Imagine one fine morning in a peaceful village, three friends – Aryan, Bella, and Zoya – went

to their local maths tutor under a big banyan tree. They had just returned from school, and

their heads were spinning with numbers, symbols, and equations.

Aryan was the most puzzled. “Sir,” he said, “I got this question in our assignment. It's like

solving three puzzles at once!”

Let us take each equation and make things easier by inverting both sides. This trick is often

useful when the variable expression is in the denominator.

Let’s use this logic for the equations.

Oops! This seems confusing – both (ii) and (iii) are about A + C, but they give different

results: 11 and 10. That means one of these must be different.

So how can they be different? That’s not possible unless there's a mistake in question or

and calmness.

loved solving mysteries, not with clues, but with numbers. One day, his teacher gave him a

scroll with three puzzles written on it.

“If you solve these,” the teacher smiled, “you’ll become a master of Arithmetic and

Geometric Series!”

Equation 1 becomes:

check it once again.

Raju finally sat down in his shop, looking at how milk sells at different prices.

city streets. Every formula, every equation, tells a story — and once you learn to listen,

math becomes your best friend.

Let’s begin our learning journey not in a classroom, but in a magical library of knowledge,

perform operations like union, intersection, difference, and symmetric difference. At the

Let’s go!

A set is simply a collection of well-defined and distinct objects. These objects are called

elements or members of the set. The best part? You can write a set using curly brackets { }.

braces, separated by commas.

A set with only one element is called a singleton set.

Set of odd prime numbers less than 3 → {2}

A set with a countable number of elements.

Example:

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

A set with uncountable/infinite elements.

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

Two sets having the same number of elements, but not necessarily the same elements.

A = {a, b, c}, B = {1, 2, 3}  A ≈ B

Symbol: A  B

A is a proper subset of B if A  B, and A ≠ B.

Symbol: A  B

A set that contains all the elements under consideration. Denoted by U.

As Aryan arranged the books, he realized that some books belonged to more than one shelf.

This is when his teacher taught him four powerful operations:

Union means combining all elements of both sets.

If some elements are repeated, we include them only once.

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

A ∩ B = {2, 3}

Symmetric difference gives elements which are in either A or B, but not in both.

Symbol: A Δ B = (A – B)  (B – A)

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

Let’s add a short story here to make this even clearer.

In a college, there are students who play Cricket (Set A) and others who play Football (Set

Sita, Karan, Meena}

Now, let’s solve the given problem step-by-step.

n(M ∩ S) = n(M) – n(M – S)

= 30 – 20 = 10

So, 10 students took both Maths and Statistics

We don’t know total who took Statistics directly, but we can find it.

n(M  S) = n(M) + n(S) – n(M ∩ S)

Let n(S) = x

n(M  S) = 70 = 30 + x – 10

=> 70 = 20 + x

=> x = 50

So, number of students who took Statistics = 50

Now:

Only Statistics = n(S – M) = n(S) – n(M ∩ S)

Aryan was overjoyed. With just the knowledge of simple set operations and understanding

the types of sets, he was able to solve a real-world problem! Just like Aryan, once we

understand the basics:

“Don’t panic if the answer looks confusing at first. Just factor it, simplify it, and let logic be

your guide.”

Before diving into definitions, let’s ask ourselves — what do we mean by “continuity” in

A function is said to be discontinuous at a point if any one of the above three conditions

This break or gap makes the function discontinuous at that location.

Before the bridge, his path was smooth (like a continuous function). But due to that missing

piece, it became like a function with a gap.

Just like Ravi needs an unbroken path to deliver mail on time, we need unbroken

(continuous) functions for smooth calculations in mathematics.

is continuous at x=2x = 2x=2

Step 1: Is f(2) defined?

So,

Think of a water pipe. If water flows smoothly through it, that’s continuity. If there’s a

blockage or hole, that’s a discontinuity.

Suppose a gardener is watering plants through a pipe. At most spots, the water flows

without issue. But at one location, there’s a small break, and water splashes out. This is just

like a discontinuous function — not all inputs give smooth outputs.

For the pipe to be functionally continuous, water must flow smoothly through every part,

just like a function that behaves well at all points in its domain.

and asked,

And so begins our math story — with three expressions, each needing a different approach

to differentiate them with respect to x.

This is a product of two functions, so we must use the Product Rule.

Substitute values:

That's your final differentiated form. You can keep it factored for simplicity.

Now find the derivatives:

Now simplify the numerator:

You can leave the answer as:

Or, if required, further simplify.

The teacher said to them,

“Differentiation is like being a detective — your job is to look at the structure of the function

and figure out which rule fits best. Once you know your tools — Product Rule, Quotient

Rule, Chain Rule, Logarithmic Differentiation — you’ll never be afraid again.”

Imagine a young entrepreneur named Arjun who runs a small online business selling

So, rearrange the equation: