✅ Correct Answer Explanation

At intersection: \(110+2x=120\) Solving: \(2x=10\) Thus: \(x=5\) Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Gives value below 120.
Option B → Still below intersection.
Option D → Occurs after the intersection point.

🎯 Strategy

Used

Substitution

Application:

Equate both linear equations.

Final Logic:

Equality occurs at x = 5.

🧠 Memory Trick

"Intersection → Equal Values"

📚 Class XI – Mathematics – Chapter: Linear Equations – Topic: Solving Linear Equations –

15 The rate of change of a poverty index is given by:
\(\frac{dI}{dt}=-2t\)
with initial condition
\(I(0)=150\)
Using integration, find the value of the poverty index at t = 5.
Given:
\(I(t)=150-t^{2}\)

A 100

B 140

C 130

D 125

💡 Short Explanation

Integrate rate equation Use initial condition Substitute t = 5

✅ Correct Answer Explanation

Integrating: \(I(t)=150-t^{2}\) At t = 5: \(I(5)=150-25\) Thus: \(I(5)=125\) Therefore, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect substitution.
Option B → Arithmetic mistake.
Option C → Miscalculated square value.

🎯 Strategy

Used

Substitution

Application:

Substitute the given value directly into the function.

Final Logic:

Poverty index at t = 5 equals 125.

🧠 Memory Trick

"150 − 25 = 125"

📚 Class XII – Mathematics Part I – Chapter: Differential Equations – Topic: Integration of Differential Equations –

16 Why are index numbers increasingly used in the social sphere?

A To quantify complex composite measures like prosperity

B To perfectly calculate absolute global warming rates

C To define vector derivatives in advanced calculus

D To find the exact atomic weight of commodities

💡 Short Explanation

Index numbers measure social conditions Prosperity and poverty are composite concepts Used in socio-economic analysis

✅ Correct Answer Explanation

Index numbers help quantify broad socio-economic measures such as prosperity, poverty, and standards of living. These are composite measures that cannot be expressed directly through a single raw value. Thus, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Global warming calculations use scientific climate models.
Option C → Vector derivatives are unrelated.
Option D → Atomic weights are scientific constants.

🎯 Strategy

Used

Odd One Out

Application:

Identify the socially relevant application.

Final Logic:

Only prosperity measurement belongs to social index usage.

🧠 Memory Trick

"Social Index = Prosperity Measure"

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

17 Using the method of least squares for 6 observations (n = 6), given:
\(\sum Y=35.6, \sum XY=9.2, \sum X^{2}=70\)
Find the value of constant a in the trend line:

A 0.13

B 5.93

C 6.10

D 7.12

💡 Short Explanation

Use least squares formula Divide total Y by n Obtain intercept a

✅ Correct Answer Explanation

The formula is: \(a=\frac{\sum Y}{n}\) Substituting: \(a=\frac{35.6}{6}\) Thus: \(a\approx 5.93\) Therefore, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect division.
Option C → Overestimation.
Option D → Arithmetic error.

🎯 Strategy

Used

Substitution

Application:

Apply the direct least squares formula.

Final Logic:

Intercept value equals 5.93.

🧠 Memory Trick

"a = Total Y ÷ n"

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

18 Given:
\(\vec{Q_{0}}=100\hat{i}+50\hat{j}\)
\(\vec{P_{n}}=3.8\hat{i}+45\hat{j}\)
The dot product
\(\vec{P_{n}}⋅\vec{Q_{0}}\)
gives:

A 2500

B 2600

C 2630

D 2730

💡 Short Explanation

Multiply corresponding components Add the products Dot product gives scalar value

✅ Correct Answer Explanation

Dot product: \(\left(3.8)(100)+(45)(50\right)\) Calculating: \(380+2250=2630\) Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Ignores one component.
Option B → Arithmetic error.
Option D → Overcalculated value.

🎯 Strategy

Used

Substitution

Application:

Apply the vector dot product formula.

Final Logic:

Scalar result equals 2630.

🧠 Memory Trick

"Dot Product = Multiply & Add"

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

19 Policy makers construct 4 index numbers using the following methods:
• Simple Aggregate
• Laspeyres
• Paasche
• Fisher
If one index is selected at random, what is the probability that it satisfies the time-reversal test?

A 1.00

B 0.75

C 0.50

D 0.25

💡 Short Explanation

Only Fisher satisfies the test Total methods = 4 Probability = favorable/total

✅ Correct Answer Explanation

Among the four methods listed, only Fisher's Ideal Index satisfies the time-reversal test. Thus: \(P=\frac{1}{4}=0.25\) Therefore, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Not all methods satisfy the test.
Option B → More than one method does not qualify.
Option C → Exactly half do not satisfy it.

🎯 Strategy

Used

Elimination

Application:

Identify the only valid method.

Final Logic:

One successful case out of four.

🧠 Memory Trick

"Only Fisher Passes"

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

20 In decision making, moving averages smooth out fluctuations in data.
If the original data oscillates strictly between:
\(y=30\)
and
\(y=50\)
then the 3-year moving average trend line will:

A Always lie within the interval 30 ≤ y ≤ 50

B Exceed y = 50 at peaks

C Drop below y = 30 at troughs

D Remain exactly a horizontal line at y = 40

💡 Short Explanation

Moving averages smooth data Average stays within bounds Extremes are reduced

✅ Correct Answer Explanation

A moving average is computed from nearby observations. If all observations lie between 30 and 50, then their averages must also remain within that interval. Thus, the trend line cannot exceed 50 or fall below 30. Therefore, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Averages cannot exceed the maximum value.
Option C → Averages cannot fall below the minimum value.
Option D → Moving averages are smoother, not necessarily constant.

🎯 Strategy

Used

Elimination

Application:

Remove impossible average behaviors.

Final Logic:

Average values remain bounded by original data limits.

🧠 Memory Trick

"Average Stays Within Limits"

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

CUET UG Applied Mathematics Booster Test 2 - Index Number Basics

Test Complete!

Answer Review