Question 1

Addition rule for any A and B is:

Accepted Answer

General addition rule subtracts intersection.

Question 2

If interest is compounded quarterly, number of periods in 2 years is:

Accepted Answer

Quarterly: 4 periods/year; 2 years = 8.

Question 3

A matrix of order 2×3 has:

Accepted Answer

m×n means m rows and n columns.

Question 4

Two linear equations in two variables generally have:

Accepted Answer

If not parallel/coincident, they intersect at one point.

Question 5

If r = −1, correlation is:

Accepted Answer

r = −1 indicates perfect negative correlation.

Question 6

If an article is sold at 20% profit on cost, then SP equals:

Accepted Answer

Profit 20% on CP means SP = CP × (1 + 20/100) = 1.20×CP.

Question 7

A value rises from 500 to 650. The percentage increase is:

Accepted Answer

Increase = 150; % increase = (150/500)×100 = 30%.

Question 8

Addition of matrices is possible only when:

Accepted Answer

Same order is required to add corresponding elements.

Question 9

In an ordinary annuity, payments are made at:

Accepted Answer

Ordinary annuity pays at period end.

Question 10

Non-negativity restriction means:

Accepted Answer

Decision variables represent quantities, so x≥0, y≥0.

Question 11

Conditional probability P(B|A) is:

Accepted Answer

Definition: P(B|A)=P(A∩B)/P(A).

Question 12

Risk refers to a situation where:

Accepted Answer

Risk means probabilities can be assigned.

Question 13

Linear programming is used to:

Accepted Answer

LPP optimizes an objective with resource constraints.

Question 14

In LPP, optimal solution (if exists) occurs at:

Accepted Answer

Corner point theorem: optimum occurs at a vertex.

Question 15

For A = [[a,b],[c,d]], |A| equals:

Accepted Answer

2×2 determinant is ad−bc.

Question 16

Matrix multiplication is generally:

Accepted Answer

In general AB ≠ BA.

Question 17

The feasible region is:

Accepted Answer

Feasible region is intersection of all constraint half-planes.

Question 18

Arithmetic mean is:

Accepted Answer

Mean = Σx/n.

Question 19

Leontief solution for output is:

Accepted Answer

From (I−A)X=D, X=(I−A)^{-1}D if inverse exists.

Question 20

A ratio compares:

Accepted Answer

Ratio is a comparison of two quantities of the same kind (same unit).

Question 21

Define standard deviation.

Accepted Answer

Standard deviation is the square root of variance and measures spread around mean.

Question 22

What is correlation?

Accepted Answer

Correlation measures degree and direction of relationship between two variables.

Question 23

State formula of CV.

Accepted Answer

CV = (SD/Mean) × 100.

Question 24

What is percentage?

Accepted Answer

Percentage means “per hundred” and is written as x% = x/100.

Question 25

Define a linear equation.

Accepted Answer

A linear equation is an equation of first degree (highest power of variable is 1).

Question 26

What is feasible region?

Accepted Answer

Feasible region is the set of all points satisfying all constraints including non-negativity.

Question 27

State break-even condition.

Accepted Answer

Break-even occurs when total revenue equals total cost (TR = TC).

Question 28

What is EMI?

Accepted Answer

EMI is the equated monthly instalment paid every month to repay a loan with interest.

Question 29

State the condition of proportion.

Accepted Answer

If a:b = c:d, then a×d = b×c.

Question 30

Give the quadratic formula.

Accepted Answer

x = (−b ± √(b^2 − 4ac)) / (2a).

Question 31

Compute EMV: High demand p=0.6 payoff 50,000; Low demand p=0.4 payoff 10,000.

Accepted Answer

EMV = 0.6×50,000 + 0.4×10,000 = 30,000 + 4,000 = 34,000.
So expected payoff = ₹34,000.

Question 32

Profit is 20% on selling price. Find profit percent on cost price.

Accepted Answer

Let selling price (SP) = 100.
Profit = 20% of SP = 20.
So cost price (CP) = SP − Profit = 100 − 20 = 80.

Profit % on CP = (Profit/CP) × 100
= (20/80) × 100
= **25%**.

Therefore, 20% profit on selling price is equivalent to **25% profit on cost price**.

Question 33

Write the matrix form of system: 2x+y=11 and x+y=7.

Accepted Answer

Coefficient matrix A=[[2,1],[1,1]], variable matrix X=[[x],[y]], constant matrix B=[[11],[7]].
So AX = B.

