Easy2Siksha

GNDU Question Paper-2022

B.A 1

Semester

QUANTITATIVE TECHNIQUES

(Quantitative Techniques-1)

Time Allowed: Three Hours Maximum Marks: 100

Note: Attempt Five questions in all, selecting at least One question each from Sections A,

B, C and D. The Fifth question may be attempted from any Section. All questions carry

equal marks.

SECTION-A

1. (i) The demand and supply equations are q

= 25 - 2p and q

= - 2 + p respectively, find

equilibrium price and quantity before and after imposition of tax of Rs. 75 per unit.

(ii) Solve:



   







 





(iii) Solve 4x

- 4x

- 7x

- 4x + 4 = 0

2.(i) Find sum of n terms of an A.P., where p

term is (3p)/4 + 1

(ii) The Geometric Means between two numbers is 16, if one number is 8, what is the

other number?

(iii) If a, b, c, d are in GP, prove that a - b b - c c - d are in GP.

Easy2Siksha

SECTION-B

3. (i) The fixed costs are Rs. 700, and the estimated cost of 200 units is Rs. 1,900, find total

cost y of producing x units, assuming it to be a linear function of x.

(ii) Explain the concepts of permutations and combinations.

(iii) Show that the points (2, 0), (4, -2) and (5, -3) lie on the same st. line and find its

equation.

4. (i) Define sets and explain various types of sets.

(ii) Explain Union, Intersection and Complement of sets with their respective properties.

(iii) Sets A and B are such that A has 25 members, B has 20 members and A B has 35

members. Find the number of members in AOB.

SECTION-C

5. (i) Define function and explain various types of functions.

(ii) Explain the concepts of limit and continuous function.

6. Find the derivative of log x and ex by first principle method.

SECTION-D

7. (i) Differentiate log 󰇣





󰇤w.r.t. x.

(ii) Evaluate log(log 





󰇜

(iii) Differentiate w.r.t. x, a

+ x

+ e

8. (i) Given the Law of demand when  





find price elasticity of demand when p = 3.

Easy2Siksha

(ii) Given the Total Cost function y = 20 + 20x + 0.5x

where y is total cost and x is

quantity, find AC and MC.

GNDU Answer Paper-2022

B.A 1

Semester

QUANTITATIVE TECHNIQUES

(Quantitative Techniques-1)

Time Allowed: Three Hours Maximum Marks: 100

Note: Attempt Five questions in all, selecting at least One question each from Sections A,

B, C and D. The Fifth question may be attempted from any Section. All questions carry

equal marks.

SECTION-A

1. (i) The demand and supply equations are q

= 25 - 2p and q

= - 2 + p respectively, find

equilibrium price and quantity before and after imposition of tax of Rs. 75 per unit.

(ii) Solve:



   







 





(iii) Solve 4x

- 4x

- 7x

- 4x + 4 = 0

Ans: 󹴡󹴵󹴣󹴤 Question 1 (i): Finding Equilibrium Before and After Tax

Let’s start with a real-world inspired story…

󹴮󹴯󹴰󹴱󹴲󹴳 A Market Tale: Understanding Demand and Supply

Imagine a small village market where people come to buy and sell handmade wooden toys.

The price of these toys is determined by the balance between how many people want them

(demand) and how many toys are available for sale (supply).

󼩕󼩖󼩗󼩘󼩙󼩚 Given:

Easy2Siksha

• Demand function:

qd=25−2p

(This means as price increases, quantity demanded decreases)

• Supply function:

qs=−2+p

(This means as price increases, quantity supplied increases)

To find the equilibrium (where supply = demand), we set qd=qsq_d = q_sqd=qs because at

equilibrium, the amount people want to buy is equal to what producers want to sell.

󷃆󼽢 Step 1: Find Equilibrium Before Tax

Set qd=qsq_

25−2p=−2+p

Now solve for p (price):

Add 2p2p2p to both sides:

25=−2+3p

Add 2 to both sides:

27=3p

Divide both sides by 3:

p=9

Now plug this value of p = 9 into either equation to find q (quantity).

Use qd=25−2p

q=25−2(9)=25−18=7

󹻀 Equilibrium Before Tax:

• Price (p): ₹9

• Quantity (q): 7 units

󷃆󼽢 Step 2: After Imposing a Tax of ₹75 per unit

When the government imposes a tax, it increases the cost to the buyer or reduces the price

received by the seller. In this case, we assume the tax affects the supply side, i.e., suppliers

now receive (p - 75) instead of p.

Easy2Siksha

So, we modify the supply equation:

Original:

qs=−2+p

After tax:

qs=−2+(p−75)=−2+p−75=−77+p

Now set new supply equal to demand:

25−2p=−77+p

Add 2p to both sides:

25=−77+3p

Add 77 to both sides:

102=3p

Divide by 3:

p=34p

Now plug back into demand equation:

q=25−2(34)=25−68=−43

Wait! 󽅂 Quantity cannot be negative.

Let’s double-check the interpretation. Actually, the tax is imposed on the sellers, so the

sellers receive (p - 75) — this means the supply equation changes to:

Let’s rephrase properly:

New Supply Function:

qs=−2+(p−75)=−2+p−75=p−77

Now solve:

25−2p=p−77

Add 2p:

25=3p−77

Add 77:

102=3pp=34

Now find quantity:

q=25−2(34)=25−68=−43

Easy2Siksha

Again, negative quantity! This implies that tax is too high, and the market collapses — no

transactions take place. Demand goes negative. This could happen in real life if tax is too

large — buyers refuse to pay high prices.

󼨐󼨑󼨒 Concept Recap:

• Equilibrium occurs where demand = supply.

• Tax shifts supply curve (or demand) depending on who pays.

• Large taxes can eliminate the market if people are not willing to pay.

󹴡󹴵󹴣󹴤 Question 1 (ii): Solve:

Let’s solve this step-by-step, but with a clear understanding.

󷃆󼽢 Step 1: Understand the Domain

We are dealing with square roots, so the expressions inside them must be non-negative:

So final domain: x≥5

󷃆󼽢 Step 2: Use Substitution

Let’s put:

Now, equation becomes:

But this seems complex.

Try solving the original equation by trial:

Let’s try x = 5:

Easy2Siksha

Too big. Try x = 1:

Not in domain. Try x = 6:

Still big.

Try x = 0.5:

Not in domain.

Try x = 5.2:

Still too big. Try x = 0.01:

Still invalid.

Try plotting or squaring both sides:

Let’s isolate:

Now square both sides:

Now subtract x:

Now square both sides:

Now check by plugging into original equation:

Conclusion: No real solution.

Try graphically? But even better:

Easy2Siksha

Let’s use substitution:

Try x = 0.086 (Using WolframAlpha or calculator), we get:

So the solution is approximately 0.086, but since it's not in domain (x ≥ 5), the equation has

no real solution.

󹴡󹴵󹴣󹴤 Question 1 (iii): Solve:

Let’s solve this using Rational Root Theorem and Factorization.

󷃆󼽢 Step 1: Use Rational Root Theorem

Try possible factors of constant (±1, ±2, ±4) over leading coefficient (±1, ±2, ±4).

Try x = 1:

4(1)

– 4(1)

– 7(1)

– 4(1) + 4 = 4 – 4 -7 4 = 4 =-7  0 Tryx = 2 :4(16) – 4(8) – 7(4) – 4(2) +4 =

toolargeTryx = - 1: \[4(-1)

– 4(-1)

- 7 (-1)

-4(-1) + 4 = 4 + 4 = 7 + 4 + 4 = 9  

= - 2 :\[4(16) = 4(=8) = 7(4) – 4(-2) + 4 = checkagain……Eventually,you find: Let’s susegron

Try Factor Theorem or use synthetic division.

Try x = 1:

4 – 4 – 7 – 4 + 4 = -7 Tryx = 2 : \[64 – 32 – 28 – 8 + 4 + Not0Eventually, Weget : Use

factorizationoronlinesolver : \[4x

– 4x

– 7x

- 4x + 4 = (x

– x - 1) (4x

+ 4x - 4)

Now solve each quadratic:

󼨐󼨑󼨒 Final Answers:

These are four real roots, all irrational.

Easy2Siksha

󷃆󼽢 Conclusion (Summary)

In this session, we explored:

1. Equilibrium of demand and supply and how taxation affects it — with real-life

market application.

2. Solving a radical equation with square roots by checking the domain and simplifying

— concluding that no real solution exists.

3. Solving a quartic polynomial by factorization and applying the quadratic formula.

Each question helped strengthen core math concepts relevant to economics, algebra, and

real-world applications. Keep practicing to sharpen these skills further!

2.(i) Find sum of n terms of an A.P., where p

term is (3p)/4 + 1

(ii) The Geometric Means between two numbers is 16, if one number is 8, what is the

other number?

(iii) If a, b, c, d are in GP, prove that a - b b - c c - d are in GP.

Ans: 󹴡󹴵󹴣󹴤 Introduction: The Story of Sequences

Imagine a wise old mathematician who loved to observe patterns. He would line up

numbers and observe how they grow or shrink. Over time, he discovered two magical ways

numbers can form beautiful patterns:

1. Arithmetic Progression (A.P.) – where numbers increase or decrease by adding or

subtracting a fixed number.

2. Geometric Progression (G.P.) – where numbers grow or shrink by multiplying or

dividing by a fixed number.

These two patterns helped him solve real-world problems like counting steps, calculating

savings, measuring growth, and much more. Let’s step into his shoes and understand how to

deal with problems involving A.P. and G.P.

󼩕󼩖󼩗󼩘󼩙󼩚 PART (i): Sum of n Terms of an A.P., Given the p-th Term

Problem Statement:

You are told that the p-th term of an A.P. is given by the formula:

And you are asked to find the sum of the first n terms of this A.P.

Easy2Siksha

󹸯󹸭󹸮 Step 1: Understand What Is Given

The p-th term of an A.P. is:

This means if you substitute p = 1, p = 2, etc., you get the 1st term, 2nd term, and so on.

Let’s find the first term:

Now find the second term:

Find the third term:

Now, let’s find the common difference d.

In any A.P.:

So the A.P. is:

󽄡󽄢󽄣󽄤󽄥󽄦 Step 2: Use A.P. Sum Formula

The sum of first n terms of an A.P. is given by the formula:

Where:

• a = first term =





• d = common difference =





Easy2Siksha

Plug into the formula:

Simplify:

So the final answer is:

󹴡󹴴󹴣󹴤 PART (ii): Geometric Mean Between Two Numbers

Problem Statement:

The geometric mean between two numbers is 16. If one number is 8, what is the other?

󹸯󹸭󹸮 Step 1: Understand Geometric Mean

In a Geometric Progression, the Geometric Mean (G.M.) of two numbers a and b is:

You are given:

• G.M. = 16

• One number (say a) = 8

• Find the other number (say b)

󽄡󽄢󽄣󽄤󽄥󽄦 Step 2: Use Formula

󷃆󼽢 Answer: The other number is 32.

Easy2Siksha

So the two numbers are 8 and 32, and their geometric mean is 16.

Let’s verify:

󹴡󹴶󹴣󹴤 PART (iii): If a, b, c, d are in G.P., Prove That a − b, b − c, c − d Are in G.P.

Problem Statement:

Given:

a,b,c,d are in Geometric Progression.

We need to prove that:

a−b, b−c, c−d

are in Geometric Progression too.

󹸯󹸭󹸮 Step 1: Express G.P. in Terms of a and r

Let the first term be a and common ratio be r.

Then the G.P. terms are:

• a

• ar

Now compute:

So now the three terms are:

This is a G.P. because each term is multiplied by the common ratio r.

Let’s check:

Easy2Siksha

󷃆󼽢 Hence, the three terms are in G.P.

󷇴󷇵󷇶󷇷󷇸󷇹 Summary of All Three Concepts

Part

Concept

Key Idea

Final Answer

(i)

A.P. sum from p-th

term

Use formula Tp=a+(p−1)d to find a and d, then use

sum formula







󰇛

  

󰇜



(ii)

Geometric Mean

Use ab=









(iii)

GP property

Show expressions a−b, b−c, c−d form a G.P.

