How to Use This Page
This page provides complete exercise solutions for Chapter: Mathematical Methods for Economics from the Samacheer Kalvi Class 11 Economics textbook.
Recommended approach: 1. Read the textbook chapter thoroughly and learn the concepts 2. Attempt every exercise question on your own first 3. Use this page to verify your answers and understand the reasoning 4. Focus on the step-by-step methods, not just the final answers 5. Review the Common Mistakes and Exam Checklist sections before your exam
Key Concepts (Quick Recap)
Functions in Economics
| Concept | Definition | Example |
|---|
| Function | A rule that assigns exactly one output (y) to each input (x) | Qd = 100 - 2P |
| Demand function | Relates quantity demanded to price | Qd = a - bP |
| Supply function | Relates quantity supplied to price | Qs = c + dP |
| Total Cost function | Total cost as a function of output | TC = 50 + 5Q |
| Total Revenue function | Total revenue as a function of quantity | TR = P x Q |
| Consumption function | Relates consumption to income | C = C0 + bY |
Equations and Their Types
| Type | Standard Form | Example |
|---|
| Linear equation | ax + b = 0 | 3x + 6 = 0 |
| Simultaneous equations | a1x + b1y = c1; a2x + b2y = c2 | 2x + y = 10; x - y = 2 |
| Quadratic equation | ax² + bx + c = 0 | Q² - 5Q + 6 = 0 |
Differentiation Rules
| Rule | Formula | Economic Use |
|---|
| Power rule | d/dx (xⁿ) = nxⁿ⁻¹ | Marginal cost, marginal revenue |
| Constant rule | d/dx (c) = 0 | Fixed costs do not change with output |
| Sum/difference rule | d/dx (f ± g) = f' ± g' | Differentiating TC = FC + VC |
| Product rule | d/dx (uv) = u'v + uv' | TR = P x Q when P is also a function |
| Chain rule | d/dx [f(g(x))] = f'(g(x)) . g'(x) | Composite economic functions |
Elasticity
| Type | Formula | Meaning |
|---|
| Price elasticity of demand | ed = (dQ/dP) x (P/Q) | Responsiveness of demand to price change |
| Relatively elastic | ed > 1 | % change in Qd > % change in P |
| Relatively inelastic | ed < 1 | % change in Qd < % change in P |
| Unitary elastic | ed = 1 | % change in Qd = % change in P |
| Perfectly elastic | ed = ∞ | Horizontal demand curve |
| Perfectly inelastic | ed = 0 | Vertical demand curve |
I. Choose the Best Answer (MCQs)
1. If the demand function is Qd = 80 - 4P, what is the quantity demanded when P = 10?
Answer
(a) 80 (b) 120 (c) 40 (d) 44
Answer(c) 40
ExplanationSubstituting P = 10 in Qd = 80 - 4P: Qd = 80 - 4(10) = 80 - 40 = 40 units
2. The derivative of Total Cost with respect to quantity gives:
Answer
(a) Average Cost (b) Marginal Cost (c) Total Revenue (d) Elasticity
Answer(b) Marginal Cost
ExplanationBy definition, Marginal Cost (MC) = d(TC)/dQ. The derivative of total cost with respect to output gives the additional cost of producing one more unit, which is marginal cost.
3. If y = x³ + 2x² + 5, then dy/dx is:
Answer
(a) 3x² + 4x (b) 3x² + 2x (c) 3x + 4 (d) x³ + 4x
Answer(a) 3x² + 4x
ExplanationApplying the power rule to each term:
- d/dx(x³) = 3x²
- d/dx(2x²) = 4x
- d/dx(5) = 0
- Therefore, dy/dx = 3x² + 4x
4. Equilibrium price is found when:
Answer
(a) Qd > Qs (b) Qd < Qs (c) Qd = Qs (d) Qd = 0
Answer(c) Qd = Qs
ExplanationMarket equilibrium occurs where demand equals supply. At this point, the quantity consumers want to buy equals the quantity producers want to sell, determining the equilibrium price.
5. If the demand function is Q = 50 - P and the supply function is Q = 10 + P, the equilibrium price is:
Answer
(a) 20 (b) 30 (c) 40 (d) 25
Answer(a) 20
ExplanationAt equilibrium, Qd = Qs: 50 - P = 10 + P 50 - 10 = P + P 40 = 2P P = 20
Substituting back: Q = 50 - 20 = 30 units
6. Price elasticity of demand is calculated as:
Answer
(a) (dP/dQ) x (Q/P) (b) (dQ/dP) x (P/Q) (c) (Q/P) x (dP/dQ) (d) P/Q
Answer(b) (dQ/dP) x (P/Q)
ExplanationPrice elasticity of demand measures the percentage change in quantity demanded divided by the percentage change in price. Using calculus, it is expressed as (dQ/dP) x (P/Q), where dQ/dP is the derivative of the demand function.
7. If TR = 100Q - 2Q², then Marginal Revenue (MR) is:
Answer
(a) 100 - 2Q (b) 100 - 4Q (c) 100Q - 4Q (d) 100 - Q²
Answer(b) 100 - 4Q
ExplanationMR = d(TR)/dQ = d(100Q - 2Q²)/dQ = 100 - 4Q
Using the power rule: derivative of 100Q is 100, derivative of -2Q² is -4Q.
8. A function f(x) is said to be increasing when:
Answer
(a) f'(x) < 0 (b) f'(x) = 0 (c) f'(x) > 0 (d) f'(x) = 1
Answer(c) f'(x) > 0
ExplanationA function is increasing at a point when its first derivative is positive. When f'(x) > 0, the function's value rises as x increases. When f'(x) < 0, the function is decreasing, and f'(x) = 0 indicates a stationary point (maximum, minimum, or inflection).
9. If TC = 100 + 10Q + 0.5Q², the marginal cost at Q = 4 is:
Answer
(a) 10 (b) 14 (c) 18 (d) 100
Answer(b) 14
ExplanationMC = d(TC)/dQ = d(100 + 10Q + 0.5Q²)/dQ = 10 + Q
At Q = 4: MC = 10 + 4 = 14
10. The slope of a straight line demand curve is:
Answer
(a) Always positive (b) Always negative (c) Zero (d) Undefined
Answer(b) Always negative
ExplanationA demand curve shows the inverse relationship between price and quantity. As price rises, quantity demanded falls. This means the slope (dP/dQ or dQ/dP) is always negative for a demand function.
II. Fill in the Blanks
1. A function is a rule that assigns exactly one value of y for each value of x.
2. If Qd = 200 - 5P, then when P = 20, Qd = 100.
3. The derivative of a constant is zero/0.
4. Market equilibrium is achieved when quantity demanded equals quantity supplied.
5. Marginal Cost is the derivative of Total Cost with respect to output.
6. If y = 7x⁴, then dy/dx = 28x³.
7. Price elasticity of demand is measured by the formula ed = (dQ/dP) x (P/Q).
8. When ed = 1, demand is said to be unitary elastic.
9. If TR = 50Q - Q², then MR = 50 - 2Q.
10. The point where the demand curve and supply curve intersect is called equilibrium point.
11. If Qs = 20 + 3P and Qd = 80 - 5P, the equilibrium price is P = 7.5.
12. The derivative of x⁵ is 5x⁴.
13. When f'(x) > 0, the function f(x) is increasing.
14. Average Cost (AC) = Total Cost / Quantity (Q).
III. True or False (with corrections for false statements)
1. Mathematics has no role in economic analysis.
AnswerFalse
Correction: Mathematics plays a crucial role in economic analysis. It helps express economic relationships precisely through functions and equations, derive results using calculus, and analyse data quantitatively. Concepts like marginal cost, marginal revenue, and elasticity all require mathematical tools.
2. A demand function shows the relationship between quantity demanded and price.
AnswerTrue
ExplanationThe demand function, typically written as Qd = f(P), shows how the quantity demanded of a good varies with its price. It captures the law of demand — as price increases, quantity demanded decreases, and vice versa.
3. The derivative of Total Revenue gives Average Revenue.
AnswerFalse
Correction: The derivative of Total Revenue with respect to quantity gives Marginal Revenue (MR), not Average Revenue. Average Revenue = Total Revenue / Quantity.
4. If the slope of a demand curve is negative, the demand curve slopes downward from left to right.
AnswerTrue
ExplanationA negative slope means that as we move along the x-axis (quantity increases), the y-value (price) decreases. This gives the demand curve its characteristic downward slope from left to right, reflecting the inverse relationship between price and quantity demanded.