Question 34

Formulate an LPP from a word problem: define variables, objective and constraints (outline).

Accepted Answer

To formulate: define decision variables (x,y), write objective function (Max/Min Z), write constraints from resources (≤/≥/=) and add x≥0, y≥0. This converts the word problem into a mathematical LPP.

Question 35

Explain difference between independent and dependent events with one example each.

Accepted Answer

Independent: one event does not change probability of the other. Example: head on first toss and head on second toss (different tosses).
Dependent: probability changes due to earlier event. Example: drawing two cards without replacement (first draw affects second).

Question 36

Explain present value and future value. Show their relationship with formulas and one numerical example.

Accepted Answer

Present value (PV) is today’s value of a future amount found by discounting. For annual compounding: PV = FV / (1 + R/100)^T.

Future value (FV) is the value of a present sum after earning interest: FV = P (1 + R/100)^T.

Relationship: FV = PV × (1 + R/100)^T and PV = FV / (1 + R/100)^T.

Example: PV of ₹20,000 due after 2 years at 10% is PV = 20,000/(1.10)^2 = 20,000/1.21 ≈ ₹16,529. So ₹16,529 today becomes ₹20,000 after 2 years at 10%.

Question 37

Define ratio and proportion. Explain their importance in business calculations with examples.

Accepted Answer

### Ratio
A **ratio** is a comparison of two like quantities expressed as a:b.
Example: Profit ₹20,000 and sales ₹1,00,000 ⇒ profit:sales = 20,000:1,00,000 = 1:5.

### Proportion
A **proportion** is an equality of two ratios.
If a:b = c:d, then **a×d = b×c**.

### Importance in business calculations (any 4–5 points)
- **Allocation and distribution:** Divide salary/bonus/budget among departments or partners in a fixed ratio.
- **Costing and production planning:** Maintain raw material mix in a required ratio (e.g., 2:3 blend).
- **Financial and performance analysis:** Ratios like current ratio, gross profit ratio, debt–equity help decision-making.
- **Sales targets and scaling:** Use proportion to scale targets when working days/sales force/output changes.
- **Trade calculations:** Many discount, markup, commission and tax problems use proportional reasoning.

**Conclusion:** Ratio and proportion give a simple, accurate way to compare quantities and make business decisions using numbers.

Question 38

Explain demand and supply functions and show how equilibrium is determined with a numerical example.

Accepted Answer

Demand function shows how quantity demanded depends on price, usually with negative relation (e.g., Qd = a − bP). Supply function shows how quantity supplied depends on price, usually with positive relation (e.g., Qs = c + dP).

Equilibrium is where Qd = Qs.
Example: Qd = 100 − 2P, Qs = 20 + 3P.
Set equal: 100 − 2P = 20 + 3P ⇒ 80 = 5P ⇒ P = 16.
Then Q = 100 − 2(16) = 68.

So equilibrium price is 16 and equilibrium quantity is 68. This point balances market demand and supply.

Question 39

Explain expected value (EMV) method for decision making under risk with an example.

Accepted Answer

Expected Monetary Value (EMV) is the probability-weighted average payoff of a decision.

Steps:
1) List possible states of nature (e.g., high/low demand) with probabilities.
2) List payoffs for each decision under each state.
3) EMV(decision) = Σ(probability × payoff).
4) Choose decision with highest EMV (risk-neutral).

Example:
Prob(high)=0.6, Prob(low)=0.4
Decision 1 payoff: 50,000 (high), 10,000 (low)
Decision 2 payoff: 40,000 (high), 20,000 (low)
EMV1 = 0.6×50,000 + 0.4×10,000 = 34,000
EMV2 = 0.6×40,000 + 0.4×20,000 = 32,000
So Decision 1 is preferred by EMV.

Question 40

Explain standard deviation and show its calculation with an example.

Accepted Answer

Standard deviation (SD) is an important measure of dispersion that shows how much observations deviate from the mean.

Steps for ungrouped data:
1) Find mean (x̄).
2) Find deviations (x − x̄).
3) Square deviations and sum them: Σ(x − x̄)^2.
4) Variance = Σ(x − x̄)^2 / n.
5) SD = √variance.

Example: Data 2, 4, 6.
Mean = 4. Deviations: −2,0,+2. Squares: 4,0,4. Sum = 8.
Variance = 8/3 = 2.667.
SD = √2.667 ≈ 1.633.

Interpretation: Higher SD means higher variability and less stability/riskier series.

Business Mathematics — Question Paper

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