󷃆󼽢 Proved

󷕘󷕙󷕚 For University Students: Why This Is Important

These topics are crucial in mathematics for several reasons:

1. Pattern Recognition: Helps you spot structures in number sequences.

2. Problem Solving: You’ll use A.P. and G.P. concepts in algebra, finance (like

compound interest), and computer science (like algorithm complexity).

3. Logical Thinking: Deriving formulas and proving properties enhances mathematical

reasoning.

Real-life Examples:

• A.P. is like depositing ₹100 more every month in a savings account. Each deposit

increases linearly.

• G.P. is like bacteria doubling every hour — exponential growth.

• Patterns like in part (iii) help in code optimization and digital circuits.

󽄻󽄼󽄽 Final Thoughts

Math is not just about numbers; it’s about relationships between numbers. Arithmetic and

Geometric Progressions teach us how numbers behave, grow, and connect. Whether you

are solving a finance problem, designing an algorithm, or just playing with numbers —

sequences are everywhere.

Easy2Siksha

By understanding these concepts deeply, you're not just preparing for exams — you're

preparing your analytical brain for life!

SECTION-B

3. (i) The fixed costs are Rs. 700, and the estimated cost of 200 units is Rs. 1,900, find total

cost y of producing x units, assuming it to be a linear function of x.

(ii) Explain the concepts of permutations and combinations.

(iii) Show that the points (2, 0), (4, -2) and (5, -3) lie on the same st. line and find its

equation.

Ans: Understanding Mathematical Concepts through Real-Life Applications

Let us imagine that you are a student trying to start your own small business of making and

selling handmade candles. Through this journey, we will explore three mathematical

concepts:

1. Linear cost functions

2. Permutations and combinations

3. The equation of a straight line using three points

Let’s walk through each of these one by one — just like real life, where math is all around

us.

Part (i): Finding the Total Cost Function y = f(x)

Given:

• Fixed costs = ₹700

• Estimated cost for 200 units = ₹1900

• Assume cost varies linearly with number of units x

• Need to find a general cost function y for any x units.

󹰤󹰥󹰦󹰧󹰨 What are Fixed and Variable Costs?

Suppose you’re running your candle-making business. Whether you make 1 candle or 100,

some costs remain the same — like rent of your small factory, electricity bills, or equipment.

These are called fixed costs.

Easy2Siksha

Then there are costs that change with the number of units you produce — like wax,

fragrance oils, packaging, etc. These are called variable costs. If each candle costs ₹5 to

make, making 100 candles would cost ₹500.

In math, when costs increase in a straight line as production increases, we call this a linear

cost function.

󷃆󼽢 Step 1: Cost Function Formula

A linear function has the form:

y = mx + c

Where:

• y is the total cost for producing x units

• m is the variable cost per unit (slope)

• c is the fixed cost (intercept)

󷃆󼽢 Step 2: Plug in Given Values

We are given:

• Fixed cost c = 700

• At x = 200 units, total cost y = 1900

Now plug into the equation:

1900 = m * 200 + 700

Subtract 700 from both sides:

1200 = 200m

Now solve for m:

m = 1200 / 200 = 6

So, the variable cost per unit is ₹6.

󷃆󼽢 Step 3: Final Cost Equation

Now plug back m = 6 and c = 700 into the linear equation:

󷃆󼽢 y = 6x + 700

Easy2Siksha

This is your total cost function. So, if you want to make x candles, your total cost will be ₹(6x

+ 700).

󹳴󹳵󹳶󹳷 Application Example

Let’s say you want to produce 500 units. Plug in x = 500:

y = 6(500) + 700 = 3000 + 700 = ₹3700

So, to produce 500 units, your cost will be ₹3700.

Part (ii): Permutations and Combinations

Imagine you have 5 different scented candles: Rose, Lavender, Vanilla, Orange, and Coffee.

You want to:

• Display them in a row (arrangement matters)

• Or choose 3 to give to a friend (order doesn't matter)

Here enters the world of Permutations and Combinations.

󷃆󽄿 What is Permutation?

Permutation refers to the number of ways to arrange items when order matters.

Example:

If you arrange 3 candles out of 5 on a shelf, and order matters, it's a permutation.

󷃆󹹳󹹴󹹵󹹶 Permutation Formula

If you have n items, and you want to arrange r of them, the formula is:

• P(n, r) = n! / (n - r)!

Where n! (n factorial) means n × (n–1) × (n–2)... × 1

Example:

How many ways to arrange 3 candles out of 5?

P(5, 3) = 5! / (5 - 3)! = 5 × 4 × 3 = 60 ways

󷃆󽄿 What is Combination?

Combination is about selecting items, where order doesn’t matter.

Easy2Siksha

Example:

You want to gift any 3 candles out of 5. The order doesn’t matter. Rose-Vanilla-Coffee is the

same as Coffee-Vanilla-Rose.

󷃆󹹳󹹴󹹵󹹶 Combination Formula

C(n, r) = n! / [r!(n - r)!]

Example:

How many ways to choose 3 candles out of 5?

C(5, 3) = 5! / [3! × (5 - 3)!] = 5 × 4 × 3 / (3 × 2 × 1) = 10 ways

󼨐󼨑󼨒 Story-Based Summary

Think of permutations as how you arrange books on a shelf (the order of books matters).

Think of combinations as how you select friends for a trip (the order you choose them

doesn’t matter).

Part (iii): Do the Points Lie on the Same Straight Line?

You’re trying to draw a straight road on a map that goes through 3 cities. The cities have

coordinates:

• A(2, 0)

• B(4, –2)

• C(5, –3)

Let’s check if they lie on the same straight line, and if so, find the equation of that line.

󷃆󼽢 Step 1: Find the Slope Between A and B

Slope between two points (x₁, y₁) and (x₂, y₂) is:

m = (y₂ – y₁) / (x₂ – x₁)

Slope of AB:

m = (–2 – 0) / (4 – 2) = –2 / 2 = –1

󷃆󼽢 Step 2: Find the Slope Between B and C

m = (–3 – (–2)) / (5 – 4) = (–1) / 1 = –1

Easy2Siksha

Since slope of AB = slope of BC = –1, they lie on the same straight line.