5. When the price elasticity of demand is zero, demand is perfectly elastic.
AnswerFalse
Correction: When the price elasticity of demand is zero (ed = 0), demand is perfectly inelastic, meaning quantity demanded does not change at all regardless of price changes. Perfectly elastic demand corresponds to ed = infinity.
6. Simultaneous equations can be solved to find equilibrium price and quantity.
AnswerTrue
ExplanationEquilibrium price and quantity are found by setting the demand function equal to the supply function (Qd = Qs) and solving the resulting simultaneous equations.
7. The second derivative of a function helps determine whether a stationary point is a maximum or minimum.
AnswerTrue
ExplanationThe second derivative test: if f''(x) < 0 at a stationary point, it is a maximum; if f''(x) > 0, it is a minimum. This is important in economics for finding profit maximisation or cost minimisation points.
8. Marginal Revenue can be negative.
AnswerTrue
ExplanationMarginal Revenue can be negative when the firm must reduce price so much to sell an additional unit that the total revenue actually decreases. This occurs in the elastic portion of the demand curve beyond the midpoint.
IV. Match the Following
| Column A | Column B |
|---|
| 1. Demand function | (c) Qd = a - bP |
| 2. Supply function | (d) Qs = c + dP |
| 3. Marginal Cost | (f) d(TC)/dQ |
| 4. Marginal Revenue | (g) d(TR)/dQ |
| 5. Price elasticity of demand | (a) (dQ/dP) x (P/Q) |
| 6. Equilibrium condition | (e) Qd = Qs |
| 7. Profit maximisation | (b) MR = MC |
| 8. Slope of demand curve | (h) Negative |
Matched pairs: 1 → c (Demand function has the form Qd = a - bP, showing inverse relation with price) 2 → d (Supply function has the form Qs = c + dP, showing direct relation with price) 3 → f (Marginal Cost is the derivative of Total Cost w.r.t. quantity) 4 → g (Marginal Revenue is the derivative of Total Revenue w.r.t. quantity) 5 → a (Price elasticity of demand formula uses derivative dQ/dP) 6 → e (Equilibrium requires demand = supply) 7 → b (Profit is maximised when MR = MC) 8 → h (Demand curve has negative slope due to inverse price-quantity relation)
V. Answer Briefly (2-mark questions)
1. Define a function. Give an example from economics.
Answer
A function is a mathematical relationship that assigns exactly one output value (y) to each input value (x). It is written as y = f(x).
Economic example: The demand function Qd = 100 - 2P assigns a unique quantity demanded for each price level. When P = 10, Qd = 80; when P = 20, Qd = 60.
2. What is meant by equilibrium price? How is it determined?
Answer
Equilibrium price is the price at which the quantity demanded equals the quantity supplied in a market.
It is determined by setting the demand function equal to the supply function: Qd = Qs
For example, if Qd = 50 - P and Qs = 10 + P: 50 - P = 10 + P 40 = 2P P = 20 (equilibrium price)
3. Distinguish between Marginal Cost and Average Cost.
Answer
| Aspect | Marginal Cost (MC) | Average Cost (AC) |
|---|
| Definition | Additional cost of producing one more unit | Total cost per unit of output |
| Formula | MC = d(TC)/dQ | AC = TC/Q |
| Nature | Calculus-based (derivative) | Arithmetic (division) |
| Use | Deciding whether to produce more | Assessing overall cost efficiency |
4. What is price elasticity of demand? State its formula.
Answer
Price elasticity of demand measures the degree of responsiveness of quantity demanded to a change in the price of the commodity.
Formulaed = (Percentage change in quantity demanded) / (Percentage change in price)
Using calculus: ed = (dQ/dP) x (P/Q)
- If ed > 1 → Relatively elastic demand
- If ed < 1 → Relatively inelastic demand
- If ed = 1 → Unitary elastic demand
5. Write the meaning of a consumption function with an example.
Answer
The consumption function expresses the functional relationship between total consumption and total income. It shows how consumption expenditure changes with changes in income.
General form: C = C0 + bY, where C = total consumption, C0 = autonomous consumption (consumption at zero income), b = marginal propensity to consume, Y = income.
Example: C = 500 + 0.8Y. If Y = 10,000, then C = 500 + 0.8(10,000) = 500 + 8,000 = Rs. 8,500.
