UG Applied Mathematics Booster Test 3 - Time Series Analysis -

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UG Applied Mathematics Booster Test 3 - Time Series Analysis

⏱️ 30 Minutes

❓ 20 Questions

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

UG Applied Mathematics Booster Test 3 - Time Series Analysis

1 / 20

QUESTION 1 OF 20

Sequential data over a period follows a differential growth trend
\(\frac{dy}{dt}=2t+1\)
If the starting value is \(y(0)=5\), evaluate the exact forecasted data value \(y(3)\)using definite integration.

Answer Review

1 Sequential data over a period follows a differential growth trend
\(\frac{dy}{dt}=2t+1\)
If the starting value is \(y(0)=5\), evaluate the exact forecasted data value \(y(3)\)using definite integration.

A 14

B 15

C 17

D 19

💡 Short Explanation

Integrate the differential equation Apply the initial condition Substitute the required time value

✅ Correct Answer Explanation

Given: \(\frac{dy}{dt}=2t+1\) Integrating both sides: \(y=\int (2t+1)dt=t^{2}+t+C\) Using \(y(0)=5\): \(0+0+C=5C=5\) Thus, \(y=t^{2}+t+5\) At \(t=3\): \(y(3)=9+3+5=17\) Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Misses one component during substitution.
Option B → Incorrect constant handling.
Option D → Arithmetic error after integration.

🎯 Strategy

Used

Substitution

Application:

Integrate first and then substitute initial values.

Final Logic:

Differential equation + initial condition uniquely determines the function.

🧠 Memory Trick

"Integrate → Add Constant → Substitute"

📚 Class XII – Mathematics – Chapter: Differential Equations – Topic: Solution Using Integration –

2 Time-based tracking coordinates for two variables form vectors
\(\vec{P}=2\hat{i}+\hat{j},\vec{Q}=3\hat{i}+x\hat{j}\)
If these dataset vectors are completely orthogonal (independent direction), what must be the value of \(x\)?
\(\vec{P}⋅\vec{Q}=0\Rightarrow (2)(3)+(1)(x)=0\)

A 6

B -3

C 0

D -6

💡 Short Explanation

Orthogonal vectors have zero dot product Apply dot product formula Solve linear equation

✅ Correct Answer Explanation

For perpendicular vectors: \(\vec{P}⋅\vec{Q}=0\) \((2)(3)+(1)(x)=0\Rightarrow 6+x=0\Rightarrow x=-6\) Hence, the correct value is \(-6\). Therefore, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Gives dot product \(12\neq 0\).
Option B → Gives dot product \(3\neq 0\).
Option C → Gives dot product \(6\neq 0\).

🎯 Strategy

Used

Substitution

Application:

Use orthogonality condition directly.

Final Logic:

Perpendicular vectors always satisfy dot product zero.

🧠 Memory Trick

"Perpendicular ⇒ Dot Product Zero"

📚 Class XI – Mathematics – Chapter: Vector Algebra – Topic: Orthogonal Vectors –

3 The data mapping of a parabolic time series curve
\(y=4-x^{2}\)
and the x-axis outlines a bounded domain. Find the total area representing cumulative metrics over this domain.

A \(\frac{32}{3}\)

B \(\frac{16}{3}\)

C \(\frac{64}{3}\)

D \(\frac{8}{3}\)

💡 Short Explanation

Find intersection with x-axis Integrate the curve Compute bounded area

✅ Correct Answer Explanation

Curve: \(y=4-x^{2}\) Set \(y=0\): \(4-x^{2}=0\Rightarrow x=\pm 2\) Area: \(\int_{-2}^{2}\,(4-x^{2}) dx\) \(\int_{-2}^{2}\,(4-x^{2})dx={\left[4x,\ \frac{x^{3}}{3}\right]}_{-2}^{2}=\frac{32}{3}\) Thus, total area equals \(\frac{32}{3}\).

❓ Why Other Options Are Incorrect

Option B → Half-area calculation mistake.
Option C → Doubled the correct result.
Option D → Incorrect integration.

🎯 Strategy

Used

Substitution

Application:

Determine limits from intercepts and integrate.

Final Logic:

Area under curve equals definite integral over bounded interval.

🧠 Memory Trick

"Area = Integral Between Roots"

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

4 A cross-sectional survey of 100 industrial firms reveals 40 exhibit high growth. If a researcher randomly selects 2 distinct firms without replacement, what is the probability that both display high growth?

