CBQ Practice›Class 12 Applied Mathematics

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Class 12 Applied Mathematics
CBQ Practice

Competency Based Questions · 6 chapters · 12 CBQ sets

Question types:Case StudySource BasedAssertion–Reason

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Ch 1

Numbers, Matrices and Quantitative Aptitude

2 sets

CBQ 1Case StudyMixture and Alligation in a Tea Factory4 marks

Read the passage

A tea factory blends two varieties of tea — Variety A costs ₹120 per kg and Variety B costs ₹80 per kg. The factory wants to create a blend that costs ₹96 per kg to sell at a profit. The manager uses the alligation rule: draw a cross with the mean price in the centre, dearer price on the top-left, cheaper price on the bottom-left. The differences from the mean give the ratio of cheaper to dearer variety. If the factory needs 60 kg of the blend, it must calculate exactly how many kilograms of each variety to use.

Using the alligation rule, what is the ratio of Variety A (₹120) to Variety B (₹80) in the ₹96 blend?

(A)1 : 3

(B)2 : 3

(D)1 : 2

How many kilograms of Variety B (cheaper) are needed for 60 kg of the blend?

(A)15 kg

(B)36 kg

(D)24 kg

If the factory uses the blend at ₹96/kg cost and sells at ₹120/kg, the profit percentage is:

(A)20%

(B)25%

(D)15%

Explain modulo arithmetic. If a factory operates 7 days a week and today is Day 1 (Monday), what day is Day 100?

CBQ 2Assertion–Reason1 mark

Assertion

Matrix multiplication is not commutative — AB ≠ BA in general.

Reason

The product AB requires the number of columns in A to equal the number of rows in B; reversing the order gives BA, which may have different dimensions or values, making commutativity impossible in general.

(A) Both A and R are true and R is the correct explanation of A

(B) Both A and R are true but R is not the correct explanation of A

(D) A is false but R is true

Ch 2

Calculus — Business Applications

2 sets

CBQ 1Case StudyProfit Maximisation in a Manufacturing Unit4 marks

Read the passage

A small manufacturer produces x units of a product daily. The total cost function is C(x) = 0.01x³ − 0.3x² + 10x + 200 (in ₹). The total revenue function is R(x) = 30x. The profit function is P(x) = R(x) − C(x). To maximise profit, the manufacturer must find where P'(x) = 0 and verify it is a maximum using P''(x) < 0. The marginal cost is the cost of producing one additional unit — given by C'(x). The marginal revenue is R'(x).

The marginal revenue R'(x) from the function R(x) = 30x is:

(A)30x

(B)30

(D)15x²

The profit function P(x) = R(x) − C(x) = 30x − (0.01x³ − 0.3x² + 10x + 200) simplifies to:

(A)P(x) = −0.01x³ + 0.3x² + 20x − 200

(B)P(x) = −0.01x³ + 0.3x² − 20x − 200

(C)P(x) = 0.01x³ − 0.3x² + 20x + 200

(D)P(x) = −0.01x³ − 0.3x² + 20x − 200

At profit maximum, which condition must hold?

(A)Marginal cost = 0

(B)Marginal revenue = 0

(C)Marginal cost = Marginal revenue

(D)P''(x) > 0

Explain how second derivative test is used to determine maximum profit in business calculus.

CBQ 2Assertion–Reason1 mark

Assertion

In economics, the demand curve slopes downward.

Reason

As price increases, the quantity demanded decreases — this inverse relationship between price and demand is represented mathematically by a negative first derivative of the demand function with respect to price.

(A) Both A and R are true and R is the correct explanation of A

(B) Both A and R are true but R is not the correct explanation of A

(D) A is false but R is true

Ch 3

Index Numbers and Time Series

2 sets

CBQ 1Case StudyPrice Index Comparison for a Household Survey4 marks

Read the passage

An economist tracks prices of four essential commodities for 2020 (base year) and 2024 (current year). Commodity prices (per unit): Rice — ₹40 (2020), ₹60 (2024); Dal — ₹100 (2020), ₹150 (2024); Oil — ₹120 (2020), ₹200 (2024); Sugar — ₹40 (2020), ₹50 (2024). The quantities consumed (base year): Rice 10 kg, Dal 4 kg, Oil 5 litres, Sugar 3 kg. Laspeyre's Price Index uses base year quantities; Paasche's Price Index uses current year quantities. Fisher's Ideal Index = √(Laspeyre × Paasche).

To calculate Laspeyre's Price Index, we use the formula:

(A)Σ(p₁q₁) / Σ(p₀q₁) × 100

(B)Σ(p₁q₀) / Σ(p₀q₀) × 100

(C)Σ(p₀q₀) / Σ(p₁q₀) × 100

(D)√[Σ(p₁q₀)/Σ(p₀q₀) × Σ(p₁q₁)/Σ(p₀q₁)] × 100

Fisher's Ideal Index is called 'ideal' because:

(A)It uses only current year prices

(B)It is the geometric mean of Laspeyre's and Paasche's indices, satisfying both time reversal and factor reversal tests

(C)It uses only base year quantities

(D)It gives the lowest value among all price indices

If Laspeyre's Index = 150 and Paasche's Index = 148, Fisher's Ideal Index is approximately:

(A)149

(B)298

(D)148.5

Distinguish between secular trend and seasonal variation in time series analysis. Give one example of each.

CBQ 2Assertion–Reason1 mark

Assertion

Consumer Price Index (CPI) is more useful for wage policy than Wholesale Price Index (WPI).

Reason

CPI measures the change in prices of goods and services consumed by households, directly reflecting the cost of living for workers, whereas WPI measures prices at the production/wholesale level before reaching consumers.