VI. Answer in Detail (5-mark questions)
1. Explain the role of mathematical methods in economics with suitable examples.
Answer
Mathematics is an indispensable tool in modern economic analysis. Its role can be explained under the following heads:
a) Expressing Economic Relationships: Mathematics helps express economic relationships in precise, unambiguous terms using functions and equations. For example, the law of demand is expressed as Qd = a - bP, clearly showing the inverse relationship between price and quantity demanded.
b) Deriving Marginal Concepts: Using differentiation (calculus), we derive marginal values:
- Marginal Cost (MC) = d(TC)/dQ — additional cost of producing one more unit
- Marginal Revenue (MR) = d(TR)/dQ — additional revenue from selling one more unit
- Marginal Product (MP) = d(TP)/dL — additional output from one more unit of labour
c) Finding Equilibrium: Simultaneous equations are solved to determine equilibrium price and quantity by equating demand and supply functions.
d) Measuring Elasticity: Calculus helps compute price elasticity, income elasticity, and cross elasticity precisely. For instance, price elasticity of demand = (dQ/dP) x (P/Q).
e) Optimisation: Firms maximise profit (where MR = MC) and minimise cost using first and second derivative tests. The first derivative identifies stationary points, and the second derivative confirms whether they are maxima or minima.
f) Forecasting and Trend Analysis: Mathematical functions and statistical methods help economists forecast demand, estimate growth trends, and make policy recommendations.
Thus, mathematical methods bring rigour, clarity, and quantitative precision to economic analysis.
2. Given the demand function Qd = 100 - 5P and supply function Qs = -20 + 5P, find the equilibrium price, equilibrium quantity, and price elasticity of demand at equilibrium.
Answer
Step 1: Find Equilibrium Price
At equilibrium, Qd = Qs: 100 - 5P = -20 + 5P 100 + 20 = 5P + 5P 120 = 10P P = 12
Step 2: Find Equilibrium Quantity
Substituting P = 12 in either the demand or supply function: Qd = 100 - 5(12) = 100 - 60 = 40
Verification: Qs = -20 + 5(12) = -20 + 60 = 40 ✓
Therefore, Equilibrium Price = 12 and Equilibrium Quantity = 40 units.
Step 3: Calculate Price Elasticity of Demand at Equilibrium
GivenQd = 100 - 5P
dQd/dP = -5 (the coefficient of P in the demand function)
At equilibrium (P = 12, Q = 40):
ed = (dQ/dP) x (P/Q) = (-5) x (12/40) = (-5) x (0.3) = -1.5
Taking absolute value: ed = 1.5
Interpretation: Since ed = 1.5 > 1, demand is relatively elastic at the equilibrium point. A 1% increase in price leads to a 1.5% decrease in quantity demanded.
3. Explain the concepts of Total Revenue, Average Revenue, and Marginal Revenue. Derive MR from TR = 100Q - 2Q².
Answer
Total Revenue (TR): Total Revenue is the total amount received by a firm from selling a given quantity of output. It is the product of price and quantity sold: TR = P x Q.
Average Revenue (AR): Average Revenue is the revenue per unit of output sold. It equals the price of the commodity: AR = TR/Q = P.
Marginal Revenue (MR): Marginal Revenue is the additional revenue earned from selling one more unit of output. It is the first derivative of Total Revenue with respect to quantity: MR = d(TR)/dQ.
Relationship between AR and MR:
- When AR (price) falls, MR falls faster than AR
- When AR is constant (perfect competition), AR = MR
- MR can be negative, but AR is always positive
Derivation of MR from TR = 100Q - 2Q²:
MR = d(TR)/dQ = d(100Q - 2Q²)/dQ
Applying the power rule:
- d(100Q)/dQ = 100
- d(2Q²)/dQ = 4Q
Therefore, MR = 100 - 4Q
Verification at Q = 10:
- TR = 100(10) - 2(10)² = 1000 - 200 = 800
- TR at Q = 11 = 100(11) - 2(11)² = 1100 - 242 = 858
- MR = 858 - 800 = 58
- Using formula: MR = 100 - 4(10) = 100 - 40 = 60
(Note: The slight difference is because MR gives the instantaneous rate, while the discrete calculation uses a unit change.)