A \(\frac{16}{100}\)

B \(\frac{26}{165}\)

C \(\frac{1}{4}\)

D \(\frac{39}{165}\)

💡 Short Explanation

Use probability without replacement Multiply sequential probabilities Simplify fraction

✅ Correct Answer Explanation

Probability first firm has high growth: \(\frac{40}{100}\) Probability second firm also has high growth: \(\frac{39}{99}\) Thus, \(\frac{40}{100}\times \frac{39}{99}\) \(\frac{40}{100}\times \frac{39}{99}=\frac{26}{165}\) Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect simplification.
Option C → Assumes independent equal probabilities.
Option D → Misses first-step probability adjustment.

🎯 Strategy

Used

Substitution

Application:

Apply multiplication rule for dependent events.

Final Logic:

Without replacement changes the denominator.

🧠 Memory Trick

"No Replacement → Change Denominator"

📚 Class XI – Mathematics – Chapter: Probability – Topic: Dependent Events –

5 A massive pooled dataset is pre-processed into raw totals over consecutive periods. If the combined data totals for the first three periods are
120, 150, 180,
what is the 3-period moving average of these aggregated totals?

A 120

B 180

C 150

D 450

💡 Short Explanation

Add the values Divide by number of periods Moving average smooths fluctuations

✅ Correct Answer Explanation

3-period moving average: \(\frac{120+150+180}{3}\) \(\frac{120+150+180}{3}=150\) Thus, the moving average equals 150.

❓ Why Other Options Are Incorrect

Option A → Uses only first observation.
Option B → Uses only last observation.
Option D → Gives moving total, not average.

🎯 Strategy

Used

Substitution

Application:

Apply moving average formula directly.

Final Logic:

Moving average = total ÷ number of periods.

🧠 Memory Trick

"Average = Sum ÷ Count"

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

6 Arrange the dataset combinations logically in order of their typical structural and dimensional complexity (lowest to highest parameters):
1. Multivariate time series (multiple variables over time)
2. Univariate time series (one variable over time)
3. Pooled data (multiple variables, multiple separate cross-sections over time)

A 3, 2, 1

B 2, 3, 1

C 1, 2, 3

D 2, 1, 3

💡 Short Explanation

Univariate has one variable Multivariate has many variables Pooled combines cross-section and time

✅ Correct Answer Explanation

Complexity increases as follows: • Univariate → one variable • Multivariate → several variables • Pooled data → several variables across many entities and time periods Thus: \(2\rightarrow 1\rightarrow 3\) Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Starts with most complex dataset.
Option B → Places pooled data too early.
Option C → Incorrect order of complexity.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Compare dimensional structure of datasets.

Final Logic:

More variables and dimensions increase complexity.

🧠 Memory Trick

"Uni → Multi → Pooled"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Types of Data – 262

7 Match the following regarding quantitative forecasting:

List I	List II
1. Future prediction base	a. Extrapolating consistent past trends
2. Cause-and-effect relationship	b. Dependence on interacting systemic variables
3. Moving average application	c. Technique applied for smoothing random variations
4. Historical availability	d. Baseline requirement for executing quantitative methods

A 1–a, 2–b, 3–c, 4–d

B 1–d, 2–c, 3–b, 4–a

C 1–b, 2–a, 3–d, 4–c

D 1–c, 2–d, 3–a, 4–b

💡 Short Explanation

Forecasting uses past trends Moving averages smooth variations Quantitative methods need data

✅ Correct Answer Explanation

Correct matches: • Future prediction base → extrapolation of trends • Cause-and-effect → interacting variables • Moving average → smoothing technique • Historical availability → necessary for quantitative analysis Thus, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrectly swaps core definitions.
Option C → Mismatched forecasting concepts.
Option D → Incorrect logical associations.

🎯 Strategy

Used

Option Grouping

Application:

Match standard forecasting terminology.

Final Logic:

Each term has a textbook-defined purpose.

🧠 Memory Trick

"Trend → Predict Future"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Forecasting Techniques –

8 Assertion (A): Qualitative forecasting methods are strictly preferred when numerical history is widely abundant and highly structured.
Reason (R): Qualitative forecasting fundamentally relies on expert judgment and analytical opinion when historical data cannot be accurately quantified.