(A) Both A and R are true and R is the correct explanation of A

(B) Both A and R are true but R is not the correct explanation of A

(D) A is false but R is true

Ch 4

Financial Mathematics

2 sets

CBQ 1Case StudyHome Loan EMI Calculation4 marks

Read the passage

Mr. Sharma takes a home loan of ₹20,00,000 from a bank at 8% per annum interest, to be repaid in 20 years (240 monthly instalments). The EMI (Equated Monthly Instalment) formula is: EMI = P × r(1+r)ⁿ / [(1+r)ⁿ − 1], where P = principal, r = monthly interest rate = annual rate/12, n = number of instalments. Monthly rate r = 8/1200 = 0.00667. The bank also offers a flat rate option at 6% flat per year, where total interest = P × rate × years. Mr. Sharma must choose between the two.

In the EMI formula, the monthly interest rate for 8% per annum is:

(A)8%

(B)0.8%

(D)0.08%

Under the flat rate method at 6% for ₹20,00,000 over 20 years, total interest paid is:

(A)₹12,00,000

(B)₹20,00,000

(C)₹24,00,000

(D)₹6,00,000

The reducing balance method (EMI) is preferred over flat rate because:

(A)Total interest paid is higher under reducing balance

(B)Under reducing balance, interest is charged only on the outstanding principal, which decreases each month as EMI payments are made

(C)EMI amount changes every month in reducing balance

(D)Flat rate always gives lower total repayment

Explain the concept of annuity and distinguish between ordinary annuity and annuity due.

CBQ 2Assertion–Reason1 mark

Assertion

Compound interest grows faster than simple interest over the same period.

Reason

In compound interest, the interest earned in each period is added to the principal and becomes the base for calculating interest in the next period — so the effective principal grows over time, unlike simple interest where the base remains constant.

(A) Both A and R are true and R is the correct explanation of A

(B) Both A and R are true but R is not the correct explanation of A

(D) A is false but R is true

Ch 5

Probability Distributions and Statistics

2 sets

CBQ 1Case StudyQuality Control in a Light Bulb Factory4 marks

Read the passage

A factory produces light bulbs and knows from experience that 10% of bulbs are defective. A quality inspector randomly selects 5 bulbs from a batch. The number of defective bulbs follows a Binomial Distribution with n = 5 and p = 0.10 (probability of defect). The probability of exactly k defectives is P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ. The factory also tracks the daily number of machine breakdowns, which follows a Poisson Distribution with mean λ = 2 breakdowns per day.

The mean (expected value) of the Binomial Distribution for defective bulbs is:

(A)5

(B)0.5

(D)2.5

The variance of this Binomial Distribution (n = 5, p = 0.10) is:

(A)0.5

(B)0.45

(D)0.25

For machine breakdowns following Poisson with λ = 2, the probability of exactly 0 breakdowns in a day is e⁻² ≈ 0.135. This means:

(A)Breakdowns never occur

(B)There is a 13.5% chance of no breakdowns in a day, meaning on average the factory has breakdown-free days about 1 in 7 days

(C)The mean number of breakdowns is 0.135

(D)λ = 0.135 for this distribution

Distinguish between Binomial and Poisson distributions. When is each appropriate?

CBQ 2Assertion–Reason1 mark

Assertion

Standard deviation is preferred over mean deviation as a measure of dispersion.

Reason

Standard deviation uses the squares of deviations from the mean, making it algebraically tractable and consistent — it supports further statistical calculations like correlation and regression, whereas mean deviation uses absolute values which are mathematically less convenient.

(A) Both A and R are true and R is the correct explanation of A

(B) Both A and R are true but R is not the correct explanation of A

(D) A is false but R is true

Ch 6

Linear Programming

2 sets

CBQ 1Case StudyProduction Planning for a Furniture Company4 marks

Read the passage

A furniture company makes chairs and tables. Each chair requires 2 hours of carpentry and 1 hour of painting. Each table requires 3 hours of carpentry and 2 hours of painting. Available weekly: 120 hours carpentry, 70 hours painting. Profit: ₹500 per chair, ₹800 per table. Let x = chairs produced, y = tables produced. Objective: Maximise Z = 500x + 800y. Constraints: 2x + 3y ≤ 120 (carpentry), x + 2y ≤ 70 (painting), x ≥ 0, y ≥ 0. The feasible region is bounded by these constraints. The optimal solution occurs at a corner point of the feasible region.

The objective function for this Linear Programming Problem is:

(A)Z = 2x + 3y

(B)Z = 500x + 800y

(C)Z = x + 2y

(D)Z = 500y + 800x

Which theorem guarantees that the optimal solution to a bounded LPP lies at a corner point of the feasible region?

(A)The Fundamental Theorem of Arithmetic

(B)The Corner Point (Extreme Point) Theorem for Linear Programming

(C)The Intermediate Value Theorem

(D)The Simplex Optimality Theorem

If the corner points of the feasible region are (0,0), (60,0), (30,20), (0,35) with Z = 500x + 800y, at which corner is Z maximum?

(A)(0, 35) — Z = 28,000

(B)(60, 0) — Z = 30,000

(C)(30, 20) — Z = 31,000

(D)(0, 0) — Z = 0

What is a 'feasible region' in Linear Programming? What does it mean if the feasible region is unbounded?

CBQ 2Assertion–Reason1 mark

Assertion

Linear Programming problems always have a unique optimal solution.

Reason

If the objective function is parallel to one of the constraint boundary lines forming the feasible region, every point on that segment of the boundary is optimal, giving infinitely many optimal solutions.

(A) Both A and R are true and R is the correct explanation of A

(B) Both A and R are true but R is not the correct explanation of A

(D) A is false but R is true

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