󷃆󼽢 Step 3: Find the Equation of the Line

We know:

• Slope m = –1

• One point on the line: A(2, 0)

Use point-slope form:

y – y₁ = m(x – x₁)

So:

y – 0 = –1(x – 2)

=> y = –x + 2

󷃆󼽢 Final Answer:

Yes, the points lie on the same straight line.

Equation of the line is: y = –x + 2

󼩎󼩏󼩐󼩑󼩒󼩓󼩔 Real-Life Application

Imagine you're planning to build electric poles at these three points. Since they lie in a

straight line, you can draw wires directly without needing extra support or turns.

󼪺󼪻 Summary: Everything in One Place

1. Cost Function

• Fixed cost = ₹700

• Cost for 200 units = ₹1900

• Equation: y = 6x + 700

2. Permutations vs Combinations

Concept

Formula

Order Matters?

Example

Permutations

P(n, r) = n! / (n – r)!

Yes

Arranging books on a shelf

Easy2Siksha

Concept

Formula

Order Matters?

Example

Combinations

C(n, r) = n! / [r!(n – r)!]

Choosing players for a team

3. Line through 3 Points

• Points: (2,0), (4,–2), (5,–3)

• Same slope  Same line

• Equation: y = –x + 2

󽄡󽄢󽄣󽄤󽄥󽄦 Final Thought for Students

Math isn’t just numbers — it’s a language. It helps us model real situations, solve problems,

and make smarter decisions. Whether you're running a business, planning a project, or

organizing a team, math helps you find patterns, optimize resources, and work efficiently.

So don’t just “do” the math. Live the math. See it in your everyday life, and you’ll not only

remember the formulas — you’ll understand their meaning.

4. (i) Define sets and explain various types of sets.

(ii) Explain Union, Intersection and Complement of sets with their respective properties.

(iii) Sets A and B are such that A has 25 members, B has 20 members and A B has 35

members. Find the number of members in AOB.

Ans: Introduction: Understanding Sets through Real-Life Stories

Imagine you're in a classroom. The teacher looks around and says, “Raise your hand if you

play football.” A few students raise their hands. Then she says, “Raise your hand if you play

cricket.” Some other hands go up. Then she asks, “How many of you play both?”

Now imagine writing down the names of the football players in one notebook and cricket

players in another. What you have just created are sets. A set is a way to group and organize

similar items, people, or objects so that we can study and understand them better.

Let’s now enter the world of sets step by step and explore what they are, the types,

operations like union and intersection, and finally solve the numerical question using these

concepts.

(i) Definition of Sets and Types of Sets

What is a Set?

Easy2Siksha

In mathematics, a set is a well-defined collection of distinct objects, considered as an object

in its own right.

• Well-defined means there is a clear rule to decide whether an item belongs to the

set or not.

• Distinct means no repetitions are allowed.

The objects in a set are called elements or members.

Examples:

• Set of vowels in the English alphabet: {a, e, i, o, u}

• Set of even numbers less than 10: {2, 4, 6, 8}

• Set of colors in the rainbow: {red, orange, yellow, green, blue, indigo, violet}

We usually represent sets using capital letters (A, B, C, etc.), and elements using curly

braces.

Types of Sets:

Let’s now understand the various types of sets with real-life examples.

1. Empty Set (Null Set):

A set that contains no elements is called an empty or null set.

Symbol: φ or {}

Example:

Let A = Set of students in your class who are 200 years old.

Clearly, A = {} since no one is that old.

2. Finite Set:

A set which contains a countable number of elements.

Example:

B = Set of days in a week = {Monday, Tuesday, ..., Sunday}

3. Infinite Set:

A set which has uncountably many elements (goes on forever).

Example:

C = Set of all natural numbers = {1, 2, 3, 4, ...}

Easy2Siksha

4. Equal Sets:

Two sets are said to be equal if they contain exactly the same elements, regardless of the

order.

Example:

A = {1, 2, 3}, B = {3, 2, 1}

Here, A = B

5. Equivalent Sets:

Two sets that have the same number of elements, though the elements may be different.

Example:

X = {a, b, c}, Y = {1, 2, 3}

They both have 3 elements  Equivalent.

6. Subset:

Set A is a subset of B if every element of A is also an element of B.

Symbol: A  B

Example:

If A = {1, 2}, B = {1, 2, 3}, then A  B

7. Proper Subset:

A proper subset is a subset that is not equal to the original set. That is, A  B means A is a

subset of B but A ≠ B.

8. Universal Set:

The universal set contains all elements under consideration. All other sets are subsets of this

universal set.

Example:

If we are studying numbers between 1 and 10, then U = {1, 2, 3, ..., 10}

9. Power Set:

The set of all possible subsets of a given set is called its power set.

Easy2Siksha

Example:

If A = {1, 2}, then P(A) = { {}, {1}, {2}, {1, 2} }

10. Disjoint Sets:

Two sets are disjoint if they have no elements in common.

Example:

A = {1, 2}, B = {3, 4}  A ∩ B = φ

(ii) Set Operations: Union, Intersection, and Complement

Once we have sets, we can do various operations with them, just like adding or subtracting

numbers.

1. Union of Sets (A  B):

The union of sets A and B is a set that contains all elements that are in A or in B or in both.

Example:

A = {1, 2, 3}, B = {3, 4, 5}

A  B = {1, 2, 3, 4, 5}

Properties:

• Commutative: A  B = B  A

• Associative: (A  B)  C = A  (B  C)

• Idempotent: A  A = A

• Identity: A  φ = A

2. Intersection of Sets (A ∩ B):

The intersection of A and B is a set containing only the elements that are in both A and B.

Example:

A = {1, 2, 3}, B = {3, 4, 5}

A ∩ B = {3}

Properties:

• Commutative: A ∩ B = B ∩ A

• Associative: (A ∩ B) ∩ C = A ∩ (B ∩ C)

• Idempotent: A ∩ A = A

Easy2Siksha

• Identity: A ∩ φ = φ

3. Complement of a Set (Aᶜ):

The complement of a set A includes all elements in the universal set U that are not in A.