A Both A and R are false

B A is false, R is true

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

D A is true, R is false

💡 Short Explanation

Qualitative methods use judgment Quantitative methods use numerical history Assertion contradicts forecasting principles

✅ Correct Answer Explanation

Assertion is false because when structured numerical data exists, quantitative methods are preferred. Reason is true because qualitative forecasting depends on expert opinion where reliable data is absent. Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Reason is true.
Option C → Assertion is incorrect.
Option D → Assertion is not true.

🎯 Strategy

Used

Elimination

Application:

Check truth values separately.

Final Logic:

Quantitative methods dominate when data is available.

🧠 Memory Trick

"Numbers Present → Quantitative"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Forecasting Methods –

9 The Cartesian plot of a specific univariate time series displays time \(t\) on the x-axis and variable \(y\) on the y-axis. If the points
\(\left(1,3),(2,5),(3,7\right)\)
are collinear, what is the geometric slope representing the constant rate of change?

A 1

B 3

C 2

D 4

💡 Short Explanation

Slope measures rate of change Use two-point formula Collinear points share same slope

✅ Correct Answer Explanation

Slope formula: \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) Using \(\left(1,\ 3\right)\)and \(\left(2,\ 5\right)\): \(m=\frac{5-3}{2-1}=2\) -10-8-6-4-2246810-10-5510A(1, 3)B(2, 5)m = 2.00 Thus, slope equals 2.

❓ Why Other Options Are Incorrect

Option A → Underestimates change.
Option B → Incorrect difference calculation.
Option D → Overestimates slope.

🎯 Strategy

Used

Substitution

Application:

Apply slope formula directly.

Final Logic:

Constant rise per unit time equals 2.

🧠 Memory Trick

"Slope = Rise ÷ Run"

📚 Class XI – Mathematics – Chapter: Straight Lines – Topic: Slope of Line –

10 Which characteristic(s) correctly distinguish a multivariate series?
I. Simultaneously tracking independent macro-variables like population and employment
II. Utilizing a multi-dimensional array of variables over corresponding time markers
III. Purposely tracking only a single isolated metric to avoid statistical complexity

A I and III only

B II and III only

C III only

D I and II only

💡 Short Explanation

Multivariate means multiple variables Variables are tracked together over time Single metric implies univariate series

✅ Correct Answer Explanation

Statement I is correct because multivariate series tracks multiple variables. Statement II is also correct because it involves multi-dimensional observations. Statement III is incorrect because single-variable tracking defines a univariate series. Therefore, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Includes incorrect statement III.
Option B → Statement III is false.
Option C → Ignores valid multivariate definitions.

🎯 Strategy

Used

Elimination

Application:

Remove options containing univariate description.

Final Logic:

Multivariate requires more than one variable.

🧠 Memory Trick

"Multi = Many Variables"

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

11 Identify the incorrect statement regarding the secular trend component:

A It is primarily driven by sudden, unanticipated events like local floods or labor strikes

B It represents mathematically smooth, regular, and long-term variations

C The fundamental trend movement can be classified as either linear or non-linear in nature

D The method of least squares is a widely used statistical approach to fit these trend lines

💡 Short Explanation

Secular trend represents long-term movement Sudden events belong to irregular variation Least squares helps fit trend lines

✅ Correct Answer Explanation

Secular trend refers to the smooth and long-term movement in a time series over many years. Option A is incorrect because sudden events such as floods, strikes, wars, or calamities are examples of irregular variation, not secular trend. Option B correctly defines secular trend as smooth and regular long-term movement. Option C is correct because trend lines may be linear or nonlinear. Option D is also correct because the method of least squares is commonly used for trend fitting. Hence, Option A is the incorrect statement.

❓ Why Other Options Are Incorrect

Option B → Correctly explains the nature of secular trend.
Option C → Trend movement can indeed be linear or nonlinear.
Option D → Least squares is a standard statistical trend-fitting method.

🎯 Strategy

Used

Odd One Out

Application:

Identify which statement belongs to irregular variation instead of trend analysis.

Final Logic:

Sudden unpredictable events are irregular components, not secular trends.

🧠 Memory Trick

"Trend = Long-Term, Irregular = Sudden"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Secular Trend –

12 Seasonal variation frequently occurs within one-year cyclic periods. If a retail index spikes predictably by 20% every winter (Quarter 4), and the base index in Quarter 3 is exactly 150, what is the expected numerical index value in Quarter 4?