4. Explain the application of differentiation in economics with examples.
Answer
Differentiation is one of the most important mathematical tools in economics. Its applications include:
a) Marginal Analysis:
- Marginal Cost (MC) = d(TC)/dQ: If TC = 100 + 10Q + Q², then MC = 10 + 2Q. At Q = 5, MC = 10 + 10 = 20.
- Marginal Revenue (MR) = d(TR)/dQ: If TR = 50Q - Q², then MR = 50 - 2Q. At Q = 10, MR = 50 - 20 = 30.
- Marginal Propensity to Consume (MPC) = dC/dY: From C = 200 + 0.75Y, MPC = 0.75.
b) Elasticity Calculation: Price elasticity of demand uses differentiation: ed = (dQ/dP) x (P/Q). If Q = 50 - 2P, then dQ/dP = -2. At P = 10, Q = 30: ed = (-2) x (10/30) = -0.67 (inelastic).
c) Profit Maximisation: Profit (π) = TR - TC. Profit is maximised when dπ/dQ = 0, i.e., MR = MC, and d²π/dQ² < 0 (second-order condition for maximum).
d) Cost Minimisation and Revenue Maximisation: Similar to profit maximisation, the first derivative set to zero gives the stationary point, and the second derivative test confirms whether it is a maximum or minimum.
e) Identifying Increasing and Decreasing Functions:
- If f'(x) > 0, the function is increasing (e.g., total cost increases with output)
- If f'(x) < 0, the function is decreasing (e.g., average cost may decrease initially)
f) Finding Rate of Change: Differentiation helps determine how fast one economic variable changes with respect to another — for example, how quickly demand changes when price changes, or how rapidly population grows over time.
Common Mistakes to Avoid
1. Confusing dQ/dP with dP/dQ: Remember, price elasticity uses (dQ/dP) x (P/Q), not (dP/dQ). The derivative is taken of Q with respect to P.
2. Sign errors in demand functions: The demand function always has a negative slope. If Qd = 100 - 5P, then dQ/dP = -5, not 5. Always retain the negative sign in elasticity calculations.
3. Forgetting to substitute equilibrium values into elasticity: Price elasticity must be calculated at a specific point. Always substitute the equilibrium P and Q values into the formula.
4. Mixing up average and marginal values: Average Cost = TC/Q (division), while Marginal Cost = d(TC)/dQ (differentiation). These are fundamentally different operations.
5. Ignoring the second derivative test: Finding where the first derivative is zero only gives stationary points. You must check f''(x) to determine if it is a maximum (f''(x) < 0) or minimum (f''(x) > 0).
6. Wrong substitution in simultaneous equations: When finding equilibrium, always substitute the equilibrium price back into BOTH the demand and supply functions to verify they give the same quantity.
7. Confusing the constant term in differentiation: The derivative of a constant is zero. In TC = 100 + 10Q, the derivative of 100 is 0, not 100.
8. Sign error in supply function differentiation: Supply functions have a positive slope (Qs = c + dP, where d > 0), while demand functions have a negative slope (Qd = a - bP, where b > 0).
Exam Checklist
- [ ] Can you define a function and distinguish it from an equation?
- [ ] Can you write and interpret demand and supply functions?
- [ ] Can you solve simultaneous equations to find equilibrium price and quantity?
- [ ] Do you know the power rule, constant rule, and sum/difference rule for differentiation?
- [ ] Can you derive Marginal Cost, Marginal Revenue, and Marginal Product using differentiation?
- [ ] Can you calculate price elasticity of demand at a given point using the calculus formula?
- [ ] Can you interpret elasticity values: ed > 1 (elastic), ed < 1 (inelastic), ed = 1 (unitary)?
- [ ] Can you identify whether a function is increasing or decreasing from its first derivative?
- [ ] Can you use the second derivative test to confirm maximum or minimum points?
- [ ] Can you apply the profit maximisation condition MR = MC to solve numerical problems?
- [ ] Do you remember the formulas: MC = d(TC)/dQ, MR = d(TR)/dQ, AC = TC/Q, AR = TR/Q?
- [ ] Can you avoid sign errors when differentiating demand functions?
- [ ] Have you practised numerical problems involving demand, supply, and equilibrium?
- [ ] Can you distinguish between slope and elasticity — they are different concepts?