Example:

If U = {1, 2, 3, 4, 5}, and A = {1, 2, 3}, then Aᶜ = {4, 5}

Properties:

• A  Aᶜ = U

• A ∩ Aᶜ = φ

• (Aᶜ)ᶜ = A

(iii) Problem Solving using Set Operations

Given:

• Number of elements in set A = n(A) = 25

• Number of elements in set B = n(B) = 20

• Number of elements in A ∩ B = n(A ∩ B) = 35

We are to find n(A  B) = Number of elements in A union B

Solution:

We use the following basic formula of set theory:

n(AB)=n(A)+n(B)−n(A∩B)n(A  B) = n(A) + n(B) - n(A ∩ B)n(AB)=n(A)+n(B)−n(A∩B)

Substitute the values:

n(AB)=25+20−35=10n(A  B) = 25 + 20 - 35 = 10n(AB)=25+20−35=10

Answer:

There are 10 members in A  B (i.e., the union of sets A and B).

What Does This Mean?

Although there are 25 elements in A and 20 in B, there is an overlap of 35 elements. This

means many elements are common between A and B, and hence when we combine them,

the actual number of distinct members is less.

Easy2Siksha

This is like if a group of 25 students play football, 20 play cricket, but 35 students play both

— it means some students are counted twice when we simply add the numbers. So, we

subtract the repeated ones (the intersection) to find the correct total in the union.

Why Are Sets Important?

Sets form the foundation of modern mathematics, including areas like:

• Logic and reasoning

• Probability

• Statistics

• Database systems

• Programming and software development

Understanding sets helps students to improve structured thinking, organize data properly,

and analyze situations logically.

Conclusion: Key Takeaways

• A set is a collection of distinct, well-defined elements.

• Types of sets include empty, finite, infinite, subset, universal, and disjoint sets.

• Union combines all elements from both sets.

• Intersection includes only the common elements.

• Complement includes all elements not in the set.

• Using the formula n(A  B) = n(A) + n(B) - n(A ∩ B) helps solve real-world problems

related to surveys, groups, and analysis.

Understanding these concepts not only helps in exams but also in developing analytical

thinking. Always remember, mathematics is not just about numbers; it's about thinking

clearly and logically. Sets help you build that mental clarity.

SECTION-C

5. (i) Define function and explain various types of functions.

(ii) Explain the concepts of limit and continuous function.

Ans: Part I: Functions and Their Types

Easy2Siksha

What is a Function?

Let’s begin with a story.

Imagine you are in a school. You give your roll number to the teacher, and she gives you

your marks. Every student gives a unique roll number, and they receive their unique marks.

No two roll numbers are the same, and each roll number gives one result: marks.

This is exactly how a function works in mathematics.

A function is like a machine that takes an input, processes it, and gives exactly one output.

In simple terms:

• You put a value into the machine (input).

• The machine follows a specific rule.

• You get one and only one output.

Mathematically, a function is defined as:

A relation f from a set X to a set Y is called a function if every element x in X has one and

only one image y in Y.

It is written as:

󷵻󷵼󷵽󷵾 f: X → Y, where f(x) = y

Here,

• X is called the domain (input set).

• Y is the codomain (possible output set).

• f(x) is the value of the function at x (actual output).

Real-life Example of Function

Let’s take another example. Imagine a vending machine:

• You press 1 → you get chips.

• You press 2 → you get chocolate.

• You press 3 → you get soda.

But if you press the same number twice, you still get the same thing. You cannot press 2 and

get both chocolate and soda. That’s what makes it a function — one input gives exactly one

output.

Easy2Siksha

Different Types of Functions

Now that we understand what a function is, let’s explore the different types of functions.

1. One-to-One Function (Injective Function)

In this type of function, each input has a different output. No two inputs share the same

output.

󹻂 Example:

f(x) = x + 2

If x = 1 → f(1) = 3

x = 2 → f(2) = 4

x = 3 → f(3) = 5

All outputs are different.

󹳴󹳵󹳶󹳷 In this function, every person gets a unique gift.

2. Onto Function (Surjective Function)

In an onto function, every element in the output set Y is covered by some element from the

input set X.

That means nothing is left out in the codomain.

󹻂 Example:

If f(x) = x² and domain is {−2, −1, 0, 1, 2}

Then codomain is {0, 1, 4}

All elements in codomain are “hit” by inputs — it is onto.

3. One-to-One and Onto (Bijective Function)

This is the perfect function. It is both one-to-one and onto.

󹻂 Example:

f(x) = x + 1

Domain: {1, 2, 3}

Codomain: {2, 3, 4}

Here, each input has a unique output, and all codomain elements are covered.

󹳴󹳵󹳶󹳷 It’s like every student gets a unique seat, and all seats are filled.

4. Constant Function

A function that gives the same output no matter what the input is.

Easy2Siksha

󹻂 Example:

f(x) = 5

f(1) = 5, f(2) = 5, f(100) = 5

No matter what x is, f(x) = 5

󹳴󹳵󹳶󹳷 Like a vending machine that always gives you water, no matter what button you press!

5. Identity Function

This function returns the same value as input.

󹻂 Example:

f(x) = x

f(2) = 2, f(5) = 5

It’s like a mirror — what you give is what you get.

6. Polynomial Function

These are functions involving powers of x, like:

f(x) = x² + 2x + 3

They are smooth and continuous curves.

7. Rational Function

When one polynomial is divided by another, we get a rational function.

󹻂 Example:

f(x) = (x² + 1) / (x + 2)

These can have holes or asymptotes (points where the function is undefined).

8. Exponential Function

Functions where the variable is in the exponent.

󹻂 Example:

f(x) = 2^x

As x increases, the output increases very quickly.

9. Logarithmic Function

These are the opposite of exponential functions.

Easy2Siksha

󹻂 Example:

f(x) = log(x)

They grow slowly and are only defined for positive x.

10. Trigonometric Functions

These involve angles and periodic behavior.

󹻂 Examples:

f(x) = sin(x), f(x) = cos(x)

These functions repeat over intervals.

Part II: Limits and Continuous Functions

Now, let’s move to the second part of the question.

What is a Limit?

Let’s say you are walking towards a wall. Every time, you cover half the remaining distance

to the wall. You will never touch the wall, but get closer and closer.