A 170

B 180

C 160

D 190

💡 Short Explanation

Seasonal increase is percentage based Increase = 20% of 150 Add increase to base value

✅ Correct Answer Explanation

Seasonal increase: \(20\% of 150\) \(150+\frac{20}{100}\times 150=150+30=180\) Thus, expected Quarter 4 index value equals 180. Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Adds only 20 instead of 20%.
Option C → Incorrect percentage calculation.
Option D → Overestimates seasonal rise.

🎯 Strategy

Used

Substitution

Application:

Apply percentage increase formula directly.

Final Logic:

Seasonal rise = original value + percentage increment.

🧠 Memory Trick

"20% of 150 = 30"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Seasonal Variations –

13 A cyclical variation completes one full sequence of predictable oscillation known strictly as a "cycle". According to time series categorization, what duration inherently separates cyclical from seasonal variation?

A Cyclical takes less than a month

B Cyclical takes exactly one year

C Cyclical takes more than a year

D Cyclical takes less than a year

💡 Short Explanation

Seasonal variation repeats within a year Cyclical variation spans many years Business cycles are long-term oscillations

✅ Correct Answer Explanation

Seasonal variations occur repeatedly within one year due to weather, customs, or festivals. Cyclical variations are long-term fluctuations associated with business cycles and usually extend over periods greater than one year. Thus, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Cyclical variations are not short monthly movements.
Option B → Exactly one year describes seasonal variation.
Option D → Less than one year also corresponds to seasonal movement.

🎯 Strategy

Used

Elimination

Application:

Separate yearly seasonal effects from long-term cycles.

Final Logic:

Cyclical variations extend beyond one year.

🧠 Memory Trick

"Seasonal = Within Year, Cyclical = Many Years"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Cyclical Variations –

📄 Passage

"Method of least squares is a technique for finding the linear equation
\(y=a+bx\)
which best fits a given set of observations. According to the principle of least squares, the best fitting equation is obtained by minimizing the sum of squares of differences between actual and calculated values. By taking the middle time period as the central origin, it mathematically reduces the sum of the time variable \(X\) to exactly zero. Consequently, the equations simplify significantly to:
\(\sum Y=naand\sum XY=b\sum X^{2}.\)
"

Based on the passage, what specific parameter is mathematically minimized to ascertain the line of best fit?

A The direct sum of the actual variables

B The cumulative sum of the recorded times

C The moving average variances

D The sum of squares of differences

💡 Short Explanation

Least squares minimizes error Deviations are squared Best-fit line has minimum squared error

✅ Correct Answer Explanation

The principle of least squares states that the best fitting line minimizes the sum of squares of deviations between actual and estimated values. \(\sum (Y-\hat{Y})^{2} is minimized\) Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Actual values themselves are not minimized.
Option B → Time totals are unrelated to least squares minimization.
Option C → Moving average variance is not the least squares criterion.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Match the passage definition directly with least squares theory.

Final Logic:

Least squares always minimizes squared deviations.

🧠 Memory Trick

"Least Squares = Least Error Squares"

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

📄 Passage

According to the passage, explicitly taking the middle time period as the data origin mathematically achieves which critical computational simplification?

A ∑X=0

B ∑Y=0

C a=0

D b=0

💡 Short Explanation

Central origin simplifies calculations Sum of deviations becomes zero Trend equations become easier

✅ Correct Answer Explanation

When the middle time period is taken as origin: \(X=x-A\) the positive and negative deviations balance out. Thus, \(\sum X=0\) This simplifies normal equations in least squares. Therefore, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Sum of Y-values does not become zero.
Option C → Constant \(a\) is generally nonzero.
Option D → Slope \(b\) is not automatically zero.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Use direct statement from passage.

Final Logic:

Central origin balances positive and negative X-values.

🧠 Memory Trick

"Middle Origin ⇒ ΣX = 0"

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

16 If the method of least squares analysis yields the final trend equation
\(y=5.9+0.13x\)
calculate the definite integral
\(\int_{-1}^{1}\,(5.9+0.13x) dx\)
representing the total cumulative yield over this interval.

A 5.9

B 11.8

C 12.0

D 10.6

💡 Short Explanation

Integrate the trend equation Odd function term cancels Evaluate definite integral limits

✅ Correct Answer Explanation

Evaluate: \(\int_{-1}^{1}\,(5.9+0.13x) dx\) \(\int_{-1}^{1}\,(5.9+0.13x)dx={\left[5.9x,\ 0.065x^{2}\right]}_{-1}^{1}=11.8\) The \(0.13x\) term cancels over symmetric limits. Thus, the result equals 11.8.

❓ Why Other Options Are Incorrect

Option A → Gives only constant term.
Option C → Arithmetic approximation error.
Option D → Incorrect integration evaluation.

🎯 Strategy

Used

Substitution

Application:

Integrate term-by-term and apply limits.

Final Logic:

Symmetric odd terms cancel over \(\left[-1,1\right]\).

🧠 Memory Trick

"Odd Terms Cancel on Symmetric Limits"

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

17 In calculating a 4-year moving average, a dataset creates sequential 4-year totals. If the first recorded moving total is 1730 and the second moving total is 1762, what is the value of the first uncentered 4-year moving average?

A 440.5

B 437.75

C 432.5

D 443.0

💡 Short Explanation

Moving average = total ÷ periods Period count is 4 Use first moving total only

✅ Correct Answer Explanation

First uncentered 4-year moving average: \(\frac{1730}{4}\) \(\frac{1730}{4}=432.5\) Thus, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect division.
Option B → Uses second total instead.
Option D → Arithmetic mistake.

🎯 Strategy

Used

Substitution

Application:

Apply moving average formula directly.

Final Logic:

Moving average equals moving total divided by number of years.

🧠 Memory Trick

"4-Year Average = Total ÷ 4"

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

18 For analyzing odd-even periods, when applying an even-period (n) moving average, how is the final "centered average" mathematically derived from the initial moving averages?

A By summing four discrete 4-year moving averages

B By subtracting consecutive 4-year moving averages

C By dividing the raw data points directly by 2

D By averaging two consecutive n-year moving averages

💡 Short Explanation

Even-period averages fall between years Centering aligns averages properly Two adjacent averages are averaged again

✅ Correct Answer Explanation

For even-period moving averages, the calculated values lie between periods. To align them correctly, centered averages are computed by averaging two consecutive moving averages. Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Not the centering procedure.
Option B → Subtraction is unrelated.
Option C → Raw observations are not directly halved.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Match standard moving-average centering procedure.

Final Logic:

Even periods require centering via adjacent averages.

🧠 Memory Trick

"Even MA → Average the Averages"

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

19 To calculate the linear trend equation systematically for an even number of years (e.g., n=6), which operations are applied to avoid cumbersome decimal tracking?
I. The middle origin A is taken as the numerical average of the two central years
II. The shifted time X is computed as
\(X=\frac{x_{i}-A}{0.5}\)
to scale the intervals appropriately
III. The raw data is exclusively converted into a 3-year moving average prior to squares calculation

A I and II only

B II and III only

C I and III only

D All I, II, and III

💡 Short Explanation

Even years require central averaging Time scaling avoids decimals Moving averages are not compulsory

✅ Correct Answer Explanation

Statement I is correct because for even years, the origin is placed midway between the two central years. Statement II is also correct because scaling by 0.5 converts fractional deviations into integers. Statement III is incorrect because least squares can be directly applied without converting data into moving averages. Thus, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Statement III is false.
Option C → Omits correct scaling procedure.
Option D → Includes incorrect statement III.

🎯 Strategy

Used

Elimination

Application:

Remove options containing incorrect procedural steps.

Final Logic:

Moving averages are separate from least squares estimation.

🧠 Memory Trick

"Even Years → Mid-Origin + Scaling"

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

20 Using the previously computed best fit linear estimation
\(y=5.9+0.13x\)
which utilizes an origin \(A=1998.5\) and time scaling calculation
\(X=\frac{x_{i}-1998.5}{0.5}\)
compute the estimated trend target for the year 2002.

A 6.29

B 6.81

C 7.15

D 6.55

💡 Short Explanation

Compute scaled \(X\) value Substitute into trend equation Evaluate final estimate

✅ Correct Answer Explanation

First compute scaled value: \(X=\frac{2002-1998.5}{0.5}=7\) Substitute into trend equation: \(y=5.9+0.13(7)\) \(y=5.9+0.13(7)=5.9+0.91=6.81\) Thus, estimated trend value equals 6.81.

❓ Why Other Options Are Incorrect

Option A → Uses incorrect X-value.
Option C → Arithmetic overestimation.
Option D → Partial multiplication error.

🎯 Strategy

Used

Substitution

Application:

Compute transformed variable and substitute.

Final Logic:

Trend estimate comes directly from fitted equation.

🧠 Memory Trick

"Find X → Put in y = a + bx"

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

UG Applied Mathematics Booster Test 3 - Time Series Analysis

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