That’s what a limit is!

A limit tells us what value a function is approaching, even if it never reaches that value.

Example:

Let’s say,

f(x) = (x² - 1)/(x - 1)

Try plugging in x = 1:

→ (1² - 1)/(1 - 1) = 0/0 = 󽅂 undefined!

But try values closer to 1:

• f(0.9) = (0.81 - 1)/(-0.1) = -1.9

• f(0.99) = (0.9801 - 1)/(-0.01) = -1.99

• f(1.01) = (1.0201 - 1)/(0.01) = 2.01

We see the function is approaching 2 from both sides.

So,

󷵻󷵼󷵽󷵾 Limit of f(x) as x → 1 is 2, written as:

Easy2Siksha



  





 

  

 

Even though f(1) is undefined, the limit exists.

Left-hand and Right-hand Limits

• Left-hand limit: approaching the value from the left (smaller than x)

• Right-hand limit: approaching the value from the right (greater than x)

If both are equal, the overall limit exists.

What is Continuity?

Now comes the idea of continuous function.

Let’s say you are drawing a graph of a function without lifting your pen. That means the

graph is continuous — it has no breaks, jumps, or holes.

A function is continuous at a point if:

1. f(x) is defined at that point.

2. The limit exists at that point.

3. The value of the function equals the limit at that point.

Mathematically:



  



󰇛



󰇜

 󰇛󰇜

Types of Discontinuity

Sometimes functions are not continuous, which means there is a discontinuity.

1. Jump Discontinuity

Function “jumps” from one value to another.

󹳴󹳵󹳶󹳷 Example: A function that is 1 for x < 0 and 2 for x ≥ 0.

2. Infinite Discontinuity

The function goes to infinity at some point.

Easy2Siksha

󹳴󹳵󹳶󹳷 Example: f(x) = 1/x at x = 0.

It goes to infinity, so not continuous at x = 0.

3. Removable Discontinuity

The function has a hole — it could have been continuous if we just “filled the gap.”

󹳴󹳵󹳶󹳷 Example:

f(x) = (x² - 1)/(x - 1), at x = 1

This simplifies to f(x) = x + 1, but is undefined at x = 1.

So the graph has a hole there.

Why Limits and Continuity Are Important?

Limits and continuity are foundation concepts in Calculus.

• They help define derivatives (rates of change).

• They explain real-life behaviors (temperature changes, population growth).

• In engineering and physics, they help model systems without sudden shocks.

Summary

Let’s sum it up with a human story:

In life, we often deal with cause and effect — press a button, and something happens.

That’s a function.

Functions can be of many types — some are predictable (identity), some are fixed

(constant), some are complex (polynomials, rational), and some are natural (trigonometric,

exponential).

But what if you are not allowed to fully reach your goal, only get closer and closer? That’s

where limits come in — they tell us what value a function is approaching, even if it never

actually gets there.

And what if we want life to go on smoothly, without breaks or surprises? That’s the beauty

of continuous functions.

Final Thoughts

Understanding functions, limits, and continuity is like understanding the behavior of life

itself — rules, progression, and smooth transitions.

Easy2Siksha

So next time you press a button on a vending machine, or draw a curve without lifting your

pencil — remember, you are seeing functions and continuity in action.

6. Find the derivative of log x and ex by first principle method.

Ans: 󷇴󷇵󷇶󷇷󷇸󷇹 Introduction: The Story of Change

Imagine you are walking up a mountain trail. The steepness of the path at every step tells

you how hard it is to climb—that steepness is like the derivative in mathematics. Just like

steepness tells you how fast your height is changing, a derivative tells how fast a function is

changing at any given point.

But how do we find this "steepness" of a function?

The most authentic and foundational way to find it is by using the First Principle of

Derivatives. This is the most basic and original method, from which all shortcut rules like

power rule, product rule, and chain rule are developed.

In this lesson, we will take a deep dive into finding the derivative of two important

functions:

1. f(x)=logx

2. f(x)=e

But first, let’s understand what the First Principle of Derivatives means.

󷉃󷉄 What is the First Principle of Derivatives?

The First Principle of Derivatives is based on the concept of limits. It is the mathematical

definition of a derivative.

󼨐󼨑󼨒 Formula:

This formula finds the instantaneous rate of change of a function at any point xxx. You take

two very close points on the curve: xxx and x+h, and find how much the function changes

between them. Then, let the distance h become infinitely small—this gives you the exact

rate of change (slope) at point x.

Now, let’s apply this principle to our two functions.

󽄬󽄭󽄮󽄯󽄰 Part 1: Derivative of log x from First Principle

We want to find the derivative of f(x)=logx using:

Easy2Siksha

Step 1: Apply Logarithmic Properties

We know that:

So,

Let’s make it easier to understand by changing variables.

Step 2: Let





 

Then h=xt , and as h→0 , then t→0 too.

Substitute:

Step 3: Use the Known Limit

It is a known result in calculus that:

So,

󷃆󼽢 Final Answer:

󼨐󼨑󼨒 Why is this Important?

The logarithmic function is used in many areas of science, economics, and engineering.

Whether you’re measuring sound intensity (decibels), growth of bacteria, or compound

interest, logarithms often come up. So knowing how fast logx is changing at any point helps

us predict behavior in such systems.

Easy2Siksha

󹸯󹸭󹸮 Let’s Understand it With a Real-Life Analogy:

Imagine you're filling up a bottle with water using a funnel. At first, the bottle fills up

quickly. But as it fills, the rate at which the water level rises becomes slower and slower.

This behavior is similar to the graph of logx : it grows quickly for small values of x, but the

rate of growth (slope or derivative) decreases as xxx increases.

So, the formula





exactly tells us this: for small x, the slope is steep (fast growth), and for

large x, it’s shallow (slow growth).

󽄬󽄭󽄮󽄯󽄰 Part 2: Derivative of e

from First Principle

Now we’ll find the derivative of the exponential function f(x) = e

This is one of the most magical functions in mathematics. Why? Because its rate of change is

exactly equal to itself! That is:

But let’s prove this from first principles.

Step 1: Use the First Principle

Use the property of exponents:

So,

Step 2: Use the Known Limit

Therefore:

󷃆󼽢 Final Answer:

Easy2Siksha

󼨐󼨑󼨒 Why is This Important?

The function e

shows up in natural growth and decay processes. For example:

• Radioactive decay

• Population growth

• Compound interest

• Heat transfer

In all these processes, the rate of change depends on the present amount. If you have more

people, population grows faster. If you have more money, interest accumulates faster. This

is exactly the behavior that e

models.

󹳣󹳤󹳥 Graphical Understanding

Let’s understand both functions on a graph.

• Only defined for x>0x > 0x>0

• Grows slowly as xxx increases

• Derivative 1x\frac{1}{x}x1 gets smaller as xxx gets bigger

Defined for all real x

• Grows rapidly for larger x

• Derivative e

increases as x increases — meaning faster and faster growth

Easy2Siksha

󷃆󹸊󹸋 Summary of Key Concepts

Function

First Principle Formula

Simplified Result

Log x

  



󰇛

  

󰇜

 







  





 









󼨻󼨼 Final Thoughts

The First Principle method may seem long and tricky at first, but it's the foundation of all

derivative rules you use later. It builds your mathematical thinking and makes you

understand:

• Why derivatives mean instantaneous change

• How functions behave locally around a point

• What makes exponential and logarithmic functions so unique

Once you've practiced this principle, you’ll better understand why shortcut rules (like

derivative of logx being





) work the way they do.

󹲹󹲺󹲻󹲼󹵉󹵊󹵋󹵌󹵍 Practice Exercises for You

1. Prove the derivative of ln x

using the first principle (hint: use chain rule too).

2. Show that





󰇛



󰇜

log a using the first principle.

3. Compare the growth rates of e

and logx at x=1, x=10x , and x=100.

󼨐󼨑󼨒 Remember:

Mathematics is not about memorizing formulas—it's about understanding the ideas behind

them. The First Principle shows you that even the most complex-looking rules come from

simple, elegant ideas.

Easy2Siksha

SECTION-D

7. (i) Differentiate log 󰇣





󰇤w.r.t. x.

(ii) Evaluate log(log 





󰇜

(iii) Differentiate w.r.t. x, a

+ x

+ e

Ans: Understanding the Topic: Logarithmic and Exponential Differentiation

Differentiation is one of the most important tools in calculus. It helps us understand how a

quantity changes with respect to another. In this topic, we will explore how to differentiate

expressions that contain logarithmic (log) and exponential (eˣ) functions.

Let’s go step by step through each question you’ve provided, but before that, let’s build a

strong foundation.

󷇴󷇵󷇶󷇷󷇸󷇹 Basic Concepts You Need to Know First

1. What is a logarithm?

A logarithm is the inverse operation of exponentiation.

If:

Then,

log

b=x

The logarithm tells us how many times we must multiply a (called the base) by itself to get

Common Types of Logs:

• log₁₀(x) or just log(x) is the common logarithm (base 10).

• ln(x) is the natural logarithm, meaning log base e, where e ≈ 2.718.

2. Important Rules of Logarithms:

These help simplify log expressions:

• log(a × b) = log a + log b

• log(a / b) = log a – log b

• log(aᵇ) = b log a

Easy2Siksha

3. Rules of Differentiation for Logs and Exponentials:

Let’s now list some key formulas for differentiation:

Function

Derivative

log x

1 / x

ln x

1 / x

eˣ

aˣ

aˣ ln a

log[f(x)]

f'(x)/f(x)

Now, let’s solve your three questions step by step.

󷃆󼽢 Question 7 (i): Differentiate

log [(cx + d)/(ax + b)] w.r.t. x

Let’s understand this with logic.

Let:

Using the log rule for division:

Now differentiate both terms:

Using the chain rule for log differentiation:

• Derivative of log(f(x)) is f'(x)/f(x)

So:

•

\frac{d}{dx}[\log(cx + d)] = \frac{c}{cx + d}]

Easy2Siksha

•

\frac{d}{dx}[\log(ax + b)] = \frac{a}{ax + b}]

Therefore:

󷃆󼽢 Final Answer:

󷃆󼽢 Question 7 (ii): Evaluate

log(log e^√(2x))

Let’s break this into steps. This is not a derivative, but a simplification problem.

Let:

We simplify it step by step:

Step 1: Understand e^{√(2x)}

Using the identity:

So:

Step 2: Now take log again:

We know that:

Easy2Siksha

Now apply log rule:

Apply log multiplication rule:

󷃆󼽢 Final Answer:

󷃆󼽢 Question 7 (iii): Differentiate

ax + xx + e^x w.r.t. x

Let’s take the function:

We will differentiate each term separately.

Step 1: Differentiate ax

This is linear:

Step 2: Differentiate x

This is tricky because both base and power are x.

We use logarithmic differentiation.

Let:

Take natural log both sides:

Easy2Siksha

Differentiate both sides:

So:

Multiply both sides by y1=x

Step 3: Differentiate e

This is basic:

Putting it all together:

󷃆󼽢 Final Answer:

󼨐󼨑󼨒 Deep Understanding: Why These Rules Work?

Now that we’ve solved each problem, let’s take a moment to understand why these rules

work. It will help you apply them to other problems in the future.

Easy2Siksha

󷃆󹸃󹸄 Chain Rule in Logarithmic Differentiation

The chain rule tells us that if a function is nested inside another function, we differentiate

from the outside in.

That’s why when we differentiate:

Because log is the outer function, and f(x) is the inner.

󹳴󹳵󹳶󹳷 Logarithmic Differentiation – When to Use?

We use logarithmic differentiation when:

1. The base and exponent are both variables (like x

)

2. The expression is very complicated (products/quotients/powers)

󹴷󹴺󹴸󹴹󹴻󹴼󹴽󹴾󹴿󹵀󹵁󹵂 Real-Life Use of These Concepts

You might wonder – where are these used?

1. Physics: Exponential and log functions describe radioactive decay, population

growth, charging/discharging of batteries, etc.

2. Economics: Used in compound interest, inflation, and logarithmic growth models.

3. Computer Science: Algorithms like binary search use logarithmic time complexity.

4. Engineering: Signal processing and sound intensity use logarithmic scales (like

decibels).

󽄡󽄢󽄣󽄤󽄥󽄦 Summary of Final Answers

Let’s quickly review:

(i)

(ii)

Easy2Siksha

(iii)

󷉃󷉄 Final Thought for Students

Differentiation might seem difficult at first, especially when logs and exponentials are

involved. But once you understand the basic rules and how to break down expressions step

by step, it becomes easier.

The key to mastering this is practice and patience. Start from basic problems, and slowly

challenge yourself with more complex ones like x^x or log(log(...)).

8. (i) Given the Law of demand when  





find price elasticity of demand when p = 3.

(ii) Given the Total Cost function y = 20 + 20x + 0.5x

where y is total cost and x is

quantity, find AC and MC.

Ans: Economics Simplified: Price Elasticity of Demand and Cost Functions

Let’s take a walk through two very important concepts in microeconomics that help

businesses, economists, and policymakers make smart decisions. These two are:

1. Price Elasticity of Demand

2. Average Cost (AC) and Marginal Cost (MC)

Let’s understand each part one by one—like a story, from the ground up.

󷆰 Part 1: Price Elasticity of Demand

Imagine you're running a business—a bakery. You sell cupcakes. You notice something

curious. When you increase the price of a cupcake, fewer people buy them. When you

decrease the price, more people come running in. This is a law of demand in action.

󹴡󹴵󹴣󹴤 What is the Law of Demand?

The Law of Demand says:

“When the price of a good rises, the quantity demanded falls; and when the price falls, the

quantity demanded rises—other things being equal.”

This is basic human behavior. People prefer to pay less and get more.

Easy2Siksha

Now economists wanted to measure this behavior. So they came up with a tool called:

󹳨󹳤󹳩󹳪󹳫 Price Elasticity of Demand (PED)

Let’s understand this idea simply.

Imagine price goes up by 10%. If demand drops by only 2%, we say the demand is inelastic

(not very responsive).

But if demand drops by 20%, we say it’s elastic (very responsive).

So, Price Elasticity of Demand is defined as:

Where:

• d

is the derivative (rate of change of quantity with respect to price)

• p is the price

• q is the quantity demanded

󼨐󼨑󼨒 Question (i): Given the Law of Demand q=20p+1q = \frac{20}{p + 1}q=p+120, find the

price elasticity of demand when p=3p = 3p=3.

Let’s go step by step:

󷃆󼽢 Step 1: Find q when p = 3

Given:

So when p=3p = 3p=3:

󷃆󼽢 Step 2: Find





We differentiate the function:

Easy2Siksha

Using the chain rule:

At p=3p = 3p=3, this becomes:

󷃆󼽢 Step 3: Use the formula for Elasticity

󹸯󹸭󹸮 Interpretation of Result:

We found:

Since the absolute value is less than 1, demand is inelastic at p=3p = 3p=3. This means:

• A 1% increase in price results in less than 1% decrease in quantity.

• Customers are not very sensitive to price changes at this point.

󹵲󹵳󹵴󹵵󹵶󹵷 Real Life Relevance:

Let’s return to our cupcake shop. If cupcakes are a necessity or luxury, and people don’t

easily find substitutes, then even if prices rise, they may still buy.

This insight helps businesses price their products wisely.

󼿝󼿞󼿟 Part 2: Total Cost, Average Cost, and Marginal Cost

Now imagine you run a factory that makes shoes. Your cost of making shoes includes:

• Fixed Costs – like rent, salaries. They don’t change with output.

• Variable Costs – like materials and electricity. These increase with production.

When economists analyze costs, they use a cost function to understand how costs change

with output.

Easy2Siksha

󼪺󼪻 Total Cost (TC)

You're given the total cost function:

Where:

• y is total cost

• x is quantity of output

Let’s break this down:

• 20: Fixed cost (doesn’t change with production)

• 20x: Linear variable cost

• 0.5x

: Increasing variable cost (due to inefficiencies, overload, etc.)

󹴡󹴵󹴣󹴤 What is Average Cost (AC)?

Average Cost (AC) is:

So,

Let’s simplify:

This tells us the cost of producing one unit on average. As x increases, the fixed part





decreases, which makes sense—you spread your fixed costs over more items.

󼿝󼿞󼿟 What is Marginal Cost (MC)?

Marginal Cost (MC) is the additional cost of producing one more unit.

Mathematically, it is:

Easy2Siksha

So we differentiate the cost function:

Hence,

This means the marginal cost increases as you produce more. Why? Because of diminishing

returns—at some point, producing more becomes harder and more expensive.

󼨐󼨑󼨒 Let’s Summarize with an Example:

Suppose you make 4 units (x=4x = 4x=4).

Step 1: Total Cost

Step 2: Average Cost

Or using the formula:

Step 3: Marginal Cost

󷙎󷙐󷙏 What Does It All Mean?

When you're running a business, understanding these costs helps you decide:

• How many units to produce?

• What price to set?

• Where you get maximum efficiency?

Easy2Siksha

For example, if your selling price is less than marginal cost, you're losing money on that

extra unit.

󹴷󹴺󹴸󹴹󹴻󹴼󹴽󹴾󹴿󹵀󹵁󹵂 Key Takeaways

󹰤󹰥󹰦󹰧󹰨 Price Elasticity of Demand

• Tells us how sensitive demand is to price changes.

• A value less than 1 (in absolute terms) means inelastic demand.

• Formula:

󹰤󹰥󹰦󹰧󹰨 Average and Marginal Cost

• AC helps find the cost per unit.

• MC shows how total cost changes with one more unit.

• These are critical for pricing, profit analysis, and cost efficiency.

󼨐󼨑󼨒 Final Thought for University Students

Economics is not just about graphs and formulas. It’s about how humans behave, how

businesses function, and how decisions affect the world. Whether you're managing a start-

up, analyzing a country’s budget, or just shopping online—demand, cost, and elasticity are

part of everyday life.

So next time you see a product with a “price drop” or “limited edition,” remember—

someone is using elasticity and cost analysis to influence your behavior. And now—you

understand the science behind it.

“This paper has been carefully prepared for educational purposes. If you notice any mistakes or

have suggestions, feel free to share your feedback.”